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You are here: Home / Cancer Champions / Ronald Reagan
Cancer Type – Colon
Bio – Ronald Reagan – We followed him from Hollywood Star to California Governor to President, but did you know he had skin and colon cancer?
Short Story – He served as the 40th President of the United States from 1981-1989, after a career in Hollywood. During his time in the oval office he had a skin cancer lump removed from his nose (1987), which returned six months later. It was removed again, and all of his cancer was gone.
What most don't realize about Ronald Reagan, is during his presidency he battled colon cancer. A routine screening found a small polyp in 1984, however this was ruled benign. In March of 1985 another polyp. The polyp was removed in an endoscopic procedure and colonoscopy revealed a second, more invasive tumor known as villous adenoma.
Two feet of the right side of Reagan's colon was moved. The good news, it had not spread, however it had invaded the muscle of the colon but was confined to the bowel wall, something physicians refer to as "Duke's B."
The President chose not to undergo chemotherapy treatments, even though doctors warned cancer cells could spread to other organs. Instead he followed his wife Nancy's request to undergo laetrile treatments. For thirteen months, President Reagan received daily IV laetrile treatments in the Oval Office.
The conservative leader was a colon cancer survivor, although he lost his daughter Maureen to brain cancer in 2001. In his late years he suffered from Alzheimer's disease. Before his passing in 2004, he wrote an optimistic note , ""I now begin the journey that will lead me into the sunset of my life. I know that for America there will always be a bright dawn ahead."
Ronald Reagan Colon Cancer Update
Ronald Reagan dealt with skin cancer in 1982, which brought significant attention to skin cancer for the first time. The day after he had the spot removed, he had to go on television; when asked about the bandage, Ronald replied, "it is a billboard to stay out of the sun." he went on to explain the Mohs surgery was successful.
Shortly after the Reagan reply, Dr. Perry Robins, the founder of The Skin Cancer Foundation, went on television and explained the warning signs, early detection importance, and effective skin cancer treatments.
Ronald's beloved wife, First Lady Nancy Reagan, also had skin cancer and breast cancer. It was the only time Ronald Reagan worried about cancer was during Nancy's breast cancer treatment. She chose to have a modified radical mastectomy. She was a 29-year survivor as her breast cancer never returned.
In the end, Ronald Reagan passed away from Alzheimer's disease and not cancer; he was a cancer survivor.
The Ronald Reagan Scholarship Foundation awards scholarships to students each year who demonstrate consummate leadership, determination, honesty, and citizenship.
In office (January 20, 1981 – January 20, 1989)
Vice President George H. W. Bush
Preceded by Jimmy Carter
Succeeded by George H. W. Bush
33rd Governor of California
In office (January 2, 1967 – January 6, 1975)
Lieutenant Robert Finch, Edwin Reinecke, John L. Harmer
Preceded by Pat Brown
Succeeded by Jerry Brown
President of the Screen Actors Guild
In office November 16, 1959 – June 12, 1960
Preceded by Howard Keel
Succeeded by George Chandler
In office (November 17, 1947 – November 9, 1952)
Preceded by Robert Montgomery
Succeeded by Walter Pidgeon
Born Ronald Wilson Reagan, February 6, 1911, Tampico, Illinois, U.S.
Died June 5, 2004 (aged 93), Bel Air, California, U.S.
Cause of death Pneumonia complicated by Alzheimer's disease
Resting place Ronald Reagan Presidential Library and Center
Political party Republican
Other political affiliations Democratic (before 1962)
Spouse(s) Jane Wyman (m. 1940–49), Nancy Davis (m. 1952)
Relations Neil Reagan (brother)
Parents Jack Reagan, Nelle Wilson Reagan
Alma mater Eureka College
Profession Actor, politician, sports, commentator
Awards Presidential Medal of Freedom, Congressional Gold Medal
Allegiance United States
Service/branch U.S. Army Air Forces
Years of service 1937–1945
Rank Captain
Unit 18th AAF Base Unit
Image Credits – See page for author [Public domain], via Wikimedia Commons


text:
Ибиан ()  —  муниципалитет в Бразилии, входит в штат Санта-Катарина. Составная часть мезорегиона Запад штата Санта-Катарина. Входит в экономико-статистический  микрорегион Жоасаба. Население составляет 1873 человека на 2006 год. Занимает площадь 147,329 км². Плотность населения — 12,7 чел./км².

Статистика
 Валовой внутренний продукт на 2003 составляет 27.618.455,00 реалов (данные: Бразильский институт географии и статистики).
 Валовой внутренний продукт на душу населения на 2003 составляет 14.452,36 реалов (данные: Бразильский институт географии и статистики).
 Индекс развития человеческого потенциала на 2000 составляет 0,809 (данные: Программа развития ООН).

Муниципалитеты штата Санта-Катарина

redpajama_set_name: RedPajamaWikipedia


The Gold Florin of Charles I (1301-1342) UNC Au - Hungarian Mint Ltd.
MNB, the Central Bank of Hungary plans to issue a gold collector coin series entitled "Gold Florins of Medieval Hungary". The first piece of the series depicts the golden florin issued by Charles I., born of the House of Anjou in Naples and called Caroberto. With the termination of the House of Árpád, he was crowned King Charles I of Hungary in1301. Among his monetary reforms, he introduced the gold coinage in Hungary. His gold coins were patterned after the gold of Florence with the fleur de lis motif (fiorino d'oro, also source of the Hungarian word 'FORINT') and with the inscription of KAROLV REX (King Charles) on the front, and an image of the parton saint of Florence, Saint John the Baptist on the back. The front and back of this historical coin are based on the gold collector coin, designed by E.Tamás Soltra. The technical parameters of the collector coin are matching those of the historical piece, plus quadruple-weight piefort versions of it were issued in a very limited mintage.


