With its natural harbour and surrounding geography, Plymouth has a rich and important maritime history. From this spot you get a wonderful view of Plymouth Sound and the landscapes on both the Devon and Cornwall sides of the Tamar.
Plymouth was one of the three principal historic dockyards of the Royal Navy, along with Portsmouth and Chatham, and was set up in the 1690s. These dockyards were crucial for the maintenance of the Fleet, which was Britain's first line of defence. They also became important industrial sites, often being technically innovative. Because of this, their protection from attack was vital.
Many attempts have been made on Plymouth Sound and the dockyard including the Spanish Armada in 1588, Napoleon in 1803-5 and German forces in 1939-45. Plans were drawn up to protect Plymouth from bombardment from the sea and from enemy forces attacking landwards. As the range of destructive power of guns increased, so the successive rings of fortifications became more complex.
The main period of fortress building was in the 19th century, especially after the Royal Commission on the Defences of the United Kingdom report in 1860, but there is a sequence of surviving defence works going back to the 15th century.
Drakes Island was fortified as far back as the 1540s to protect the approach channel to the harbour and continued to be modified up until 1945. 24 forts were built encircling Plymouth in two rings, to protect from seaward and landward attacks. They were planned and constructed during the 1860s at a cost of around £3,000,000.
There were three forts covering the seaward approaches to Plymouth. Fort Picklecombe, in Cornwall covered the western side, the Breakwater Fort covered the central area and Fort Bovisand guns covered the eastern side of the breakwater and entrance to the Sound.
The breakwater was designed by John Rennie and completed by his son Sir John Rennie in 1844. The opening was celebrated with a horse-drawn omnibus driven from end to end, with a full complement of passengers and accompanied by a military band!
It protected ships from prevailing winds and sheltered the entrance to the harbour. It took over 30 years to build, using sround 4 million tons of rock at the then colossal cost of £1.5 million (equivalent to £87.2 million today). It is 1,560 metres (1,710 yd) long and 13 metres (43 ft) wide at the top. The base is 65 metres (213 ft) wide and it lies in about 10 metres (33 ft) of water.
Jennycliff is at the furthest coastal edge of the South Devon Area of Outstanding Natural Beauty (AONB) towards Plymouth and is an important area for conservation.
It is a County Wildlife Site because of the presence of four nationally scarce plants; pale St. John's wort, maidenhair fern, dwarf elder and round-leaved crane's bill. The woodlands and grassland are important habitats for insects and birds. Jennycliff is part of the Plymouth Sound, Shores and Cliffs Site of Special Scientific Interest.
It covers the eastern side of Plymouth Sound from Mountbatten Point to Andurn Point, just beyond Bovisand Bay. It is important for its rocky foreshore, coastal grassland and underlying geology.
The coast around Wembury and Plymouth Sound is very special and is a Marine Conservation Area. The sea cliffs provide nesting and roosting sites for seabirds. The beaches and sea caves provide a haven for marine wildlife and underwater are cliffs and reefs which can rival any of more tropical climes! Plymouth Sound is also protected under the European designation, Special Area of Conservation. This is the highest level of protection given to nature conservation in the UK and is testament to the amazing variety of marine life in the seas just offshore here.
This map shows Jennycliff and Plymouth Sound where an information panel is installed for you to soak up the stories about this area in person.
Explore Wembury: Jennycliff & Plymouth Sound 6 MB pdf.


text:
RFP: Funding Available for Solutions that Address Gender Inequity in AI Technology
By Alexander Riabov
Tags: AI Gender Challenge • AI • USAID
As the use of artificial intelligence (AI) proliferates, more instances of the inequitable design, use, and impact of AI-enabled tools in developing countries are coming to light. Many of these tools and approaches have the capacity for inequitable outcomes across genders due to humans unconsciously embedding bias in AI technology during data collection, model design or end-use applications. As a result, these tools often pose the greatest risk of harm and missed opportunities to those already marginalized in low- and middle-income countries. To foster an equitable and inclusive digital ecosystem, more efforts are needed to identify innovative and creative approaches to help decision-makers address actual and potential gender biases and inequitable outcomes resulting from AI technology.
Equitable AI Challenge: Sourcing Approaches to Prevent and Respond to Gender Inequity in AI
The U.S. Agency for International Development (USAID), through Digital Frontiers, is launching a challenge leading to one or more grants for innovative approaches to help identify and address actual and potential gender biases in AI systems relevant in global development contexts. USAID seeks to support approaches to increase the prevention, identification, transparency, monitoring and accountability of AI systems so that their outputs do not produce gender-inequitable results. Approaches may or may not be technology-based. Digital Frontiers anticipates disbursing up to five grants using a total of $375,000 in award funding. The challenge will take place over two phases:
Phase One: This is an open call for short concept notes. Concepts will be accepted under two tracks—one track called "Prevention," for approaches to prevent AI systems from incorporating gender bias into their outputs, and one called "Response," for approaches to inspire transparency of and accountability in existing AI systems. Strong applications from Phase One will be designated semi-finalists and invited to attend a co-creation workshop in early 2022 to discuss the context in which different instances of inequity arise, showcase different interventions, and encourage collaboration and partnership among semi-finalists within and across the tracks.
Phase Two: Semi-finalists will then be invited to submit revised concept notes that reflect updates to their original concept note as relevant. Winning submissions will be awarded grant funds to design, test, scale, and/or socialize the approach as appropriate over the course of one year.
Digital Frontiers encourages proposals from:
All organizations regardless of size; technology firms, startups, small, and medium enterprises, civil society organizations, and researchers from around the world.
Diverse groups that have clear, strategic, and collaborative approaches or frameworks to tackle the complexity of the gender bias and inequity occurring as a result of AI applications, including and especially, collaboration between developers, AI tool buyers, and civil society organizations.
Organizations that present a business case for why addressing gender inequitable outcomes from AI is both valuable to their organization and will be sustained following the grant.
Proposals that represent a good faith effort from developers to serve as an industry example on the steps needed to identify and eliminate instances of gender inequity in their AI tools.
Proposals that incorporate the diversity of AI-enabled bias or inequity women face according to intersectional identities (disability, sexual minority, language minority, caste status, etc.).
Deadline for Concept Notes is January 6, 2022, by 5pm EST. A link to RFP can be found here.
The AI Age Presents New Challenges in Cybersecurity
Can Regulators Keep up with Artificial Intelligence?
Request for Information: Exploring the Role of AI and Automation on Agrifood Systems

redpajama_set_name: RedPajamaCommonCrawl


The Chambersburg Civil War Seminar series comes to Sharpsburg this month with more than 40 speakers and guides, and runs over 5 days — July 25th thru 29th. This will be the most comprehensive such event held in recent memory, and promises something for every student of the Battle of Antietam.
Featured speakers and tour guides include most of the very best on the subject including Tom Clemens, Ed Bearrs, John Schildt, John Priest, Ted Alexander, REL Krick, Dennis Frye, Ethan Rafuse, Mark Snell, Perry Jamieson, and just about the entire Battlefield staff including fellow bloggers Hoptak and Gentile.
Bonus events include a sneak-peek at Virtual Antietam with Stephen Recker, the famous Paul Chiles Artillery Hell tour, and Pat Falci does AP Hill.
See the organizer's home page for more details and registration information.
I'll be making my first trip to Antietam on Sunday. Any tips on preparations for the trip (other than reading extensively from aotw, of course) and mandatory stops? What is the best single-volume history of the campaign, in your opinion?
I'm doing some work on the cavalry breakout from Harper's Ferry, but am trying to make that stop on Friday so I can focus on the battlefield proper on Sunday.
My favorite single volume narrative for the battle is Sears' Landscape Turned Red. It suffers from some factual defects, and Sears positively foams at the mouth for McClellan, but nobody has written a better overview.
The best tour book is Luvaas and Nelson's Guide. They use OR excerpts to do the talking, and provide excellent driving directions and detail map segments. South Mountain, Harpers Ferry, and Shepherdstown actions included.
For your first time at the field, and with only one day, I'd recommend you use the Park's tour route. Taped tours (and the books above) are available in the Visitor's Center bookstore. That'll hit all the highlights including famous cornfield, sunken lane, and stone bridge. Try to catch at least one of the Ranger talks/hikes, too. See if you can find Mannie, if he's guiding on Sunday.
I've scanned and posted the current Park brochure, if you want to get a jump on it.
The Park also hosts professional tour guides, and these are really the best way to go, but there's not much time to arrange, and there is a fee. Check with Stephen Recker if interested.
Thanks very much for the tips, Brian, that helps a great deal.


Never before has an historical aircraft been successfully recreated as a Light, Ultralight and Experimental-Kit aircraft in such detail.
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A perfect 70% replica of the legendary P-51 Mustang, the ScaleWings SW-51 is one of its kind. Feel the perfection in every detail as you look at the nearly 100,000 rivets we incorporated into all surfaces in more than 60,000 man hours. The SW-51 Mustang looks like Aluminium but is a state-of-the-art all carbon aicraft. Even the mirror shine of polished aluminum has been realized to enhance the historical impression, creating awe with every observer.
A unique, true to original 70 % replica. Over 100,000 rivets and sheet metal joints were processed during over 60,000 working hours.
From instrumentation to equipment. Historical peculiarities as well as innovative techniques have been emphasized.
Although realized in a new carbon fiber technology, the shine of polished aluminum is conveyed for the historical impression.
High-quality processing in our independent ScaleWings productions. Professionals with many years of experience invest a lot of time and know-how.
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Build your dream mustang in the fastest possible time. We provide expert assistance at Scalewings locations in Krosno, Poland and Eggenfelden, Germany to customize your plane to the highest degree in less than four weeks.


I've been reading through another of my new books, the hot-off-the-presses title: The Genius of Ancient Man. I'm hoping to get some more ideas to add to my collection of posts because this topic is as big a thinking-changer as any.
So far, it seems that I've already at least touched on most of the types of things we find that tell us our earliest ancestors were the opposite of "primitive." That's encouraging, but I want to know more!
This new book does a great job discussing how carefully Jesus' followers need to handle this topic. Ignoring our ancestor's achievements is unwise, but much of their lives were focused on idolatry that we detest. I've had to deal with this as well, so my favorite posts have been the stone spheres and farming terraces where there was little or no sign of devil worship involved.
Now, I'll be the first to admit that although we can make a good case for the explanation this article presents, that doesn't force it to be the truth. I once read a book that made a good case for the Last Czar's daughter Anastasia having lived for many years after the fall of Russia. Just a few years ago we found her remains with the rest of the murdered family, so the book was proved wrong. It is equally possible to have another explanation for this rock cutting ability, but we know it must involve very high brain power!
Like others, this article points out just how high the quality of early stone work (in this case from Egypt) actually was. They were capable of carving hard granite to a smooth polish. This is not something that can be accurately done with stone chipping tools. I'm not going to cover all the likely tools the ancients would have needed to produce the stone work they left behind. You'll have to read the article for that, but there was one point that blew my mind.
I've already mentionedmany ancient civilizations had these bowl-shaped disks that could have been used for astronomy. What was to stop them from using these disks to carve and polish rock with the rays of the sun?
The Laura Lee article points out that both the Egyptians and Incas lived in places where sunshine was a predictable thing. They would have had ample opportunity to use it as a tool while they still remembered how.
We know that whatever technique these people used to do their incredible stone work was completely lost. Today we don't have any tools in our museums that could remotely explain their quality of workmanship. But if the engineer's ideas about using sunlight is accurate, we do have records of their tools having existed; they were cut in pieces and melted for their gold.
People have always been smart. They also completely rebelled against God. Without God's help and protection there is nothing to stop our great-grandchildren from forever losing the technology we have slowly built up over the centuries as well. Let us trust and honor the Giver of life who never vanishes or abandons us!
Shall I add this quote to my Ancient Optics post?
Tell that to the biologists who see proof of a Designer every day!
Well, is science wise enough to date them accurately, or isn't it? Make up your mind!


Last week we had our mission trip through World Servants in the ever so small town of Jellico, Tennessee. Jellico is located at the northern border of Tennesse, so much so if you walk four blocks north of Main Street you are now in Kentucky (not joking). Spenser, Kate and I met up with an outstanding group from First Baptist Church of Hamilton, Ohio. Throughout the week we held a sports camp/VBS type activity for the kids of the area. Over 50 kids joined in each day to learn about football, soccer, basketball or cheerleading. In addition, they were exposed to God's love and his Word.
In the afternoons, we worked on a few other projects including demolishing/rebuilding a deck, putting a new roof over an existing (leaky) roof and painting a couple houses. If was amazing! I can now say I know how to build a deck from the ground up and that it is even level and rock solid.
Aside from the work, World Servants stresses one thing: relationships. Relationships within the team, relationships with the families we are working with and relationships with the community as a whole. It was awesome. I worked on the deck of this delightful woman named Wanda. She went out of her way to make us feel welcomed. On Tuesday, both homes we were working on hosted lunch for us. Chicken and dumplings, full turkey dinners with all of the fixings, fried cabbage, corn dogs, hot dogs, cornbread and so many desserts I am still full on sugar. I have never seen such hospitality. These families gave fully of themselves and their food when they did not have money to spare. They gave because it was what they could give and they did it with such joy and warmth. You felt that you were part of the family.


text:
Q: When entering www.google.com I get www.google.co.il , why? Why when I hit www.google.com in my browser , I get www.google.co.il ?
OK , so I'm in Israel , but what's wrong with getting www.google.com ?
Does it happen because the www.google.co.il server is closer to me (Israel) than the www.google.com server ?

A: When ever You enter Google.com That will Be redirected with your Local area..
Depending On your local server it will display the server..
Weather if it is
Israel it will re direct to  www.google.co.il
Australia it will re direct to  www.google.com.au
If you want to change that will depend Up on your ISP(Internet Service Provider).
To redirect to main google site you need to change your ISP.
or else You need to USE Some third party servers. which will maintain a virtual servers like   using VPN(virtual private Network)on your system.. example(Hideman,tunnelbare,hotspot vpn)
If You want to redirect to www.google.com with out installing any software then
TRY this
at your browsr type https://www.google.com/ncr (here ncr is non-country-related)
You will be redirect to Main www.google.com
Here its https so don't worry..

A: yes Google redirects to your local Google search page to avoid this try
Worldwide Google Search address: http://www.google.com/ncr
where ncr is non-country-related

A: No. Google.com usually redirects to a local version of the Google page—like Google.in for INDIA.You can use www.google.com/ncr to use google.com. ncr means "no country redirection"

redpajama_set_name: RedPajamaStackExchange


Sandy Hook parent Neil Heslin reportedly spent April 15 in Washington, DC, lobbying for background checks. He's now facing failure to appear charges, because on that day he was supposed to be in court in his own criminal cases. He is accused of passing bad checks, larceny, and two counts of driving on a license suspended for drunk driving, which is probably a felony.
Maybe the White House needs to do its own background checks. Here's a picture of Heslin in the Rose Garden, about five feet from the Vice President.
Has he ever been elected Mayor? He sounds like Bloomberg's type.


text:
scsmartflo.com
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Host: http://scsmartflo.com/

redpajama_set_name: RedPajamaGithub


It can be challenging to stick to a healthy diet during the holiday season with all the parties, family dinners, and tempting festive desserts. Not to mention the stress around planning, shopping, and food preparation to be the perfect host. Join Stony Brook Southampton Hospital's Clinical Nutrition Manager Paula Montagna, MS, RD, CDN, as she demonstrates how to make nutritious and delicious appetizers. Each attendee will get to enjoy samples, and also receive the recipes to bring home. Limited to 30 participants. Call (631) 288-3335 for more information and to reserve your seat.


text:
Teenager shot dead after attacking passengers on train in Germany
Three people seriously injured after 17-year-old armed with axe and knife attacks passengers on train in Bavaria
Islamic State flag 'found at home of suspected German train attacker'
Philip Oltermann in Berlin and Kevin Rawlinson
Tue 19 Jul 2016 02.06 EDT First published on Mon 18 Jul 2016 17.01 EDT
Germany axe attack: police describe assault on train passengers
A teenager armed with an axe and a knife has attacked about 20 passengers on a train in northern Bavaria, according to local police.
Three people were seriously hurt and one person sustained light injuries before the attacker was shot dead by police, a police spokesman said on Monday evening.
Bavaria's interior minister, Joachim Herrmann, said the attacker was a 17-year-old Afghan asylum seeker who had arrived in Germany as an unaccompanied minor and had been living with a foster family in Ochsenfurt, south of Würzburg, for "a few months".
After passengers managed to alert the driver, the train was stopped in the Heidingsfeld district of Würzburg and the attacker initially managed to flee from the carriage on foot, Herrmann said.
A police taskforce that happened to be in the vicinity then pursued the attacker, shooting dead the teenager, who was carrying an axe and a knife when he had attacked members of the unit.
Asked whether the attack had an Islamist background, Herrmann said one witness inside the train said the teenager had shouted "Allahu Akbar" during the assault. The report was being investigated.
The man reportedly attacked passengers on the regional train travelling between the town of Treuchtlingen and Würzburg. Fourteen other passengers were reportedly in a state of shock and receiving treatment by specialists.
The interior ministry could not confirm whether some of the victims were in a life-threatening condition.
The train line between Ochsenfurt and Würzburg remained closed while police investigated. Police initially said there was no indication of a motive, and they were treating the attacker as a lone individual, citing witness reports.
There have not been any attacks with an explicit terrorist motive in Germany, which has been at the heart of the refugee crisis over the last year. In November, a football friendly between Germany and Holland in Hanover was cancelled after a terrorist attack tip-off, with the interior minister, Thomas de Maiziére, saying there had been a "concrete threat" of an attack.
In the aftermath of the terrorist attack in Brussels in March, the foreign minister, Frank-Walter Steinmeier, appealed to citizens to keep a "cool head".
"The terrorists would like to carry their war into our cities and into our heads, to create a kind of permanent siege, to force their perverse logic of violence and hatred on us", the social democrat politician said. "We would do our best not to play along with this game".
About 1.2 million refugees are estimated to be living in Germany. Latest estimates by the European commission put the number of asylum seekers in Europe by the end of 2017 at around 3 million.
According to a study published in April by Germany's Friedrich Ebert Foundation, about 154,000 Afghan citizens migrated to Germany in 2015, of whom 32,000 applied for asylum. More than 120,000 Afghan citizens remained in Germany without authorisation or had moved on to other countries, according to the study.
The study found that most Afghan migrants were young and male. The majority of family members who were interviewed for the survey said they had fled their country because of the economic situation. Political developments in Europe or the open-border policy of the chancellor, Angela Merkel, at the height of the refugee crisis had played only a minor role in encouraging Afghans to leave their country, the study found.
Germany, while initially generally welcoming refugees, has since begun to take a much tougher line, declaring it would deport Afghan asylum seekers whose applications are rejected.
Earlier in the year, Afghans were the second largest group entering the European Union, with 178,230 Afghans seeking asylum in the EU's 28 states in 2015.


text:
News / Workers At Total's North Sea Oil, Gas Platforms Plan for fifth Strike On Aug 20th
Workers At Total's North Sea Oil, Gas Platforms Plan for fifth Strike On Aug 20th
Workers at three oil and gas platforms in the North Sea operated by Total have planned for a fifth set of a 24-hour on August 20,after failure to reach an agreement in the meeting with the French major on Thursday.
Unite, UK's biggest trade union today posted on twitter;
"Unite met Total today where the union made a series of counter proposals to Total for consideration. The company will now consider these proposals and both parties will re-convene next Thursday for further discussions. The scheduled 24 hour stoppage on Monday goes ahead."
The series of strikes were a result of Total's decision to place workers at the platforms on a three-week rotation system instead of two.
"The dispute concerns the company's wage review and its plans to force workers to increase their offshore working time," Unite had said at the start of the strike.
The fields hit by the strikes account for some 10 percent of the UK's gas production. Crude oil production at Alwyn, Elgin, and Dunbar pumps up to 50,000 bpd to the Forties and Brent Blend crude streams—the key components of the Dated Brent benchmark.
Shell to invest $15 billion in Nigeria's oil and gas industry
Oil prices firm as Libya supply risks mount
Tullow Oil forecasts oil production boost in 2019
UNOC Struggles to Sell Uganda's 45000 Litters of Test Crude Oil


text:
import * as Enums from '../enums/index';
import * as Models from './index';

/**
 * The CoverLetter model represents a message sent along with an invitation to use CertExpress to
upload certificates.  An invitation allows customers to use CertExpress to upload their exemption
certificates directly; this cover letter explains why the invitation was sent.
 * @export
 * @interface CoverLetterModel
 */
 export interface CoverLetterModel {
    /**
     * @type {number}
     * @memberof CoverLetterModel
     */
   id?: number;
    /**
     * @type {number}
     * @memberof CoverLetterModel
     */
   companyId?: number;
    /**
     * @type {string}
     * @memberof CoverLetterModel
     */
   title?: string;
    /**
     * @type {string}
     * @memberof CoverLetterModel
     */
   subject?: string;
    /**
     * @type {string}
     * @memberof CoverLetterModel
     */
   description?: string;
    /**
     * @type {Date}
     * @memberof CoverLetterModel
     */
   createdDate?: Date;
    /**
     * @type {Date}
     * @memberof CoverLetterModel
     */
   modifiedDate?: Date;
    /**
     * @type {boolean}
     * @memberof CoverLetterModel
     */
   active?: boolean;
    /**
     * @type {number}
     * @memberof CoverLetterModel
     */
   pageCount?: number;
    /**
     * @type {string}
     * @memberof CoverLetterModel
     */
   templateFilename?: string;
    /**
     * @type {number}
     * @memberof CoverLetterModel
     */
   version?: number;
 }


text:
\section{Introduction} \label{sec:intro}

The accretion of gas onto disk galaxies is a fundamental part of galaxy formation and evolution, as gas within disks is continually used to form stars and must therefore be regularly replenished \citep[e.g.,][]{Putman17}. All such gas, whether pristine gas from cosmological inflows or recycled gas in the process of reaccreting, must pass through the local environment surrounding galaxies, often called the circumgalactic medium (CGM). The CGM might contain a substantial amount of angular momentum as shown by many studies of galaxy simulations \citep[e.g.,][]{Stewart11,Danovich15,DeFelippis20}. As the gas accretes onto the galaxy, the angular momentum will flow inward too, meaning the CGM is a source not just of the mass of the disk, but its angular momentum as well.