The Crown has been making headlines this week. First it was reported that Claire Foy, who played Queen Elizabeth II, was to back paid €200k after it was revealed that The Crown actress, Claire Foy, who played Queen Elizabeth II, was paid less than her co-star, Matt Smith, who starred as her husband Prince Phillip, for her role in the hit show.
The second headline hitter is the news of who will be playing (everybody's favourite) Princess Margaret in the upcoming series three of the show.
We already knew that the third series will see the show entirely recast, with Olivia Colman taking over as Queen Elizabeth II and Tobias Menzies as Prince Philip.
The Evening Standard reported earlier in January that Helena Bonham Carter and Netflix were in talks and rumours have been swirling ever since.
Finally, we have had confirmation from Helena who said in a statement that she is "terrified" to be following in the footsteps of Vanessa Kirby, whose performance as the princess has earned her universal acclaim.
"I'm not sure which I'm more terrified about – doing justice to the real Princess Margaret or following in the shoes of Vanessa Kirby's Princess Margaret. The only thing I can guarantee is that I'll be shorter (than Vanessa)," the statement read.
Bonham Carter, 51, is known for her many film roles, including her Bafta-winning peformance in The King's Speech, as well as The Wings of the Dove, Hamlet and the Harry Potter films.
Jason Watkins has also been announced to play former Prime Minister Harold Wilson.
The first series covered the period 1947 to 1955; the second 1956 to 1963 and The Crown series three will land in 2019.


Atmospheric Science deals with the physical and chemical processes occurring in the envelope of air surrounding the Earth. Research in atmospheric science uses a wide array of methods to describe and understand the phenomenon observed in our atmosphere, including clouds, rain, weather, climate, severe weather, El Nino, etc. The data for analysis of such phenomenon are obtained from instruments on the surface, balloons and planes in the atmosphere, and satellites observing from space. Using super computers these data help develop mathematical models that are used for weather and climate forecasts. These observations and models allow us to study many different natural phenomenon, some at the forefront of public interest: climate change, global warming, the ozone hole, air pollution, weather forecasting, El Nino, severe weather, hurricanes, etc. In addition, applications are related to weather forecasting for aviation (military and civilian), public health, and early warning of natural hazards such as flash floods, hurricanes, tornadoes, hail storms, etc.
The climate is defined as the long term statistics of the atmosphere and weather. For example: temperature changes, rainfall amount, and the frequency of storms. In our department we perform analysis on climatic data sets to study the changes that have occurred over the last decades. We also run climate models to try to understand how the climate may change in the future under the influence of global warming.
Meteorology, or the study of the weather, deals with the faster changes in the atmosphere on time scales of minutes to days. In our department we study the physics behind weather predictions, and run models that can predict the weather. Meteorology can be divided into three branches: physical, synoptic and dynamic meteorology.
This branch deals with the structure and composition of the atmosphere, radiative transfer, acoustic waves in the atmosphere, aerosols, cloud and rain formation, atmospheric electricity (lightning), weather modification (cloud seeding), and observational methods like radar, satellites and lidar.
This branch deals with weather forecasting. Describing, analysing and forecasting the motions in the atmosphere on the large scale, using observations and models. Surface and satellite data are merged with numerical models to help improve weather forecasting.
This branch deals with the theoretical description of the atmospheric flows (winds) and their development over time. A system of fundamental equations describing the motion of the atmosphere is used to describe the flow of air in the atmosphere on all spatial and temporal scales. These equations are the building block of the models used to predict the weather and climate in the future. Good weather forecasts are a result of merging synoptic and dynamic meteorology.
At altitudes above 20 km the physics of the atmosphere changes. In the lower atmosphere (troposphere) thermodynamics is the key driver of the processes of weather and climate, while above 20km (stratosphere, mesosphere, thermosphere) the Sun is the main driver of the atmosphere. The ozone layer is found around 20-30km altitude, and is important for blocking out harmful UV radiation from the Sun. Above 70km the gases are ionized by the Sun's radiation, forming a plasma layer we call the ionosphere. This is important for the propagation of radio waves around the globe for communications. Above about 100km we enter the realm of space.
Weather forecasting including advanced atmospheric models and developing of numerical techniques. Investigation of extreme Weather events - their causes and effects using both observations and models. Climate changes and their impacts on the Mediterranean employing observations and various climate models. Investigation of different atmospheric aerosols employing observations along with global & regional models. Regional NASA center for the Earth observations. Study of the Dead-Sea climate.
Research in my group centers on developing an understanding of the main atmospheric wind and temperature patterns, their natural variations and response to different forms of forcing, on daily, seasonal, annual and longer time scales.
Main current projects: The contribution of the atmospheric circulation to extreme weather events; The maintenance and variability of different global circulation regimes; Stratospheric planetary waves, their interaction with ozone, and the downward coupling to the troposphere; The role of upper level outflow in hurricane evolution.
Atmosphere Electricity- lightning, thunderstorms, sprites, meteors. Electromagnetic waves in the atmosphere in the frequency ranges of ULF, ELF and VLF. Global and regional climate change, global warming, El Nino, atmospheric chemistry, biomass burning, earthquakes, Big Data.
Modeling and forecast of Saharan dust transport over the Mediterranean, the Middle East, Europe, and over the Atlantic Ocean. Modeling of space-time distribution of marine aerosols over the Mediterranean Sea and their effects on the Eastern Mediterranean weather and climate. Global distribution and trends of aerosols using satellite data and ground-based measurements. Study of the Dead Sea aerosols and their effects on atmospheric radiation and Dead Sea water temperature.
Observation and monitoring of Space Weather in Hermon mountain at the Cosmic Ray and Space Weather Emilio Segre Observatory. Influence of Cosmic Ray from the solar flares and from Supernova explosions on the Earth environment (magnetosphere, ionosphere, high atmosphere).


Water refill points will available in Plymouth - and very other every major city and town in England - by 2021, to help reduce the amount of plastic waste we produce.
Industry body Water UK says the scheme, which will see free water stations in shops, cafes and businesses, could cut disposable plastic bottle use by tens of millions a year.
Whitbread, which owns Costa Coffee and Premier Inn, is the first major business to sign up to the initiative and will provide water in all of its branches from March 2018, reports the BBC.
The initiative follows a a previous scheme, the Refill campaign, which has established more than 1,600 refill stations across the UK and operates in 13 towns and cities in England.


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Quit any apps that you are not using!
Please do not install software that claims to speed up your computer, we do not recommend any of these types of applications. In our experience these applications cause more problems than they solve and can compromise your Mac and your privacy.
If you need help with any of the above please let us know how we can help speed up your Macs at your home or business.