Not all gas surrounding galaxies is inflowing though: the CGM also contains outflowing gas ejected from the galaxy by feedback from supernovae and active galactic nuclei (AGN), which is capable of affecting the way in which CGM gas eventually joins the galaxy \citep{DeFelippis17}. All of these physical processes occur concurrently and result in a multiphase environment shown in observations to have complex kinematics \citep[see][and references therein]{Tumlinson17}.

A large number of recent observations of the CGM have been accomplished through absorption line studies of background quasars through dedicated surveys \citep[e.g.,][]{Liang14,Borthakur15,Kacprzak15}. For instance, some surveys are constructed by cross-correlating quasar absorption lines with spectroscopic redshift surveys such as the Keck Baryonic Structure Survey \citep[KBSS;][]{Rakic12,Rudie12,Turner14} or with photometric surveys like the Sloan Digital Sky Survey \citep[SDSS;][]{Huang16,Lan18,Lan20}. Other CGM surveys attempt to either match individual absorption lines to known galaxies (i.e.,~are ``galaxy selected''), like the COS-Halos \citep[e.g.,][]{Tumlinson11,Werk13,Borthakur16,Burchett19}, COS-LRG \citep{Chen18,Zahedy19}, and the low-redshift Keck surveys conducted at Keck Observatory \citep{Ho17,MartinC19}, or match galaxies near known absorbers (i.e.,~``absorber selected'') such as the MusE GAs FLOw and Wind survey \citep[MEGAFLOW;][]{Schroetter16,Schroetter19,Schroetter21,Wendt21,Zabl19,Zabl20,Zabl21}. In these surveys, there is generally only one quasar sight line per galaxy, but in certain rare cases it is possible to find multiple sight lines associated with a single galaxy through multiple quasars \citep{Bowen16}, a single multiply lensed quasar \citep{Chen14,Zahedy16,Kulkarni19}, an extended lensed quasar \citep{Lopez18}, or even an extended background galaxy \citep{Diamond-Stanic16}.

The \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} ion has been a focus of many recent surveys including the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} Absorber-Galaxy Catalog \citep[MAGIICAT;][]{Chen08,Chen10a,Nielsen13a,Nielsen13b,Nielsen15}, the Magellan MagE \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} (M3) Halo Project \citep{Chen10a,Chen10b,Huang21}, the MUSE Analysis of Gas around Galaxies Survey \citep[MAGG;][]{Dutta20}, the PRIsm MUlti-object Survey \citep[PRIMUS;][]{Coil11,Rubin18}, and the aforementioned MEGAFLOW survey, as well as individual absorbers \citep[e.g.,][]{Lopez20}. These studies belong to a long history of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} $\lambda2796$ absorption line surveys \citep[e.g.,][]{Bergeron91,Bergeron92,Steidel92}, which unveiled the first galaxy--absorber pairs at intermediate redshifts. Though not the focus of this paper, \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} has also been seen in emission in extended structures around the galaxy and in the CGM \citep[e.g.,][]{Rubin11,RickardsVaught19,Rupke19,Burchett21,Zabl21}.

Along with this wealth of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} observations, researchers in recent years have found \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} kinematics to be correlated over large spatial scales. In particular, both \cite{Bordoloi11} and \cite{Bouche12} found a strong dependence of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption on azimuthal angle: specifically, more absorption near $\phi=0^{\circ}$ and $90^{\circ}$ and a lack of absorption near $45^{\circ}$. This type of absorption distribution is generally interpreted as bipolar outflows along the minor axis and inflows along the major axis. In this context, both galaxy-selected \citep[e.g.,][]{Ho17,MartinC19} and absorption-selected \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} studies \citep[e.g.,][]{Kacprzak12,Bouche13,Bouche16,Zabl19} have given support to the interpretation of accretion of gas from the CGM onto the galaxy. These \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} studies show that when sight lines are located near the major axis of the galaxy there are clear signatures of corotating cold gas with respect to the galaxy kinematics.

However, despite such extensive observational data, developing a general understanding of cold gas in the CGM from the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} line alone remains difficult due to the limited spatial information provided by the observational technique (though IFU mapping of lensed arcs in \citealp[e.g.,][]{Lopez20}, \citealp{Mortensen21}, and \citealp{Tejos21} can improve this in the future), as well as the fact that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas may not be representative of the entire cold phase of the CGM. To study more physically fundamental properties of the CGM, it is therefore necessary to turn to galaxy simulations.

In cosmological simulations (see \citealp{Vogelsberger20} for a review), the CGM has been notoriously difficult to model accurately due to the need to resolve very small structures \citep[e.g.,][]{Hummels19,Peeples19,Suresh19,Corlies20}. Nonetheless, the CGM has been shown to preferentially align with and rotate in the same direction of the galaxy, especially near the galaxy's major axis \citep{Stewart13,Stewart17,Ho19,DeFelippis20}, which is  qualitatively consistent with observations in the same spatial region of the CGM \citep[e.g.,][]{Zabl19}. However, this general qualitative agreement between simulations and observations is difficult to put on firm ground quantitatively due to the inherent differences between observations and simulations.

In this paper, we analyze a set of halos from the TNG50 simulation \citep{Nelson19,Pillepich19} using the \textsc{Trident} tool \citep{Hummels17} to model the ionization state of the CGM and then perform a quantitative comparison of the kinematics of the cool ($T \lesssim 3\times10^4 \; \rm{K}$) CGM traced by \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption to major-axis sight lines from the MEGAFLOW survey \citep{Zabl19} while attempting to match the observational selection criteria as described in Section \ref{sec:methods}. We note that our comparison to MEGAFLOW galaxies with stellar masses $M_{*} \sim 10^{10} \; M_{\odot}$ is complementary to that of both \cite{Nelson20}, who study the origins of cold CGM gas of very massive galaxies ($M_{*} > 10^{11} \; M_{\odot}$), and \cite{Nelson21}, who study properties of extended \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} emission in the CGM.

The paper is organized as follows. In Section \ref{sec:methods}, we describe the TNG50 simulation and MEGAFLOW sample used in the comparison, and we outline the analysis pipeline used to generate mock observations. In Section \ref{sec:results}, we describe our main results, first by comparing the simulated and real observations, then by analyzing the features of the simulation that give rise to the properties of the mock observations. In Section \ref{sec:discussion}, we discuss the implications of our results for the role of the CGM in galaxy formation, and we summarize our findings in Section \ref{sec:summary}.

\section{Methods} \label{sec:methods}

\subsection{Simulations}

We utilize the TNG50 simulation \citep{Nelson19,Pillepich19}, the highest-resolution version of the IllustrisTNG simulation suite \citep{Marinacci18,Naiman18,Nelson18,Pillepich18,Springel18}, which is itself based on the original Illustris simulation \citep{Vogelsberger14a,Vogelsberger14b}. TNG50 evolves a periodic $\approx (52 \; \rm{Mpc})^3$ box from cosmological initial conditions to $z=0$ with the moving-mesh code \textsc{Arepo} \citep{Springel10,Weinberger20}. It has a baryonic mass resolution of $\sim8.5\times10^4 \; M_{\odot}$ per cell, which is a factor of $\approx 16$ better than the resolution of TNG100. We discuss the effect of simulation resolution on our results later in Section \ref{sec:results}.

\subsection{Observational Data}
\label{sec:methods2}

The MEGAFLOW survey (N. Bouch\'e et al., in preparation) consists of a sample of 79 \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} $\mathrm{\lambda\lambda}2796,2803$ absorbers in 22 quasar lines of sight observed with the Multi-Unit Spectroscopic Explorer \citep[MUSE;][]{Bacon06}. The quasars were selected to have at least three \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorbers from the \cite{Zhu13} SDSS catalog in the redshift range $0.4 < z < 1.4$ such that the [\textsc{{\rm O}\kern 0.1em{\sc ii}}] $\lambda\lambda 3727,3729$ galaxy emission lines fell within the MUSE wavelength range ($4800-9300 \; \rm{\AA}$). A threshold on the rest-frame equivalent width of $\sim0.5-0.8 \; \rm{\AA}$ was also imposed on each absorber.

For this paper, we focus on a preliminary subset of the MEGAFLOW sample of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorber--galaxy pairs whose quasar location is positioned within 35$^\circ$ of the major axis of the host galaxy \citep[][]{Zabl19}. This subset consists of nine absorber--galaxy pairs with redshifts $0.5 < z < 1.4$ and impact parameters ($b$) ranging from 13 to 65 kpc with a mean of $\approx34$ kpc. \citet{Zabl19} found that the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas in these absorbers show a strong preference for corotation with their corresponding host galaxies.

The galaxies in \citet{Zabl19} are both fairly isolated by having at most one companion within $100 \; \rm{kpc}$, and star forming with [\textsc{{\rm O}\kern 0.1em{\sc ii}}] fluxes $f_{\textsc{{\rm O}\kern 0.1em{\sc ii}}}>4\times 10^{-17} \; \rm{erg \; s^{-1} \; cm^{-2}}$, i.e., star-formation rates $\gtrsim 1 \; M_{\odot} \; \rm{yr^{-1}}$. The galaxies have stellar masses $M_{*}$ ranging from $10^{9.3}-10^{10.5} \; M_{\odot}$ and halo masses $M_{\rm{vir}}$ ranging from $\approx 10^{11.4}-10^{12.2} \; M_{\odot}$, where $M_{\rm{vir}}$ is defined from the stellar mass--halo mass relation from \cite{Behroozi10}. As \cite{Zabl19} show, these halo masses generally match the \cite{Bryan98} definition of $M_{\rm{vir}}$.

\subsection{Sample selection and Forward Modeling}
\label{sec:methods3}

\begin{figure}
\fig{SFR_vs_Mvir.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{Star formation rate of the central galaxy vs. halo mass for all TNG50 halos between $10^{11} \; M_{\odot}$ and $10^{13} \; M_{\odot}$ at $z=1$. Each point is colored by the stellar mass of the halo's central galaxy. Two thick vertical lines demarcate the halo mass range of the fiducial sample.
}
\label{f:SFR_vs_Mvir}
\end{figure}

Figure \ref{f:SFR_vs_Mvir} shows the central galaxies' instantaneous star formation rates (SFR) and stellar masses of all TNG50 halos in and around the mass range of interest. Since we aim to compare the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption properties of mock line-of-sight (LOS) observations through TNG50 halos to those of major-axis sight lines of the MEGAFLOW survey, we first select a sample of simulated halos at $z=1$ in the mass range $10^{11.5} \; M_{\odot} < M_{\rm{halo}} < 10^{12} \; M_{\odot}$ using the \cite{Bryan98} definition for $M_{\rm{halo}}$, which results in a sample of 495 halos. In the remainder of this paper, we will refer to this subsample as the ``fiducial'' sample. The chosen redshift is typical for the \citet{Zabl19} sample, and the halo mass range covers the typical inferred virial masses of their halos. Nearly all of the halos in our fiducial sample host central galaxies with SFR $\gtrsim1 \; \rm{M_{\odot} \; yr^{-1}}$ and stellar masses of $\sim 10^{10} \; \rm{M_{\odot}}$, which is consistent with the MEGAFLOW subsample as described in Section \ref{sec:methods2}.

For each halo, we adjust all velocities to be in the center-of-mass frame of the stars in the central galaxy, and we rotate it so that the stellar specific angular momentum of the central galaxy points in the $+z$-direction (the $x$- and $y$-directions are both arbitrary). With this geometry we then define a sight line in the $x-z$ plane by the impact parameter $b$, the azimuthal angle $\alpha$, and the inclination angle $i$, where $b$ is the projected distance from the center of the galaxy in the $y-z$ plane (i.e.,~``sky'' plane), $\alpha$ is the angle above the rotational plane of the galaxy, and $i$ is the angle of the sight line with respect to the sky plane. In this setup, edge-on and face-on views have $i=90^{\circ}$ and $i=0^{\circ}$, respectively (see Figure 1 of \citealp{Zabl19} for a sketch of the geometry described here). In order to mimic the observations of \citet{Zabl19}, we select sight lines through each halo at values of $b$ ranging from $15 \; \rm{kpc}$ to $60 \; \rm{kpc}$, $\alpha = 5^{\circ}$ and $25^{\circ}$, and at $i = 60^{\circ}$, representing the average inclination angle of a random sight line.

In order to generate observations of our TNG50 sample, we use the \textsc{Trident} package \citep{Hummels17}, which calculates ionization parameters for outputs of galaxy simulations using properties of the simulated gas cells and \textsc{Cloudy} \citep{Ferland13} ionization tables. These tables take as input the gas temperature, density, metallicity, and cosmological redshift of each gas cell and provide ionization fractions and number densities of desired ions. We make use of the current development version of \textsc{Trident}\footnote{\url{http://trident-project.org}} (v1.3), which itself depends on the current development version of \textsc{yt}\footnote{\url{https://yt-project.org}} (v4.0). In this paper, we use a set of ion tables created assuming collisional ionization equilibrium, photoionization from a \cite{Faucher-Giguere09} UV background, and self-shielding of neutral hydrogen (for details see \citealp{Emerick19} and \citealp{Li21}), as this was the background radiation model used to evolve the TNG50 simulation. We also use the elemental abundance of magnesium in each gas cell tracked by the simulation rather than assuming a constant solar abundance pattern throughout the halo to achieve greater self-consistency with TNG50. We note, however, that our results are not particularly sensitive to either of these choices.

Since our focus is on the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} $\lambda 2796$ line,
we show in Figure \ref{f:MgII_phase} a temperature--density phase diagram of the gas in one of the TNG50 halos from our sample, colored by the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} mass probability density. From this plot, it is clear that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} is mostly formed from the coldest ($\lesssim 10^{4.5} \; \rm{K}$) and densest ($\gtrsim 0.01 \; \rm{cm^{-3}}$) gas in the halo, though some \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} mass exists at a larger range of temperatures and densities. However, contours showing the total gas mass demonstrate that despite this large range in temperature and density, essentially none of the diffuse ``hot'' phase, comparable in mass to the cold phase, contributes to \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption. We also note here that for this analysis we are excluding star-forming gas as its temperature and density are defined using an effective equation of state \citep{Springel03} and are therefore not analogous to the physical properties of non-star-forming gas. Properly modeling the physical properties of the star-forming gas (see \citealp{Padilla21} for an example of this technique) introduces a level of complexity not necessary for this analysis: we find that our results are not affected by the exclusion of this gas since our sight lines through the CGM rarely intersect any star-forming gas cells as most of them are within the galactic disk.

\begin{figure}
\fig{nT_MgII_halo225_Mgp1mass.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{Temperature--number density phase diagram of a single TNG50 halo at $z=1$, colored by the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} mass probability density per dex$^2$. Contours show the distribution of all gas mass in the halo. }
\label{f:MgII_phase}
\end{figure}

\section{Results} \label{sec:results}

\begin{figure*}
\centering
\gridline{
\fig{NMgII_halo225_noSFgas.pdf}{0.6\textwidth}{}
}
\vspace{-25pt}
\gridline{
\fig{NMgII_halo265_noSFgas.pdf}{0.33\textwidth}{}
\fig{NMgII_halo271_noSFgas.pdf}{0.33\textwidth}{}
\fig{NMgII_halo340_noSFgas.pdf}{0.33\textwidth}{}
}
\vspace{-25pt}
\caption{\textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} column density maps of four TNG50 halos from the fiducial halo mass bin of $10^{11.5} \; M_{\odot} < M_{\rm{halo}} < 10^{12} \; M_{\odot}$ at $z=1$, aligned so the angular momentum vector of the stars in the central galaxy points along the vertical axis (i.e., edge-on). The lower limit of the color bar is chosen to approximate observational detection limits. The red circle in each panel is centered on the galaxy and has a radius of twice the galaxy's stellar half-mass--radius, and the blue scale-bar shows a distance of $50 \; \rm{kpc}$ on the maps. The complexity and diversity of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} structure in the CGM of similar-mass halos are evident even in this small sample.
}
\label{f:MgII_maps}
\end{figure*}

We first present in Section \ref{sec:results1} the results of directly comparing the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} properties of TNG50 and MEGAFLOW using the analysis described in Section \ref{sec:methods}. Then, we further analyze the 3D kinematic properties of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{}-bearing gas from TNG50 in Section \ref{sec:results2} and consider evolution of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption properties with halo mass and simulation resolution in Section \ref{sec:results3}.

\subsection{Comparing TNG50 to MEGAFLOW}
\label{sec:results1}

In Figure \ref{f:MgII_maps}, we show \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} column density maps of a selection of TNG50 halos drawn from our fiducial sample at $z=1$. The halos are aligned so that the angular momentum vector of the stars in the central galaxy points along the vertical axis; thus, the view is edge-on. The strongest \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} columns are found within and very close to the galaxy, demarcated by a red circle with a radius of twice the galaxy's stellar half-mass--radius \citep[the same definition used in][]{DeFelippis20}. Beyond the galaxy, \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas consistently appears to both surround the galactic disk and be very clumpy, but the amount and morphology of such gas varies greatly. In particular, there is significant variation with azimuthal angle: the highest \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} columns generally appear in the plane of rotation, but strong columns can occur above and below the disk as well, such as in halo 265 (the bottom left panel of Figure~\ref{f:MgII_maps}). \citet{Peroux20} found the CGM gas metallicity to vary with azimuthal angle, but interestingly, they found gas near the major axis to have lower than average metallicity in the halo, indicating that large \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} columns do not necessarily correspond to metal-enriched gas. High \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} columns are much less common in the outer halo ($r \gtrsim 50 \; \rm{kpc}$), but the presence of satellite galaxies can populate that region with \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas, shown most clearly in halo 340 (the bottom right panel of Figure \ref{f:MgII_maps}).

Within our fiducial sample, it is evident that the distribution of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} varies drastically, presumably due to different halo formation histories. Sight lines through different halos will therefore likely produce different absorption profiles even for sight lines with identical geometries. This highlights the necessity of calculating population averages of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} properties from TNG50 to compare to MEGAFLOW.

\begin{figure}
\fig{EW0_vs_b_fiducial.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{Mean \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} equivalent widths of halos in our fiducial sample vs. the impact parameter of sight lines through those halos. Black and red lines and corresponding shaded regions show the mean and $\pm1\sigma$ scatter of all halos and the subset of strong absorbers (EW$_0 > 0.5 \; \rm{\AA}$), respectively. Sight lines at a constant azimuth angle of $\alpha = 5^{\circ}$ and $25^{\circ}$ are shown with solid and dotted lines, respectively. Observations of individual accretion systems from \citet{Zabl19} are shown as green squares. The fraction of strong absorbers as a function of impact parameter (blue) is shown with the right vertical axis. The cyan and orange dashed lines are log-linear fits of $z\sim1$ \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorbers from \cite{Nielsen13b} and \cite{Lundgren21}, respectively.
}
\label{f:EW_vs_b}
\end{figure}

We begin such a comparison with Figure \ref{f:EW_vs_b}, which shows the average strength of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption, represented as the rest-frame equivalent width (EW$_0$) as a function of impact parameter ($b$) for our fiducial sample. In this plot, we make an important distinction between the entire fiducial sample, shown in black, and the subset of ``strong absorbers'' in red. We define strong absorbers as sight lines through a halo that produce an absorption spectrum with EW$_0 > 0.5 \; \rm{\AA}$ \citep[the same as in][]{Zabl19}. It is this ``absorber-selected'' subset of the fiducial sample that is most directly comparable to MEGAFLOW. For easier comparison to Figure \ref{f:MgII_maps}, we find that sight lines with EW$_0 = 0.5 \; \rm{\AA}$ have \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} column densities ranging from $\approx10^{13.5}-10^{14.5} \; \rm{cm^{-2}}$, i.e., just above the lower limit of the color bar.