If you want to make sure that your customers, employees or houseguests are comfortable, then you need Kimberly-Clark Professional – KLEENEX® Hard Roll Towels, 8 x 600ft, 1 3/4" Core dia, White - 6 Rolls/Carton. Foodservice businesses like restaurants, diners and catering halls love filling its bathrooms with bulk paper towels because it allows customers to dry their hands. Likewise, this is an excellent choice for offices, law firms, hospitals and homes. And since there are so many paper towels in one box, you're guaranteed to receive enough to satisfy an entire group of people. The bulk paper towels arrive in a solid cardboard box that protects the rolls and keep them in good shape. This durable cardboard exterior makes it easy to stack the bulk paper towels in storerooms or on shelves.
What's so Good about Kimberly-Clark Professional - KLEENEX Hard Roll Towels?
Kleenex makes these bulk paper towels so that they're highly absorbent. Not only will this absorbency help you dry hands fast, but it also ensures that you'll use fewer sheets, which can save money. Additionally, this absorbency enables you to clean the mess with minimal effort. When someone spills water, juice or other liquid on the floor, you can wipe it up fast. Similarly, these paper towels can help you remove stains from plates and pans or remove dust from furniture. If your bookshelves, ceiling fan or picture frames are collecting dust, try using a moist paper towel to make them look spotless. Many customers find that these sheets are great at cleaning bathrooms, cars, dishes and more.
Where Does Kimberly-Clark Professional - KLEENEX Hard Roll Towels Originate?
These paper towels come from the Kimberly-Clark Corporation, which is an American corporation that specializes in producing personal care products. However, despite being passionate about delivering high-quality items, the company is also driven to improve society. It implements business practices that protect the environment and contributes to local communities.
Although bulk paper towels help you please customers, you can use them in several other ways, too. For example, they're great for preventing mess when you're painting. If you're painting your home and don't want to stain your tile or hardwood floor, try spreading a few sheets from your bulk paper towels on the floor. Also, art teachers enjoy using paper towels to show children how to produce creative art. Creating paper mache is a fantastic art project that kids adore. Instead of using construction paper or newspapers to make the paper mache, you can use paper towels. Likewise, pupils can practice their painting skills on paper towels. And when they're complete, you can hang the bright, colorful sheets all along the walls as a homemade decoration. These brown paper towels can even help you create holiday-specific artwork, including snowflakes during the winter and spooky masks during Halloween. And if they're innovative enough, kids may be able to form the sheets into crowns, hats and other accessories.


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SSE5 instruction set coming in 2009
Written by Phil Cogar
Tags: #bulldozer #code #instruction-set
Companies: #amd
Yesterday, AMD announced its plans to help improve the performance of applications on the x86 architecture with the introduction of SSE5 instructions.
SSE5 is designed to help software developers to simplify code and improve efficiency by giving them even more capabilities to maximize the performance of applications. The instruction set is currently being made available but will not debut in a chipset until 2009 when AMD releases its "Bulldozer" core for the Fusion line of processors.
The key components of SSE5 include:
3-Operand Instructions - A computing instruction is executed by applying a mathematical or logical function to operands, or inputs. By increasing the number of operands an x86 instruction can handle from 2 to 3, SSE5 enables the consolidation of multiple, simple instructions into a single, more effective instruction. The ability to execute 3-Operand Instructions is currently only possible on certain RISC architectures.
Fused Multiply Accumulate - The 3-Operand Instruction capability enables the creation of new instructions which efficiently execute complex calculations. The Fused Multiply Accumulate instruction combines multiplication and addition to enable iterative calculations with one instruction. The simplification of the code enables rapid execution for more realistic graphics shading, rapid photographic rendering, spatialized audio, complex vector mathematics and other performance-intense applications.
For those of you who are interested, you can get a copy of the SSE5 specification over at AMD's developer site. Go ahead and take a peak and let us know what you think about it in the comments section below or over in the forums.
AMD promises completely different CPU architecture
AMD's Technical Director of Sales and Marketing has said that its next-gen CPU architecture will be "completely different" from Phenom.
Analyst: AMD gains some market share
iSuppli Corp. has estimated that AMD has actually gained some market share during Q2 07 based on chip revenue figures.
September 13, 2007 | 08:24
AMD, IBM develop 45nm high-k metal gate transistor
Intel isn't the only one to sport high-k metal gate transistors: AMD and IBM have joined forces to develop their own high-k 45nm process.
July 30, 2007 | 13:55

redpajama_set_name: RedPajamaCommonCrawl


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Lynchburg & Roanoke Workers Compensation Lawyer Jaleh Slominski Named a "Go-To Lawyer" for Workers Comp by Virginia Lawyers Weekly
Aug 11, 2021 | Business
Virginia Lawyers Weekly, one of the best sources of legal information for practicing attorneys in the state, recently named Lynchburg & Roanoke Jaleh Slominski from Slominski Law as a "Go-To Lawyer" for Workers' Compensation Law.
LYNCHBURG, VA, August 11, 2021 /24-7PressRelease/ — Virginia Lawyers Weekly, one of the best sources of legal information for practicing attorneys in the state, recently named Lynchburg & Roanoke Jaleh Slominski from Slominski Law as a "Go-To Lawyer" for Workers' Compensation Law.
Jaleh Slominski, was selected because of her deep experience, track record of satisfied customers and successful cases.
The accolade is given to a small group of top lawyers who are well-versed in the nuances of workers' compensation case law. The selection process also looked for lawyers able to think creatively to ensure the best possible outcomes for their clients. These attorneys also come highly recommended by their peers because of their accomplishments and expertise in workers' compensation law.
About Slominski Law
Jaleh K. Slominski, Esq, one of the 21 lawyers recognized by the publication, has been practicing law for more than 25 years. She earned her bachelor's degree from the University of Kentucky and her law degree from Duquesne University in Pittsburgh, Pennsylvania.
With her years of experience in workers' compensation cases, she has been providing workers throughout Virginia with affordable and effective representation. She has made it her priority to use her extensive and diverse experience to handle each client's unique litigation needs to optimize benefits for her clients.
Slominski Law is specifically designed to be a small firm to ensure the ability to offer very high-quality legal representation combined with personalized attention. This model ensures top notch service every step of the way along with top results.
Slominski Law focuses it's practice primarily on cases involving workers' compensation injury claims but also handles regular personal injury and wrongful death claims following an auto accidents, as well as lemon law claims. It is located in Lynchburg and Roanoke, and proudly serves the entire western half of the state of Virginia.
For more information, you can contact Slominski law by calling (434) 384-9400 (Lynchburg) or (540) 554-3762 (Roanoke) or visit https://www.slominskilaw.com.
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Studies have proven that children who are enrolled in after-school programs tend to perform better both academically and socially.
Kids can make their own lightning indoors with a few simple materials.
Summer camps are a great place where children learn a lot of things, develop new friendships, and explore their creativity.
Not counting school sponsored extra-curricular activities and other weekend functions, most children spend over 180 days in school each year. Academic summer camps offer kids a different environment in which to expand their knowledge in a particular subject and improve their social skills.
For most parents who are sending their child to a summer camp for the first time, finding one that is best suited for their child is of utmost importance. Here are a few things to consider when looking for the right summer camp for your child.


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Monty M. Alger
Director, Institute for Natural Gas Research and Professor of Chemical Engineering
Monty Alger is a professor of chemical engineering at the Pennsylvania State Univ., where he is Director, Institute for Natural Gas Research. His more than 30 years of experience in the chemicals and energy industries includes positions as Vice President and Chief Technology Officer at Air Products and Chemicals, Inc., and as Senior Vice President of Research at Myriant. He spent 23 years at General Electric (GE), where he led technology development at the Global Research Center of GE Plastics, and was the General Manager of Technology for the Advanced Materials business.
Prior to GE, he was Director of the MIT Chemical Engineering Practice School Station at GE Plastics. He is a member of the National Academy of Engineering, and serves on advisory boards for organizations including the Shenhua National Institute of Clean and Low Carbon Energy and PTTGC (Thailand). He is a former member of the Council on Competitiveness Technology Leadership Strategy Initiative.
He earned SB and SM degrees from MIT and a PhD from the Univ. of Illinois at Urbana-Champaign, all in chemical engineering. He served on AIChE's Board of Directors (2010–2012), the Finance Committee, and worked on investment priorities to ensure alignment with member value.
Connect on AIChE Engage
ChEnected contributions
President's Message: A Test of Resilience
Dec 22, 2020 . by Monty M. Alger
Monty Alger reflects on the challenges and accomplishments of chemical engineers in 2020, and looks ahead to the demands and opportunities in 2021.
From the AIChE President: Career Discovery and AIChE's Institute for Learning and Innovation
Jun 23, 2020 . by Monty M. Alger
AIChE is redoubling its dedication to the success of chemical engineers everywhere with the launch of our new Institute for Learning and Innovation.
2020 AIChE President's Message
Jan 17, 2020 . by Monty M. Alger
President Monty M. Alger shares thoughts and plans for AIChE's future, and asks for your input on ways AIChE can do even more.


Cannabissearch.com Buzz Rating: While this strain is likely to seem weak in potency at first, the full effect will come on strong, after about 15 or 20 minutes. This is a true creeper strain with great potency, once it kicks in.
Unfortunately, the breeder has not chosen to disclose the exact genetics of Barrier Reefer, which is pretty understandable. While not a connoisseur-grade strain, this bud is worth safeguarding against impostors. Too many notoriously famous strains have been attempted to be recreated by sub-par breeders. One such strain is Blueberry, which has at least 8 different breeders claiming to have created it. Koala Seeds obviously did not want this to happen to Barrier Reefer and so they have not released the genetics. Just like the Barrier Reef in Australia, Barrier Reefer is large and chunky. The buds have medium density and a good layering of trichomes. The bright green color and minimal pistils makes for quite an interesting sight.
It's best to vegetate these ladies around 6 to 8 weeks, before inducing flowering. They will usually only double in height at the end of flowering, so vegetating this long is not a problem. The yield is very heavy, especially for an indica-dominant strain and it has a good resistance to most diseases, but few pests. Spider mites and whiteflies have shown to reek havoc on Barrier Reefer, which is why it is recommended to only cultivate these ladies indoors, unless extreme measures are taken against pests.
Barrier Reefer 0 out of 5 based on 0 ratings. 0 user reviews.


text:
\section{Introduction.}
\label{sect:intro}

Recent discoveries of distant planetary companions (separations of
at least tens of AU) around several nearby stars by direct imaging
(Marois et al.\ \rm 2008; Thalmann et al.\ \rm 2009)
have stimulated investigation of their formation mechanisms.
Two contending theories of the planet formation --- the core instability
(Perri \& Cameron 1974; Harris 1978; Mizuno 1980; hereafter CI)
and the gravitational
instability (Cameron 1978; Boss 1998) --- differ quite dramatically in
their ability to form planets at various distances from the star.

It is generally thought that CI should be capable of
forming giant planets close to the star, at separations
$\lesssim 10$ AU. Both theoretical modelling (Mizuno 1980;
Stevenson 1982; Pollack et al.\ \rm 1996) and the existence
of two gas giants at semi-major axes of $5.2$ AU and $9.5$ AU in
our own Solar System (coupled with some knowledge about the
evolution of the planetary orbital architecture in the early Solar
System) attest to this statement. At the same time it seems
unlikely that giant planets can be formed by the gravitational
instability in the inner parts of protoplanetary disks because
of the long local cooling time of gas (Matzner \& Levin 2005;
Rafikov 2005, 2007). On the contrary, giant planet formation by
the direct gravitational instability appears feasible at large
distances from the star (Boley 2009; Rafikov 2009; Clarke 2009;
but see Boss (2006) for an opposite view) where the disk cooling
time is short enough to permit fragmentation of the gravitationally
unstable disk (Gammie 2001).
Whether the CI can operate beyond several tens of AU
from the star is not so clear.

The well known problem faced by the CI far from the star is that
the buildup of massive refractory core (necessary for triggering
the vigorous gas accretion) is thought to take very long time,
longer than the $1-10$ Myr lifetime of the gaseous component of
protoplanetary nebula. Studies illustrating this problem are
usually based on two key assumptions (Dodson-Robinson et al.\ \rm 2009):
first, that the CI is triggered whenever a
$M_{crit}\sim 10$ M$_\oplus$ core is built by planetesimal
accretion, and, second, that the planetesimals accrete onto the growing
core at rather modest rates (Ida \& Lin 2004;
we will explain later what do we mean
by that). When these conditions are imposed it is generally
found that the CI does not commence at $a\gtrsim 20-30$ AU prior
to the nebular gas removal.

In this note we show that aforementioned assumptions are too restrictive and
neither of them need to be adopted in determining the feasibility
of the CI. First, planetesimal accretion by the growing core can
(at least potentially) be much faster than what has been previously
assumed. Second, the critical core mass itself strongly depends
on the planetesimal accretion rate. Based on these observations we
formulate in this work a novel constraint on the operation of the
CI in protoplanetary disks.