At all impact parameters, the average rest-frame EW of the ``all absorbers'' sample from TNG50 (black) is smaller than those of MEGAFLOW, as expected given the selection function. The difference ranges from a factor of only $\approx 3$ at $b \leq 20 \; \rm{kpc}$ to a factor of $\approx 30$ at $60 \; \rm{kpc}$. If, instead, we compare the average EW$_0$ of the strong absorber subset (EW$_0 > 0.5 \; \rm{\AA}$) from TNG50, which is the appropriate comparison to make, we find the mean shown in red. This is much more similar to the values from MEGAFLOW, especially for $b \geq 40 \; \rm{kpc}$, but it is still as much as a factor of $\approx 2$ lower than the observed values at $b \leq 20 \; \rm{kpc}$. However, the limited size and large scatter of the MEGAFLOW points from \citet{Zabl19} make it difficult to assess the precise level of disagreement with TNG50. Sight lines at $\alpha = 5^{\circ}$ (solid) and $\alpha = 25^{\circ}$ (dotted) produce essentially identical equivalent widths over both the entire fiducial sample and the subset of strong absorbers. With the two additional dashed lines in Figure \ref{f:EW_vs_b} we provide a point of comparison to larger samples of moderate-redshift \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorbers from \cite{Nielsen13b} and \cite{Lundgren21}. Though both of these samples have a slightly smaller equivalent-width threshold than \cite{Zabl19} ($\approx 0.2-0.3 \; \rm{\AA}$) and no selection based on the geometry of the sight line, they still bracket both the \cite{Zabl19} absorbers and the strong absorbers from TNG50, indicating that these simulated \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} EWs are also consistent with observed \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} EWs in general, given the large scatter.

The blue lines in Figure \ref{f:EW_vs_b} show the fraction of all sight lines that host strong absorbers as a function of impact parameter. At sight lines very close to the galaxy ($b = 15 \; \rm{kpc}$), strong absorbers are common and in fact represent a majority of all halos. However, by $b = 20 \; \rm{kpc}$ the strong absorber fraction drops below 50\%, and at the largest impact parameters shown, the fraction is only $\approx 1\%$. Strong absorbers are slightly more common at $\alpha = 5^{\circ}$ compared to $\alpha = 25^{\circ}$, which can be understood by noting that the sight lines with smaller $\alpha$ pass through the disk midplane closer to the galaxy's center, where gas is generally denser. However, this difference in strong absorber fraction does not affect the measured equivalent widths, indicating that the TNG50 halos' agreement with MEGAFLOW for sight lines near the galaxies' major axes is not subject to the precise geometries of the sight lines.

\begin{figure}
\fig{covering_fraction.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{\textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} covering fraction of our fiducial sample as a function of impact parameter for sight lines  with $\alpha=5^{\circ}$ (blue), $\alpha=45^{\circ}$ (orange), and $\alpha=85^{\circ}$ (green). Solid and dashed lines show sight lines that are edge-on and inclined at $i=60^{\circ}$, respectively.
}
\label{f:covering}
\end{figure}

In Figure \ref{f:covering}, we examine how \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} EWs vary throughout the entire halo in TNG50, not just near the major axis, and we find a clear trend: at all impact parameters we study, the mean EW of a perfectly edge-on sight line decreases as the azimuth angle of that sight line $\alpha$ increases. Sight lines near the minor axis (green) have EWs at least $0.35 \; \rm{dex}$ smaller than sight lines near the major axis (blue), and sight lines between both axes (orange) have EWs between the values at both axes. This represents a disagreement between TNG50 and \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} observations, which are generally observed to have a bimodal distribution of $\alpha$ near $0^{\circ}$ and $90^{\circ}$ \citep{Bordoloi11,Bouche12,Kacprzak12,MartinC19,Zabl19,Lundgren21}. The distribution of $\alpha$ in TNG50 is clearly peaked at small $\alpha$, implying that TNG50 is not producing the same kind of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} that is inferred to be outflowing in observations. It is also clear that this azimuthal angle dependence is very sensitive to the inclination angle of the sight line because it nearly disappears when the sight lines are inclined at an angle of $60^{\circ}$ with respect to the axis of rotation (dotted lines in Figure \ref{f:covering}), as would be typical for observations. This sensitivity indicates that most \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption in TNG50 comes from a gas in the vicinity of the disk midplane, where we have already seen (Figure \ref{f:EW_vs_b}) that TNG50 is consistent with observations. Therefore, for the remainder of this paper we restrict our observational comparison to sight lines near the major axis.

Having established the degree of consistency of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} equivalent widths, we now examine kinematic signatures of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} along sight lines in TNG50 and compare them to MEGAFLOW. In Figure \ref{f:MgII_sightlines}, we explicitly draw the connection between the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas cells that contribute to the column densities seen in Figure \ref{f:MgII_maps} and the velocity spectrum created from a subset of those cells that intersect a sight line through the halo. In each row, we show two orientations of one of the four halos from Figure \ref{f:MgII_maps} overlaid with a sight line with $b=30 \; \rm{kpc}$, $\alpha = 5^{\circ}$, and $i = 60^{\circ}$, and the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} velocity spectrum generated from that sight line. From these few examples it is clear that the gas producing the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption is generally not distributed uniformly along any sight line: it is usually concentrated in discrete clumps in regions of the sight line nearest to the galaxy. This is seen clearly in rows one, two, and four of Figure \ref{f:MgII_sightlines}, where the majority of gas cells have positive LOS velocities (i.e.,~corotating with the galaxy) and produce distinct kinematic components in the spectrum that are often saturated.

\begin{figure*}
\gridline{
\fig{halo225_sightline_noSFgas.pdf}{0.55\textwidth}{}
\fig{halo225_b30spec.pdf}{0.33\textwidth}{}
}
\vspace{-20pt}
\gridline{
\fig{halo265_sightline_noSFgas.pdf}{0.55\textwidth}{}
\fig{halo265_b30spec.pdf}{0.33\textwidth}{}
}
\vspace{-20pt}
\gridline{
\fig{halo271_sightline_noSFgas.pdf}{0.55\textwidth}{}
\fig{halo271_b30spec.pdf}{0.33\textwidth}{}
}
\vspace{-20pt}
\gridline{
\fig{halo340_sightline_noSFgas.pdf}{0.55\textwidth}{}
\fig{halo340_b30spec.pdf}{0.33\textwidth}{}
}
\vspace{-15pt}
\caption{Each row contains two \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} column density maps of a halo from Figure \ref{f:MgII_maps} projected along the vertical (left) and a horizontal (middle) axis. A sight line at $b=30 \; \rm{kpc}$, $\alpha = 5^{\circ}$, and $i=60^{\circ}$ is overlaid along with gas cells that intersect that sight line and have a \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} column density of at least $10^{12} \; \rm{cm^{-2}}$, which accounts for $>95\%$ of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} mass along those sight lines. The \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas cells and the resulting flux-normalized velocity spectrum (right) are colored by the velocity along the line of sight normalized by $V_{\rm{vir}}\sin(i)$, where $V_{\rm{vir}}$ is the virial velocity of the halo. Dashed circles show twice each galaxy's stellar half-mass--radius.
}
\label{f:MgII_sightlines}
\end{figure*}

It is also notable that by comparing the spectra alone it is possible to distinguish morphological differences in the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} distribution between halos. The first two halos, for example, have a prominent \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} disk that both spectra reveal to be primarily corotating. The halo in row three, however, does not have such a clear disk, and the spectrum is instead composed of a cluster of counter-rotating gas cells significantly above the plane of the galaxy. The halo in row four has a spectrum with substantial corotating and counter-rotating components, which imply \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} structure in between the ordered halos (rows one and two) and disordered ones (row three). With this small sample, we have demonstrated that the velocity spectrum, despite being composed of a very small fraction of all of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas, is capable of reflecting the potential diversity of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas kinematics in halos of similar mass, but is also fairly consistent between halos with similar morphologies. Later in the paper, we consider whether the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas reflects the kinematics of other components of the CGM.

From these results, we now compare stacked spectra from the fiducial sample to the stacked spectra presented in \citet{Zabl19}. Figure~\ref{f:spectra} shows stacked spectra for the entire TNG50 fiducial sample (black), TNG50 strong absorbers (red), and the absorbers from \citet{Zabl19} (green). The two panels correspond to two different impact parameters that allow a comparison between absorbers nearer to a galaxy and farther from a galaxy. In the left panel, showing stacked spectra at small impact parameters, there is a very clear kinematic picture. The strong absorber spectrum from TNG50 is symmetric, centered at $\approx0.6 V_{\rm{vir}}$, and has a with a full width at half maximum (FWHM) of $1.2 V_{\rm{vir}}$, the same as the spectrum of \citet{Zabl19}. Thus, qualitatively, strong \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorbers as a population generally have LOS velocities in the same direction as their corresponding galaxies' rotations. One slight difference with the stacked spectra for strong absorbers is that the TNG50 spectrum (red) is somewhat shallower than the observed spectrum (green). However, there is essentially no difference between TNG50 spectra generated from sight lines at the two azimuthal angles $\alpha = 5^{\circ}$ (solid line) and $25^{\circ}$ (dotted line).

\begin{figure*}
\fig{bclose_fiducial_boot.pdf}{0.48\textwidth}{}
\fig{bfar_fiducial_boot.pdf}{0.48\textwidth}{}
\vspace{-15pt}
\caption{The stacked \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} velocity spectra for the full fiducial TNG50 sample (black) and the subset of strong absorbers (red) for sight lines with $\alpha = 5^{\circ}$ (solid) and $25^{\circ}$ (dotted), and $b = 20 \; \rm{kpc}$ (left) and $40 \; \rm{kpc}$ (right). Spectra are normalized by $V_{\rm{vir}}\sin(i)$, where $V_{\rm{vir}}$ is the halo's virial velocity. The green line in each panel is the stacked spectrum of the four smallest (left) and largest (right) impact parameters from \citet{Zabl19}, and the green shaded region is an estimate of the error from bootstrapping.
}
\label{f:spectra}
\end{figure*}

In Figure~\ref{f:spectra} (left), the only difference between the full fiducial spectrum and the strong absorber-only spectrum is the depth, indicating that, as a population, strong absorbers are not kinematically distinct from absorbers in general at this impact parameter. The precise reason for the discrepancy in the depth is difficult to determine, but it may be sensitive to certain parameters in the TNG physics model (e.g., metal loading of outflows from supernovae). However, it could also be an effect of simulation resolution (see Section \ref{sec:results3}). So, while TNG50 potentially slightly underproduces the observed amount of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas at 20 kpc, it does possess average kinematics that are consistent with observations of the same region of the CGM.