\section{Planetesimal accretion.}
\label{sect:accr}

Onset of the CI is intimately related to the accretion of
solid material by the protoplanetary core, as discussed in
more detail in \S \ref{sect:atm}. For this reason we start
by reviewing the process of planetesimal accretion by growing cores.

It is well known that the rate at which a core accretes
planetesimals is a strong function of planetesimal random
velocities (Dones \& Tremaine 1993; hereafter DT93).
An important parameter for determining the dynamical state of
planetesimal population is the ratio $p=R_c/R_H$ of the core
radius $R_c$ to the Hill radius $R_H\equiv a(M_c/M_\star)^{1/3}$,
where $a$ and $M_c$ are the semi-major axis and the mass of
the core, and $M_\star$ is the mass of the star.
At large separations from the star
\begin{eqnarray}
p=\left(\frac{3M_\star}{4\pi \rho_c a^3}\right)^{1/3}
\approx 5\times 10^{-4}a_{10}^{-1}
M_{\star,1}^{1/3}\rho_{c,1}^{-1/3}
\label{eq:p}
\end{eqnarray}
is much less than unity, where $\rho_c$ is the core density
and we define $a_{10}\equiv a/(10~\mbox{AU})$,
$M_{\star,1}\equiv M_\star/M_\odot$, $\rho_{c,1}=\rho/1$ g cm$^{-3}$.

DT93 have demonstrated that whenever $p\ll 1$
there are four possible regimes of planetesimal accretion.
\begin{itemize}
\item When the planetesimal random velocity dispersion\footnote{For
simplicity here we do not differentiate between the horizontal
and vertical velocity dispersion.}
$\sigma\gtrsim \Omega R_H p^{-1/2}$ ($\Omega=\sqrt{GM_\star/a^3}$
is the angular frequency at the core's location) -- a regime
that we call {\it very high dispersion} -- accretion
is very slow because gravitational focussing is inefficient and
the collision cross-section is given by the geometric
cross-section of the core.
\item When
$\Omega R_H\lesssim \sigma\lesssim \Omega R_H p^{-1/2}$
({\it high dispersion} in the notation of DT93) the
relative planetesimals velocity with respect to the core
$v_{rel}\sim\sigma$. Gravitational focussing is
important and planetesimal accretion rate $\dot M$ grows as
$\sigma$ goes down.
\item When
$\Omega R_H p^{1/2}\lesssim \sigma\lesssim \Omega R_H$
({\it intermediate dispersion})
one finds that $v_{rel}$ is no longer determined by the random
motions of planetesimals but is rather set by the shear in
the differentially rotating disk, so that $v_{rel}\sim\Omega R_H$.
Gravitational focussing saturates at a constant value whenever
$\sigma\lesssim \Omega R_H$ and $\dot M$ keeps increasing as $\sigma$
decreases simply because the thickness of the disk goes down increasing
local volume density of accreting bodies.
\item Finally, when
$\sigma\lesssim\Omega R_H p^{1/2}$ ({\it very low dispersion})
disk becomes so thin that a core can accrete the whole vertical
column of material doomed for collision with it. Given that
focussing is at its maximum this situation corresponds to the
most efficient accretion regime.
\end{itemize}

The rate of accretion in
the very low dispersion case is (DT93)
\begin{eqnarray}
&& \dot M=\dot M_{max}\approx (6.47\pm 0.02)\Omega p^{1/2}
\Sigma_s R_H^2,
\label{eq:max_rate}\\
&& \approx 5.1~G^{1/2}M_c^{2/3}\Sigma_s\rho_c^{-1/6},
\end{eqnarray}
where $\Sigma_s$ is the surface density of solids. In this study we
parametrize $\Sigma_s$ as
\begin{eqnarray}
\Sigma_s=1\Sigma_1 a_{10}^{-\alpha}\,\mbox{g cm}^{-2},
\label{eq:Sig_s}
\end{eqnarray}
where $\Sigma_1=\Sigma_s/(1$ g cm$^{-2})$ and $\alpha$ is a constant
(minimum mass Solar nebula (MMSN)
corresponds to $\alpha\approx 3/2$ and $\Sigma_1\approx 1$).
Note that $\dot M_{max}$ is independent of $M_\star$.

For comparison, it is often assumed that protoplanetary cores
predominantly accrete planetesimals with the dispersion of
random velocities $\sigma=\Omega R_H$, which corresponds to the
{\it transition} between the so called shear- and dispersion-dominated
dynamical regimes (e.g. Dodson-Robinson 2009). This is what we
called a ``modest'' accretion rate before. At this transition
DT93 find
\begin{eqnarray}
\dot M_{tr}\approx (6-10)p\Sigma_s\Omega R_H^2,
\label{eq:M_tr}
\end{eqnarray}
and one immediately sees that $\dot M_{tr}\ll\dot M_{max}$ since
$p\ll 1$.

Previous studies of the CI have been limited to considering
planetesimal accretion only in the high-velocity regime
($\sigma\gtrsim \Omega R_H$) or at the boundary with the
intermediate velocity regime (i.e. for $\sigma\approx\Omega R_H$).
While large (tens to hundreds km in
size) planetesimals indeed likely accrete in these dynamical
regimes at rather slow rates it is quite possible that most
of the mass growth of the core is not due to accretion of these
large bodies. According to Rafikov (2004a) and Goldreich et al.\ \rm
(2004), as the core becomes massive enough it dynamically
excites large planetesimals in its vicinity since damping agents
such as gas drag are not effective for massive bodies.
This leads to fragmentation (rather than growth) of planetesimals
when they collide with each other. Resulting
fragmentation cascade converts significant fraction of the
solid mass into small mass debris. These fragments finally reach
small enough sizes that they are dynamically ``cooled'' by gas
drag to low velocities, which makes it possible for them to be accreted
at very high rate compatible with
$\dot M_{max}$ (Rafikov 2004a,b) in the very low dispersion regime.
Recent coagulation calculations
done by Kenyon \& Bromley (2009) confirm this general picture,
lending support to the possibility of maximally efficient accretion
in the very low dispersion regime.

In this work we are interested in obtaining a robust limit on
the operation of core instability, which means that we need
to consider all possible modes of planetesimal accretion and
actually {\it determine} the accretion regime facilitating the
onset of the CI the most (instead of selecting it ad hoc). It
is not obvious a priori which accretion regime is best for
achieving the CI within the limited amount of time: on one
hand high $\dot M$ allows to build the core of a given mass
faster, but on the other hand high $\dot M$ also implies
larger critical mass (see \S \ref{sect:mcrit}), which takes
longer to build.

We will discover in \S \ref{sect:mcrit} that fastest route
to the CI lies through the most efficient accretion at the rate
$\sim \dot M_{max}$ and for this reason we parametrize
planetesimal accretion rate as
\begin{eqnarray}
\dot M &=&\chi\dot M_{max}\approx 6.47\chi p^{1/2}
\Sigma_s\Omega a^2\left(\frac{M_c}{M_\star}\right)^{2/3},
\label{eq:mdot}\\
&\approx & 2\times 10^{-5}\mbox{M}_\oplus~\mbox{yr}^{-1}\chi
a_{10}^{-\alpha}\Sigma_1\rho_{c,1}^{-1/6}M_{c,10}^{2/3},\nonumber
\end{eqnarray}
($M_{c,10}\equiv M_c/10$ M$_\oplus$) where the parameter
$\chi$ accounts for the deviation of $\dot M$ from
$\dot M_{max}$. In general $\chi$ may be a function of $M_c$
and $a$, and it is expected that $\chi\lesssim 1$
(although in some situations $\chi\gtrsim 1$ may possible,
see \S \ref{sect:disc}). But we will show in \S \ref{sect:mcrit}
that slow accretion (i.e. $\chi\ll 1$) results in
smaller distance from the star at which planets could form
by the CI than in the case $\chi\sim 1$. It should also be
emphasized that our concentration on the highest $\dot M$
accretion regime is not only for the sake of the argument --
as mentioned before this regime may actually occur quite
naturally in the course of the core buildup.

Accretion luminosity corresponding to the $\dot M$ given
by equation (\ref{eq:mdot}) is
\begin{eqnarray}
L &=& \frac{GM_c\dot M}{R_c}\approx 8.2\chi\Sigma_s G^{3/2} M_c^{4/3}
\rho_c^{1/6}
\label{eq:lumin}\\
& \approx & 8\times 10^{-6}\mbox{L}_\odot ~\chi
a_{10}^{-\alpha}\Sigma_1\rho_{c,1}^{1/6}M_{c,10}^{4/3}.\nonumber
\end{eqnarray}

According to equation (\ref{eq:mdot}) accretion at the
rate $\dot M_{max}$ results in the core radius growing linearly
with time\footnote{One can easily see that accretion at the rate
$\dot M_{tr}$ (see Eq. [\ref{eq:M_tr}]) also leads to
$M_c^{1/3}\propto \tau$, although the
coefficient of this relation is considerably smaller than in
the case of $\dot M=\dot M_{max}$ and has different scaling with
$a$.}. Time needed for the core mass to reach a predetermined
value $M_c$ by accretion at the rate $\chi\dot M_{max}$ is
[see equation (\ref{eq:max_rate})]
\begin{eqnarray}
\tau(M_c)\approx 0.6\frac{\rho_c^{1/6}M_c^{1/3}}{\chi\Sigma_s G^{1/2}}
\approx 0.3~\mbox{Myr}~a_{10}^\alpha
\frac{\rho_{c,1}^{1/6}M_{c,10}^{1/3}}{\chi\Sigma_1}.
\label{eq:timescale}
\end{eqnarray}
It follows from this formula that a core can grow to 10 M$_\oplus$
in 3 Myr by accretion at the rate $\dot M_{max}$ even at $\sim 50$
AU in the MMSN. Previous results which did not account for the
factors leading to the fast accretion (planetesimal fragmentation
and gas drag on debris) predict much smaller distance
($\lesssim 20$ AU) at which such core would be able to form on
that time scale.

Having said all that, in the next section we show that the
formation of the 10 M$_\oplus$ core is not a prerequisite
for a CI and that the $M_{crit}$ is a function of accretion
rate.

\section{Protoplanetary atmosphere.}
\label{sect:atm}

Numerical calculations (Mizuno 1980; Pollack et al.\ \rm 1996;
Ikoma et al.\ \rm 2000) and analytical theory (Stevenson 1982) show
that CI commences when the core mass $M_{crit}$ is
so large that the mass of its atmosphere is comparable
to the core mass itself,
\begin{eqnarray}
M_{atm}(M_{crit})=\eta M_{crit},
\label{eq:instab_cond}
\end{eqnarray}
i.e. when the gaseous component of the protoplanet becomes
self-gravitating. Here $\eta$ is a parameter of order unity;
its exact value may depend on the accretion history of the core
(Ikoma et al.\ \rm 2000). Condition (\ref{eq:instab_cond}) should be
regarded as an equation for $M_{crit}$ which can be easily solved
as long as the dependence of $M_{atm}$ on $M_c$ and other
parameters of the problem is specified. Thus, calculation of
$M_{crit}$ involves understanding properties of planetary
atmospheres and calculation of $M_{atm}$ in particular.

It has been known since the works of Mizuno (1980) and
Stevenson (1982) that the critical core mass is a rather
weak function of the density $\rho_0$ and temperature $T_0$
of the surrounding nebula (see Rafikov [2006] and \S
\ref{subsect:opacity} for the discussion of the conditions
under which this behavior is expected). This has led to the
wide acceptance
of the CI idea since the interiors of the Solar System giant
planets are believed to harbor $\sim 10$ M$_\oplus$ cores
despite their different separations from the Sun.
Unfortunately, this observation
of the $M_{crit}$ invariance has shadowed the fact first
noticed by Stevenson (1982) that $M_{crit}$ is a strong function
of the planetesimal accretion luminosity $L$, or, equivalently,
planetesimal accretion rate $\dot M$. This fact has been
subsequently confirmed both numerically (Ikoma et al.\ \rm 2000;
Hori \& Ikoma 2010) and analytically (Rafikov 2006).