Figure~\ref{f:spectra} (right) compares the stacked spectra at a larger impact parameter ($b=40$ kpc). The strong absorbers from TNG50 and MEGAFLOW \citep{Zabl19} are both shallower, wider (FWHMs of $1.3 V_{\rm{vir}}$ and $2 V_{\rm{vir}}$ respectively), no longer symmetric, and significantly noisier, though both are still approximately centered at a velocity on the order of $V_{\rm{vir}}/2$. At this impact parameter, the depths of the simulated strong absorber and observed spectra are consistent with each other. However, strong absorbers no longer kinematically resemble the full fiducial sample: in addition to being much rarer at 40 kpc than at 20 kpc, the strong absorbers have larger positive velocities, indicating that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} in this region is tracing atypically faster-moving gas. As was the case at 20 kpc, the difference in the spectra between the two azimuth angles is minor. We also note here, but do not show, that the shapes and depths of individual  spectra from \citet{Zabl19} match quite well with particular individual spectra from the much larger fiducial sample from TNG50 (examples of individual spectra from TNG50 are shown in Figure~\ref{f:MgII_sightlines}).

\subsection{3D Kinematics of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} in TNG50}
\label{sec:results2}

In this section, we characterize the three-dimensional kinematics of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas in TNG50 and its relation to the observed quantities we discussed in Section~\ref{sec:results1}. We show average velocity profiles of the halos in the fiducial sample in Figure~\ref{f:vprofiles}. The top panel shows the azimuthal velocity component ($v_{\phi}$) in spherical coordinates as a function of radius. We divide gas into cold and hot components based on a temperature threshold of $3\times10^4$ K, which is chosen to separate the cold and hot clusters seen in Figure~\ref{f:MgII_phase}, although the profiles are not sensitive to the precise choice of temperature threshold. To understand the relationship of the hot and cold gas to \textsc{{\rm Mg}\kern 0.1em{\sc ii}}-bearing material we also show the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} mass-weighted profiles.

\begin{figure}
\fig{v_profiles_fiducial.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{Mean mass-weighted velocity profiles of the spherical phi-component ($v_{\phi}$, top) and r-component ($v_{r}$, bottom) for cold gas (blue), hot gas (red), and \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas (black) in spherical bins. Velocity is given in $\rm{km \; s^{-1}}$ and as a fraction of the virial velocity. A temperature of $3\times10^4 \; \rm{K}$ is used to separate ``cold'' and ``hot'' gas. Profiles are shown for gas in the entire halo (solid), gas with $\alpha > 45^{\circ}$ (dotted), and gas with $\alpha < 45^{\circ}$ (dashed). Shaded regions show the $\pm1\sigma$ scatter of the solid lines and are of similar size for all profiles.
}
\label{f:vprofiles}
\end{figure}

First, we see that the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas and the cold gas have nearly identical $v_{\phi}$ profiles throughout the halo. In the innermost regions of the CGM ($15-20 \; \rm{kpc})$, the cold gas has a mean azimuthal velocity of $80 \; \rm{km} \; \rm{s^{-1}}$ ($\approx 0.6 V_{\rm{vir}}$), while in the outermost regions ($90-100 \; \rm{kpc})$, the mean azimuthal velocity decreases to $20 \; \rm{km} \; \rm{s^{-1}}$ ($\approx 0.15 V_{\rm{vir}}$). At all radii, the $\pm1\sigma$ scatter is quite large ($\approx 100 \; \rm{km} \; \rm{s^{-1}}$), though the standard errors on this and all other mean velocities in Figure \ref{f:vprofiles} range from only $1-3 \; \rm{km} \; \rm{s^{-1}}$. Though not explicitly shown, most of the cold and \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas mass is closer to the major rather than the minor axis because the all-$\alpha$ profiles are much more similar to the $\alpha < 45^{\circ}$ (dashed) profiles than the $\alpha > 45^{\circ}$ (dotted) profiles. Hot gas has lower azimuthal velocities at all radii, a slightly shallower slope to its profile, and a smaller scatter in azimuthal velocity by a factor of $\approx 2$ but is otherwise qualitatively similar to the cold and \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas. This relationship between hot and cold gas is consistent with similar measurements of $v_{\phi}$ made from TNG100 in \cite{DeFelippis20}.

In the radial-velocity profiles (Figure \ref{f:vprofiles}, bottom), we see a gulf between the velocities of the hot and cold gas develop within $90 \; \rm{kpc}$. Above this radius, the average radial velocities of all components of the gas converge to $-20 \; \rm{km} \; \rm{s^{-1}}$ ($\approx 0.15 V_{\rm{vir}}$), though the spread of radial velocities in this region of the CGM is very large, especially for cold gas ($\pm1\sigma$ scatter of $120 \; \rm{km} \; \rm{s^{-1}}$). Moving toward smaller radii, the cold gas inflow velocities become larger, while hot gas inflow velocities decrease and then switch to a net outflow at $50 \; \rm{kpc}$. The \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas still traces the cold gas, which reaches typical inflowing velocities of $45-50 \; \rm{km} \; \rm{s^{-1}}$ in the inner CGM out to $r=40 \; \rm{kpc}$, where the spread in radial velocities is a factor of $2$ smaller than in the outer halo. The geometry of accretion and outflows is evident from this panel as well: hot gas has especially large mean outflowing velocities for $\alpha > 45^{\circ}$ while cold gas in the same region has a mean inflowing velocity in the inner halo and nearly no net radial motion in the outer halo. Most of the cold and \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas mass is moving toward the galaxy in regions surrounding the major axis out to a substantial fraction of the virial radius. It is also clear that kinematically, \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas in TNG50 is nearly identical to a simple cut on temperature and so is an excellent tracer of the kinematics of cold CGM gas. In the context of Section \ref{sec:results1}, these results indicate that mock \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} spectra are representative of the entire cold phase of the CGM along the same sight lines.

Finally, we examine the 3D velocities of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas along our sight lines. In Figure \ref{f:vcomponents}, we plot stacked spectra for \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} using the three spherical velocity components individually ($r$, $\theta$, and $\phi$), and compare those to the spectrum generated with the full velocity of our fiducial sample of halos. Both the $r$ and $\theta$ component spectra are centered at $0 \; \rm{km} \; \rm{s^{-1}}$, indicating that over the entire sample they do not contribute any net velocity shift to the gas along the sight lines. The spectrum of the $\phi$ component is remarkably similar to the spectrum of the entire velocity, both in terms of velocity shift and width. This means that for our fiducial sample, the shape of the stacked velocity spectrum along sight lines is completely determined by only the $\phi$ (i.e., rotational) component of the velocity along those sight lines.

\begin{figure}
\fig{b20vcomp_fiducial.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{Stacked \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} velocity spectra for the full fiducial TNG50 sample at a single sight line. The contributions of the three spherical components of velocity -- $v_{r}$ (dotted red), $v_{\phi}$ (dashed green), and $v_{\theta}$ (dotted-dashed blue) -- are shown, as well as the spectrum created from the total velocity (solid black).}
\label{f:vcomponents}
\end{figure}

\subsection{Effects of halo mass and resolution on \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} in TNG50}
\label{sec:results3}

We now describe how our main results vary with halo mass and mass resolution. To study the effect of halo mass, we consider two mass bins containing halos from TNG50 with $10^{11} \; M_{\odot} < M_{\rm{halo}} < 10^{11.5} \; M_{\odot}$ and $10^{12} \; M_{\odot} < M_{\rm{halo}} < 10^{12.5} \; M_{\odot}$  at $z=1$, which are above and below the fiducial mass range and contain 1130 and 167 halos, respectively. As in Section \ref{sec:results1} we calculate \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} equivalent widths and generate velocity spectra that we show in Figure \ref{f:mass_and_resolution}. For easier comparison, we also show the TNG50 fiducial sample.

As shown in the left panel of Figure \ref{f:mass_and_resolution}, at a given impact parameter, the shape of the equivalent-width distribution changes with halo mass: lower halo masses (cyan) are much more likely to host weak or nonabsorbers than higher halo masses (magenta), and they are much less likely to host strong absorbers. We find this trend to hold at all impact parameters studied in this paper. We can see the effect on observability with the vertical lines in this panel, which show the mean equivalent widths of the strong absorbers in each mass bin. Typical strong absorbers in the fiducial sample have only slightly larger equivalent widths than those those at lower halo masses, but are substantially weaker than the strong absorbers at higher halo masses. At larger impact parameters, the mean equivalent widths of all strong absorbers is $\approx 0.8 \; \rm{\AA}$, but they are exceedingly rare in lower-mass halos. Thus, the primary effects of increasing halo mass on strong absorbers are to increase their occurrence at all impact parameters, especially at large distances, and to increase the mean equivalent width of strong absorbers for halo masses $\gtrsim 10^{12} \; M_{\odot}$. We note that this result is qualitatively consistent with \cite{Chen10b}, who find a larger \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} extent in the CGM of higher-mass galaxies.

Also shown in the left panel of Figure \ref{f:mass_and_resolution} is the equivalent-width distribution of 4315 halos with the same mass as the fiducial sample from the TNG100 simulation, which has a lower baryonic mass resolution than TNG50 by a factor of $\sim 16$. Decreasing the simulation resolution lowers equivalent widths overall and steepens the distribution in the same way as decreasing the halo mass does, but the effect is weaker. The mean equivalent width of strong absorbers is largely unaffected by the change in resolution.

\begin{figure*}
\fig{EWhist_b20.pdf}{0.48\textwidth}{}
\fig{b20_spectra_comparison.pdf}{0.48\textwidth}{}
\vspace{-15pt}
\caption{Left: rest-frame equivalent-width distribution of the TNG50 fiducial sample (solid black), lower-mass halos with $10^{11} \; M_{\odot} < M_{\rm{halo}} < 10^{11.5} \; M_{\odot}$ (cyan), higher-mass halos with $10^{12} \; M_{\odot} < M_{\rm{halo}} < 10^{12.5} \; M_{\odot}$ (magenta), and the same mass halos from TNG100 (dotted black) at the same sight line of $b=20 \; \rm{kpc}$, $\alpha = 5^{\circ}$, and $i=60^{\circ}$. The mean EW$_0$ of the strong absorbers in each halo mass bin is shown with a translucent vertical line of the same color. Right: stacked velocity spectra of the same halo samples with velocities in $\rm{km \; s^{-1}}$.}
\label{f:mass_and_resolution}
\end{figure*}

In the right panel of Figure \ref{f:mass_and_resolution} we examine the effect of halo mass and resolution on the observed \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} spectrum of strong absorbers. We note that the spectra of the entire samples, as in Figure \ref{f:spectra}, have the same shape and center as their corresponding strong absorber subset, but are substantially shallower. We also plot the real velocity rather than the normalized velocity to emphasize the difference in equivalent widths, which can be more easily read off.

We see that the fiducial and lower-mass bins have remarkably similar spectra: they are both symmetric and centered at moderate positive velocities. The spectrum of the higher-mass bin is markedly different: it is much broader, asymmetric, and centered at a significantly higher velocity. It still, however, shows a preference for \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas to be corotating. We note that the difference between Figure \ref{f:mass_and_resolution} as shown and the corresponding velocity-normalized spectrum (not shown) is that the normalized higher-mass spectrum is compressed slightly and therefore appears more similar to the normalized fiducial spectrum. Additionally, while the lower-mass and fiducial spectra are both centered at $\approx 0.5 V_{\rm{vir}}$, the higher-mass spectra are peaked at $\approx V_{\rm{vir}}$. Higher halo masses ($\gtrsim 10^{12} \; M_{\odot}$) thus have substantially more \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption and more complex kinematic signatures than for the halo masses of the fiducial sample and lower.

Finally, we consider the difference that resolution makes in the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption spectrum. As was the case with equivalent widths, the difference caused by resolution is smaller than the difference caused by either increasing or decreasing the halo mass. Apart from a slight change in the depth of the spectrum, the kinematic properties of strong absorbers in TNG are essentially resolution independent (see solid vs. dotted curves in Figure \ref{f:mass_and_resolution} for TNG50 and TNG100, respectively). The effect of increasing the resolution of the simulation is therefore primarily to increase the occurrence of strong absorbers at a given halo mass.

\section{Discussion} \label{sec:discussion}

\subsection{The Role of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} in TNG}

We consider here the ramifications of the detailed analysis of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} in TNG from Section \ref{sec:results}. In Figure \ref{f:vprofiles}, we found that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas is very well approximated by a simple temperature cut. Therefore, we expect the angular momentum of cold gas in the CGM of TNG galaxies should be very similar to that of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}. \cite{DeFelippis20} found cold CGM gas in halos of this mass range and redshift to have higher angular momentum when surrounding high-angular-momentum galaxies, meaning \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} is likely tracing high-angular-momentum gas in the CGM of these halos. As the velocity spectrum's center and shape is almost completely set by the rotational velocity component (see Figure \ref{f:vcomponents}), it should therefore be possible to use \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} velocity spectra from sight lines near the major axis to estimate the angular momentum of cold gas in the CGM.

In Section \ref{sec:results3} we examined possible halo mass and resolution dependencies of our results with two main goals in mind: to establish any broad effects of the TNG feedback model on \textsc{{\rm Mg}\kern 0.1em{\sc ii}}, and to determine to what extent the cosmological simulation can capture \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} kinematics. Feedback is known to be important for regulating gas flows into, out of, and around galaxies, and therefore could have observable signatures in the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} spectra, especially at different halo masses. The results of the halo mass analysis suggest that for halos with masses between $10^{11} \; M_{\odot}$ and $10^{12} \; M_{\odot}$, the physical mechanisms affecting their CGM are similar enough to result in \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} spectra that essentially scale with the halo's virial velocity. This is presumably because feedback from supernovae is the dominant form of feedback that affects the CGM for all halo masses below $\sim10^{12} \; M_{\odot}$ and produces \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas with similar kinematic signatures. For halos above $10^{12} \; M_{\odot}$ however, \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas has stronger overall absorption, as reflected by their flatter EW distribution, and substantially larger velocities and velocity dispersions, as reflected by their very broad velocity spectra. This is likely due to the dominant form of feedback switching from stars to AGN around this halo mass. However, within the higher-mass sample, halos with larger black hole masses do not themselves have broader \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} spectra, so there is probably a combination of effects that result in a noticeable difference in the properties of the spectrum at higher masses.

\cite{Nelson20} have recently used TNG50 to study the origin of cold \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas in the CGM of very massive ($M_{*} \gtrsim 10^{11} \; M_{\odot}$) galaxies and found structures of size a few $\times 10^2 \; \rm{pc}$ that are sufficient to explain the observed covering fractions and LOS kinematics. They also note that while some fundamental properties like the number of cold gas clouds present in halos are not converged at TNG50's resolution, the total cold gas mass of such halos is converged in TNG50. This supports our findings that our kinematic results do not qualitatively change even going from TNG50 to TNG100, a factor of $\sim16$ in mass resolution (Figure \ref{f:mass_and_resolution}), because the majority of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} mass is already in the halo by TNG50's resolution. We expect higher-resolution simulations to produce more strong absorbers at a given halo mass but the rotation of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} near the major axis appears to be a resolution-independent aspect of the CGM for MEGAFLOW analogs in the TNG simulations.

\begin{figure}
\fig{j_vs_Mstars.pdf}{0.48\textwidth}{}
\vspace{-25pt}
\caption{Median-specific angular momentum vs. galactic stellar mass for the cold (blue), hot (red), and \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} (black) CGM as defined in Figure \ref{f:vprofiles}, as well as the dark matter halo (dotted orange) and the stellar component of the galaxy (purple) at $z=1$. Unlike previous figures, medians are calculated using a sample of all halos containing central galaxies with stellar masses $10^{9} \; M_{\odot} < M_{*} < 10^{11} \; M_{\odot}$. Shaded regions show the 16th and 84th percentiles of the distributions of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas (black), which is similar in size to all components except dark matter (orange), which has noticeably larger scatter. Black points show the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} specific angular momentum of the halo-mass-selected fiducial sample that is biased toward higher $j$ for $M_{*} \lesssim 10^{9.75}$. Green squares show estimations for the specific angular momentum of the major-axis absorbers using inferred rotational velocities from \cite{Zabl19}.}
\label{f:jM}
\end{figure}

Finally, in Figure \ref{f:jM}, we show the specific angular momentum ($j$) of different halo components as a function of stellar mass of their central galaxies, with the goal of contextualizing the angular momentum of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas (black line) in the CGM in relation to the rest of the gas in the CGM as well as to the other components of the halo. The slope of this $j-M_{*}$ relation for the stellar component of galaxies (purple line) is $\sim 0.6$ as generally observed \citep[e.g.,][]{Fall13}, and all other components appear to have roughly equal slopes. Most interesting are the relative positions of the CGM and dark matter (orange line) on this plane. At a given stellar mass, all components of the CGM have a slightly higher typical $j$ than that of the dark matter by $\sim 0.2 \; \rm{dex}$. There are multiple potential reasons for this. First, galaxies can remove low-angular-momentum gas from the CGM by accreting it and using it to form stars. Second, feedback from stars and/or AGN can also eject low-angular-momentum gas from the halo completely. Finally, dark matter in the halo can transfer some of its angular momentum to the gas. Regardless, it is clear that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} traces the angular momentum of the both the cold and hot components of the CGM quite well.

Also shown in Figure \ref{f:jM} are two sets of points representing \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas in individual halos: the fiducial sample in black and the \citet{Zabl19} sample in green, for which $j$ was estimated using their derived rotational velocities. The two are not directly comparable since the points from \citet{Zabl19} represent \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas along a single sight line, yet they are still able to reproduce the scatter in this relation found in TNG50, though they are somewhat biased toward higher $j$. This bias is likely due to the selection in \citet{Zabl19} of strong \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption near the major axis, which is where high-$j$ cold gas tends to reside in the CGM as shown in \citet{DeFelippis20}. Nevertheless, from Figure \ref{f:jM} we can conclude that estimations of the angular momentum content of the CGM provided by single sight lines of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} can get within $\sim0.5 \; \rm{dex}$ of typical values from TNG50 over a large range of galaxy masses.

\subsection{Comparisons to Recent Work}

We now highlight results from previous work on \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption in observations and simulations in the context of our results. Observations of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} using sight lines near the major axis of galaxies have generally found that gas is corotating with the galaxy both for small impact parameters of $< 15 \; \rm{kpc}$ \citep[e.g.,][]{Bouche16} and large impact parameters of $> 50 \; \rm{kpc}$ \citep[e.g.,][]{MartinC19}. Using a lensed system, \cite{Lopez20} observed multiple sight lines of the same CGM and measured a decreasing \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} rotation curve that is qualitatively similar to Figure \ref{f:vprofiles}. However, their absorption data only go out to $\approx30 \; \rm{kpc}$. Our work suggests \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} rotation curves should continue to decrease to at least 100 kpc, though based on the maps in Figure \ref{f:MgII_maps} the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} column densities at those distances are significantly below current observational limits.

While this paper is focused on \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas near the major axis, there are also recent results suggesting \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} outflows along the minor axis of galaxies with velocities $> 100 \; \rm{km \; s^{-1}}$ \citep[e.g.,][]{Schroetter19,Zabl20}. It is worth noting though that \cite{Mortensen21} found a lensed system with \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} on the geometric minor axis of the absorber galaxy with LOS velocities $< 100 \; \rm{km \; s^{-1}}$ and a large velocity dispersion, indicating that the kinematics of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} outflows may vary significantly. We showed in Figures \ref{f:covering} and \ref{f:vprofiles} that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption along the minor axis is weaker than along the major axis, and that there are no net \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} outflows along the minor axis in the TNG fiducial sample. This result appears to be discrepant with the previously cited observational papers, but we defer a detailed analysis to a future paper.

\cite{Ho20} recently studied similar aspects of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption in the EAGLE simulation at $z\approx 0.3$ and found results broadly consistent with ours. Specifically, they measure a rotating \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} structure around star-forming galaxies as well as a lower detection fraction of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} near the minor axis. They also find that higher-mass galaxies host detectable (i.e.,~above a fixed column density) \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} structures out to larger distances in the CGM, which we indirectly show with the EW distributions in Figure \ref{f:mass_and_resolution}, where higher-mass halos have more strong absorbers.