Stevenson's analytical (1982) results strictly apply only to cores
possessing radiative atmospheres with constant opacity.
Rafikov (2006; hereafter R06) has studied more general types of
atmospheres and has shown that they segregate into two
classes depending on whether $L$ is higher or lower than some
critical luminosity $L_{cr}$.
Whenever $L\gtrsim L_{cr}$ the intense energy release near the
core surface makes protoplanetary atmosphere and the
nebular gas in the Hill sphere of the core convectively
unstable. Such atmosphere have high entropy and rather
low mass relative to the mass of the core. In the opposite
case, when $L\lesssim L_{cr}$, protoplanetary atmosphere is
separated from the nebular gas by a roughly isothermal shell
of gas in which energy is transported radiatively and gas
entropy decreases from the nebular value to a much smaller
value characteristic for the atmosphere. Density in this
shell increases roughly exponentially towards the planet
which makes $M_{atm}$ much higher (for a given core
mass) than in the high-luminosity case.

It was demonstrated in R06 that far from the star
protoplanetary atmospheres are virtually always characterized
by $L\ll L_{cr}$, even if planetesimal accretion proceeds
at the maximally efficient rate $\dot M_{max}$. For that
reason we will consider only the low luminosity atmospheres in
this work.
The total atmospheric mass $M_{atm}$ in the low luminosity case
was computed in R06 by making simplifying assumptions about
the conditions in the deep layers of the planetary atmosphere:
either a polytropic model with constant polytropic index
(mimicking the fully convective interior) or a fully radiative
atmosphere with a simple power-law parametrization of the
opacity dependence on gas pressure and temperature.
Under these assumptions a significant fraction of the atmospheric
mass resides in the outer\footnote{This is different from the case
of an atmosphere with constant opacity considered by Stevenson
(1982).}
layer, near the inner boundary of the roughly isothermal external
radiative zone, in agreement with calculations by Mizuno (1980).
It can then be shown that (R06)
\begin{eqnarray}
M_{atm}\approx \zeta\left(\frac{G M_c\mu}{k}\right)^4
\frac{\sigma}{\kappa_0 L},
\label{eq:Matm}
\end{eqnarray}
where $k_B$ is the Boltzmann constant and $\zeta$ is a weakly
varying factor, which can be computed
exactly for a given density distribution inside the atmosphere.
This formula shows how $M_{atm}$ depends on important parameters
like $M_c$ and $\kappa_0$ and on accretion history, to which
$L$ is sensitive.

However it is not obvious that the simplifying assumptions
about the atmospheric properties
used in deriving formula (\ref{eq:Matm}) are justified given
the complexity of the physical effects encountered deep in
the atmosphere: self-gravity gradually becomes
important as $M_c$ approaches $M_{crit}$, grain sublimation changes
opacity in a non-trivial fashion, the equation of state
may be varying with depth because of molecular dissociation,
an atmosphere may have both radiative and convective regions at
the same time. These complications likely do not affect the
qualitative conclusions reached in R06 but should be important
for quantitative comparisons.

On the other hand, numerical calculations of the CI
which include the aforementioned physical effects (Ikoma et al.\ \rm 2000)
are typically
limited to exploring the dependence of $M_{crit}$ on only a
limited set of input parameters such as $\kappa_0$ and $\dot M$.
The latter is usually taken to be constant through the
calculation and this assumption significantly limits the direct
applicability of these numerical results to realistic situations
(since typically $\dot M$ increases as $M_c$ grows), including
our present study.

To be able to apply the existing numerical results for the
cases when the core accretion history is non-trivial
($\dot M\neq$ const) while at
the same time keeping the flexibility of the
theory outlined in R06 we have resorted to the following approach.
First, we still use formula (\ref{eq:Matm}) to calculate $M_{atm}$
but now we do not assume coefficient $\zeta$ to be constant as
various physical effects in the deep atmosphere introduce
additional variations of the density profile not captured by
the analytical theory of R06. Second, using equation
(\ref{eq:instab_cond}) we calculate $M_{crit}$ under the
assumption $\dot M=const$ used in Ikoma et al.\ \rm (2000). With this
information in hand we can calibrate the dependence of $\zeta$
on various physical parameters of the problem by comparing our
results with those of Ikoma et al.\ \rm (2000). Finally, after $\zeta$
has been calibrated for a particular accretion history
$\dot M=$const, equation (\ref{eq:Matm}) can be applied for
a more general situation, e.g. for the $\dot M(M_c)$ dependence
used in this work.

This approach should work well as long as the state of the
atmosphere is fully determined by the current value of $\dot M$,
and is independent of the full accretion
history. For this to be the case the thermal timescale of the
atmosphere must be short compared to the planetesimal accretion
timescale, and this has been verified in R06.
This gives us confidence that the outlined method
should be robust and justifies its application to the problem at hand.

Since $\zeta$ is
predominantly affected by the changes of the thermodynamical
state of material in the atmosphere (grain sublimation,
molecular dissociation, variations of
the equation of state depend on gas temperature and density) one
should expect $\zeta$ to depend most strongly on $M_c$ and $L$
--- variables that determine $T$ and $\rho$ deep in the envelope.
Ikoma et al.\ \rm (2000) have shown that in their case $M_{crit}$
scales with $\dot M$ and $\kappa_0$ roughly as power laws.
This motivates us to look for the dependence of $\zeta$ on $M_c$
and $L$ also in the power law form. The details of the calibration
procedure are presented in Appendix \ref{app:zeta} where we
show that
\begin{eqnarray}
&& \zeta= \zeta_0 \eta\, M_c^\delta,~~~\delta\approx 1.2,
\label{eq:zeta}\\
&& \zeta_0\approx
4\times 10^{-32}\,\mbox{g}^{-\delta}\approx 82\,M_\oplus^{-\delta}\nonumber
\end{eqnarray}
provides a reasonably good fit to the numerical results of Ikoma
et al.\ \rm (2000). Note that $\zeta$ has been found to depend on $L$
only weakly, and can be considered a function of $M_c$ only.
In deriving this result for $\zeta$ we have assumed that
the coefficient $\eta$ appearing in the relation
(\ref{eq:instab_cond}) is independent of the planetesimal accretion
history of the core and is thus the same in Ikoma et al.\ \rm work as well
as in our case.