\section{Summary} \label{sec:summary}

We have simulated \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption in the CGM of halos from TNG50 comparable to the major-axis sight lines observed in the MEGAFLOW survey by \cite{Zabl19} and compared absorption and kinematic properties of the two samples. We also examined the 3D kinematics of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} in TNG50. Our conclusions are as follows:

\begin{enumerate}
    \item The equivalent widths of absorber-selected halos (i.e.,~strong absorbers) from TNG50 match reasonably well with the equivalent widths of major-axis sight lines from \cite{Zabl19} (Figure \ref{f:EW_vs_b}).
    \item A majority of halos are strong absorbers at the smallest impact parameter studied (15 kpc), but the strong absorber fraction drops quickly as a function of distance (Figure \ref{f:EW_vs_b}).
    \item The stacked velocity spectra of TNG50 strong absorbers match the stacked spectra of \citet{Zabl19} very well, thus supporting the physical interpretation of corotation both below 30 kpc, where the spectra are strongly peaked near $\sim0.5 V_{\rm{vir}}$ and symmetric, and above 30 kpc, where the spectra are similarly peaked but are much noisier, broader, and asymmetric (Figure \ref{f:spectra}).
    \item In TNG50, \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas has velocity profiles nearly identical to gas below a temperature cutoff of $3\times10^4 \; \rm{K}$, meaning \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption is a good proxy for cold gas kinematics in general. There is substantial rotation and typical inflow velocities of up to $50 \; \rm{km} \; \rm{s^{-1}}$ out to $\sim40 \; \rm{kpc}$ in the CGM (Figure \ref{f:vprofiles}).
    \item The radial and polar velocity components by themselves do not cause any net velocity shift in the stacked spectrum, which implies that \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption kinematics alone cannot be used to measure typical inflow speeds of rotating gas in the CGM. (Figure \ref{f:vcomponents}).
    \item \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} absorption strengths and spectra are stronger and broader for halos more massive than the fiducial sample of $10^{11.5}-10^{12} \; M_{\odot}$ halos but do not change very much for halos less massive than the fiducial sample. Lowering the resolution from TNG50 to TNG100 only modestly changes any of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} kinematic properties (Figure \ref{f:mass_and_resolution}).
    \item The median-specific angular momentum of the \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} component of the CGM as a function of galactic stellar mass is very similar to that of both cold and hot CGM gas, and it is larger than that of the dark matter halo and the stars in the galaxy by $\sim0.2 \; \rm{dex}$ and $\sim0.8 \; \rm{dex}$, respectively. Estimates of the specific angular momentum of \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} from the \citet{Zabl19} data are also reasonably close to the values from TNG50 to within a factor of $\sim0.5 \; \rm{dex}$. (Figure \ref{f:jM}).
\end{enumerate}

This work demonstrates that generating mock \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} observations from TNG50 generates absorption spectra that are comparable to real data. In particular, our results are consistent with the emerging picture of rotating \textsc{{\rm Mg}\kern 0.1em{\sc ii}}{} gas found in observations and also other simulations. In future work, we plan to widen our investigation to include other ions that trace warmer and more diffuse gas, as well as follow gas at particular redshifts backward and forward through time to determine the stability of various ion structures and their role in transporting angular momentum to or from the galaxy.

\acknowledgments

We thank Johannes Zabl, \'Edouard Tollet, Joakim Rosdahl, and J\'er\'emy Blaizot for insightful and useful discussions, as well as Cameron Hummels for assistance with \textsc{Trident}. We also thank the anonymous referee for helpful comments. D.D. acknowledges support from the Chateaubriand Fellowship of the Office for Science \& Technology of the Embassy of France in the United States. N.B. acknowledges funding support from the French Agence National de la Recherche (ANR) grant ``3DGasFlows'' (ANR-17-CE31-0017). Flatiron Institute is supported by the Simons Foundation. G.L.B. acknowledges financial support from the NSF (grants AST-1615955, OAC-1835509) and computing support from NSF XSEDE. F.M. acknowledges support through the Program ``Rita Levi Montalcini'' of the Italian MUR.

\software{\textsc{Trident} \citep{Hummels17},
          \textsc{yt} \citep{Turk11},
          \textsc{Cloudy} \citep{Ferland13},
          \textsc{NumPy} \citep{vanderWalt11},
          \textsc{Matplotlib} \citep{Hunter07},
          and \textsc{IPython} \citep{Perez07}
          }

redpajama_set_name: RedPajamaArXiv


We stayed at this hotel for 3 nights. After the first night, we noticed ants in the bathroom area of the room we were in. We notified the front desk abd they took care of most of the ant issue but there were still more each of the following morning. We were on the first floor and some areas were under construction.
Our room was nicely updated and very attractive, however the bed was not very comfortable and I didn't care for the lighting in the bathroom. When we arrived and got to our room it had not been cleaned from the person before us. The water was lukewarm both days and the day we left the door wouldn't close all the way. There was also construction being done at the front entrance and we couldn't pull up there to park or check in and there was dust everywhere. Usually we have a great experience at Hampton's, but in this situation, there were a lot of things wrong that shouldn't have been.
It needs updating and need to work on the water temp in the rooms. No cold water available in the sink and the water in the shower was hot no matter how far you turned the knob. Not much available for snacks, ice machine on our floor was broken and the breakfast is ok, but not anything special. Coffee was cold! Air conditioning was not evenly controlled.


text:
Q: How to show $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \ dx$? $$\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = 4 \int_{0}^{\frac{1}{2}} \frac{\arcsin^{2}(x)}{x} \ dx.$$
Someone please show that this equation is correct !?

A: It is known that
$$
\arcsin(x)^2=\frac{1}{2}\sum_{n=1}^\infty\frac{(2x)^{2n}}{n^2\binom{2n}{n}}
$$
see Companion to Concrete Mathematics - Mathematical Techniques and Various Applications Z. A. Melzak p. 108
Then
$$
\begin{aligned}
\int_{0}^{1/2}\frac{4\arcsin(x)^2}{x}
&=\int_{0}^{1/2}\frac{2}{x}\sum_{n=1}^\infty\frac{(2x)^{2n}}{n^2\binom{2n}{n}}dx\\
&=\sum_{n=1}^\infty\frac{2^{2n+1}}{n^2\binom{2n}{n}} \int_{0}^{1/2}x^{2n-1}dx\\
&=\sum_{n=1}^\infty\frac{2^{2n+1}}{n^2\binom{2n}{n}} \frac{\left(\frac{1}{2}\right)^{2n}}{2n}\\
&=\sum_{n=1}^\infty\frac{1}{n^3\binom{2n}{n}}
\end{aligned}
$$

A: Hint: $~\displaystyle\sum_{n=1}^\infty\frac{(2x)^{2n}}{n^2\displaystyle{2n\choose n}}=2\arcsin^2x$.


Many people applaud how seriously I take my endeavors. But if I'm honest with you, I struggle with it.
There are many times where I struggle to hold my life with open hands. I fight and wrestle to make my dreams become a reality because I hold them within high regards. This is why I get anxious with my busyness and workload—I take it too seriously.
But I know that if I zoom out and view my dreams from a godly perspective, I'll see that my efforts and dreams aren't as important as I think they are.
This is a tough pill to swallow. So much of us wants to resist this truth. Yet, in our core, we know this is true: our life isn't as important as we think it is.
In the Bible, James writes that our life is a vapor. It is here and then it is gone. Some people perceive that as reason to party. Others want to make their vapor mean something.
But how we should really interpret this truth is that we shouldn't take matters of our life too seriously—not as if they are matters of utmost importance.
While we want to reject this truth, there is so much freedom behind this idea.
This means we don't have to obsess about certain parts of our life. This means we have the freedom to mess up or take another road. This means we can hit the do-over button or get started with something new, something exciting.
The key to living with meaning and joy is to live with open hands.
This idea of living life with open hands has been saving me recently. There are times when I take article writing and matters of my business too seriously—as if I have to get them done no matter what.
Yet, God doesn't call us to elevate these matters to such importance. Instead, He calls us to trust His activity in these matters. He calls us to open our hands and relinquish our control so that we can live meaningfully and He can be glorified in our activities.
Living with open hands is a foreign concept to our culture. In our culture, we constantly teach people to not give up on their dreams no matter what. We teach others to take matters of their life seriously if they want to excel at them.
But what I've found is, this is only partially true.
It's good taking your life seriously, but it's not healthy to obsess about it. And in our culture where we're obsessed with controlling our outcomes, this is a difficult idea to practically live out.
Somewhere in your aspirations, there has to be a line drawn, a point where we are willing to open our hands and let grace, kindness, forgiveness, and humility surge in.
If that line is not present in your work, build it in.
Remember: too much of anything is a bad thing. Too much concern about the matters of your life is a bad thing when it chokes out the passion to your life. Instead, learn to release things to God. Develop a godly perspective and live a more meaningful life.
Superb article, I love this idea of opening our hands and relinquishing control. At the end of the day all is dust and striving after wind, that which makes life possible is the all important grace. All to often we get lost in our pursuits and forget to acknowledge this miracle of existence. Thanks for the reminder.


text:
Bigodini - Il gioco che non fa una piega è stato un programma televisivo italiano condotto da Max Novaresi e la partecipazione del duo Gretel & Gretel (Francesca Zanni e Michela Andreozzi); è stato trasmesso su Italia 1 dal 3 aprile al 30 dicembre 2000 dalle ore 13:00 alle 13:45, venendo promosso nell'orario dell'access prime time (dalle 20:00 alle 20:40) nel periodo estivo, dal 10 al 29 luglio, in sostituzione di Sarabanda. Si trattava di un quiz a premi, con domande perlopiù basate sui pettegolezzi dei VIP tratte da famose riviste di cronaca rosa, quali Novella 2000, Gente, Chi, ed altre.

Storia
Il programma nacque ed andò in onda solo nel 2000; la prima puntata, trasmessa il 3 aprile ebbe un ascolto di 1.311.000 spettatori ed uno share del 7,52%. Tuttavia durò pochissimo, e dopo la chiusura della prima stagione il 29 luglio, riprese il 4 settembre con una nuova edizione che terminò il 30 dicembre. Vennero in tutto mandate in onda circa 200 puntate.

Il programma
Lo studio era mostrato come un salone da parrucchiere. Il quiz si svolgeva con quattro concorrenti ed era suddiviso in tre manches:

 Prima manche: i quattro concorrenti dovevano rispondere a delle domande basate sul gossip con tre opzioni di risposta. Ad ogni risposta esatta, si guadagnava un punto. Alla fine della manche, il concorrente col punteggio più basso veniva eliminato.
 Senti chi sparla: i tre concorrenti (seduti e muniti di pulsantiera), guardavano due scenette mostrate sullo schermo in cui Gretel & Gretel stavano sparlando di un/una VIP, ed essi dovevano indovinare chi era. I due che rispondevano correttamente superavano la prova, mentre l'altro/a veniva eliminato/a.
 A colpi di phon: è l'ultima manche di gioco, nella quale i due concorrenti dovevano rispondere a otto domande con tre opzioni di risposta, cui seguiva la verifica per stabilire colui/lei che veniva eliminato/a per aver risposto di meno. A questo punto l'unico/a concorrente rimasto/a, per cercare di conquistare il montepremi della puntata, poteva correggere per 30 secondi le risposte alle otto domande: per farlo doveva attivare prima il pulsante, e poi cominciare velocemente a cambiarle toccando lo schermo con la mano. Dopo ogni prima attivazione ne poteva correggere però solo tre alla volta, affinché egli/ella non ripremesse in tempo il pulsante (per bloccare il tempo), per poi ritentare. Infine, sempre l'unico/a concorrente rimasto/a, vincendo o non vincendo il montepremi poteva prendere parte alla puntata successiva.

Nella seconda edizione era stato introdotto un nuovo gioco, quello del Tabellone, in cui i concorrenti potevano scegliere appunto dal tabellone un tema su cui rispondere.

Il montepremi partiva dalla cifra di 3 milioni di lire a salire.

Curiosità
 Ad ogni eliminazione di un concorrente, veniva messa in sottofondo la canzone E salutala per me di Raffaella Carrà.

Note

Programmi televisivi degli anni 2000

redpajama_set_name: RedPajamaWikipedia