\section{Critical core mass.}
\label{sect:mcrit}

Results obtained in the previous section allow us to compute
$M_{crit}$ for $\dot M$ given by equation (\ref{eq:mdot}):
substituting $M_{atm}$ in the form (\ref{eq:Matm})
with $\zeta$ given by equation (\ref{eq:zeta}) into the
instability condition (\ref{eq:instab_cond}) and using equation
(\ref{eq:lumin}) for core's accretion luminosity we find
\begin{eqnarray}
\label{eq:Mcrit}
M_{crit} & \approx & \left[\frac{8.2}{\zeta_0}
\left(\frac{k_B}{\mu}\right)^4\frac{\chi\Sigma_s\rho_c^{1/6}\kappa_0}
{\sigma G^{5/2}}\right]^{3/(5+3\delta)}\\
& \approx & 40~\mbox{M}_\oplus\left(
\frac{\chi\Sigma_1\kappa_{0.1}\rho_{c,1}^{1/6}}{\mu_2^4}\right)^{0.35}
a_{10}^{-0.35\alpha},\nonumber
\end{eqnarray}
where $\kappa_{0.1}\equiv \kappa_0/(0.1$ cm$^2$ g$^{-1})$ and
$\mu_2\equiv \mu/(2m_H)$ ($m_H$ is the atomic hydrogen
mass).

Equation (\ref{eq:Mcrit}) shows among other things that $M_{crit}$
is indeed a function of the planetesimal accretion regime:
according to equation (\ref{eq:mdot}) we may think of the
free parameter $\chi$ as a direct measure of planetesimal
$\dot M$, and $M_{crit}$ scales roughly as $\chi^{0.35}$. Just
for illustration, if (as has been done in
Dodson-Robinson et al.\ \rm 2009) planetesimal random
velocities were kept at the level $\sigma\sim \Omega R_H$
(rather than being essentially zero as assumed in deriving equation
(\ref{eq:mdot})) the equation (\ref{eq:mdot}) could still be used
but with $\chi\approx p^{1/2}\ll 1$ (DT93; R06). At $a=40$ AU where
$p\approx 10^{-4}$ one would then find $M_{crit}$ smaller by a
factor of $5$ compared to what equation (\ref{eq:Mcrit}) predicts
for $\chi=1$ at the same location.

With equation (\ref{eq:Mcrit}) we also show for our chosen $\dot M$
behavior that $M_{crit}$ varies with the distance from the central
object: for $\chi$=const the critical mass scales
as $\approx a^{-0.53}$ in the MMSN ($\alpha=3/2$).
This makes $M_{crit}$ at 10 AU
twice as large as $M_{crit}$ at 40 AU, everything else being equal.
Thus, taking $M_{crit}$ to be
independent of the regime of planetesimal accretion and distance
from the central object is generally not justified.

\section{Limit on core instability.}
\label{sect:constr}

For the CI to happen before the nebular gas dispersal
the growth time of the core with mass $M_{crit}$ must be less
than the lifetime of the protoplanetary nebula $\tau_{neb}$, i.e.
$\tau(M_{crit})< \tau_{neb}$. Plugging in our result (\ref{eq:Mcrit})
into equation (\ref{eq:timescale}) this constraint can be rephrased in
terms of the {\it lower limit} on the surface density of solids:
\begin{eqnarray}
\chi\Sigma_s &>&\Sigma_{lim},\label{eq:sig_crit}\\
\Sigma_{lim}&=&0.74\left[
\frac{\rho_c^{(2+\delta)/2}\left(k_B/\mu\right)^4}
{\zeta_0\tau_{neb}^{5+3\delta}
G^{(10+3\delta)/2}}
\frac{\kappa_0}{\sigma}\right]^{1/(4+3\delta)}
\label{eq:Sig_lim}\\
& \approx & 0.1~\mbox{g sm}^{-2}\left(\frac{\tau_{neb}}{3\,\mbox{Myr}}
\right)^{-1.13}\frac{\rho_{c,1}^{0.21}\kappa_{0.1}^{0.13}}{
\mu_2^{0.53}}.\nonumber
\end{eqnarray}
This inequality represents the main result of this work --- a robust
lower limit on the {\it planetesimal} surface density at which
CI is capable of producing giant planets within a
protoplanetary nebula lifetime. Note that $\Sigma_{lim}$ is a
sensitive function of the nebula lifetime $\tau_{neb}$, while it
depends rather weakly on the bulk density of the core $\rho_c$,
atmospheric dust opacity $\kappa_0$, and the mean molecular
weight of the atmospheric gas $\mu$.

Now we can also determine the limiting core mass $M_{lim}$ defined
as the critical core mass for $\chi\Sigma_s=\Sigma_{lim}$, i.e.
at the very extreme of the region where CI can still occur
within $\tau_{neb}$. It is found by substituting (\ref{eq:Sig_lim})
into (\ref{eq:Mcrit}):
\begin{eqnarray}
M_{lim}&=&\left[\frac{4.9}{\zeta_0}
\left(\frac{k_B}{\mu}\right)^4\frac{\rho_c^{1/3}\kappa_0}
{\sigma G^{3}\tau_{neb}}\right]^{3/(4+3\delta)}
\label{eq:Mlim}\\
& \approx & 15~\mbox{M}_\oplus\,
\left(\frac{\tau_{neb}}{3~\mbox{Myr}}\right)^{-0.4}
\frac{\rho_{c,1}^{0.13}\kappa_{0.1}^{0.4}}{\mu_2^{1.58}}.\nonumber
\end{eqnarray}
By construction $M_{lim}$ is independent of the surface density
profile in the nebula.
This mass is to be compared with the isolation mass $M_{iso}$
(the core mass
at which it has accreted all solid mass within its feeding zone) ---
an annulus centered on core's orbit and having a full width equal
to $\xi R_H$:
\begin{eqnarray}
M_{iso}&=&\frac{\left(2\pi\xi\Sigma_s a^2\right)^{3/2}}{M_\star^{1/2}},
\label{eq:M_iso}\\
& \approx &2\,\left(\xi/5\right)^{3/2}\Sigma_1^{3/2}a_{10}^{3/4}
\,M_\oplus\nonumber
\end{eqnarray}
(an MMSN-like density profile was used in making numerical estimate).
At $44$ AU one finds $M_{iso}\approx 6$ M$_\oplus$, which is smaller
than $M_{lim}$. However, a modest radial displacement of the core due
to some type of migration can easily expose it to additional fresh
material allowing $M_c$ to reach $M_{lim}$ (Alibert et al.\ \rm
2005). Alternatively, increasing $\Sigma_s$ (boosting up $\Sigma_1$)
by a factor of 2 makes $M_{iso}\approx M_{lim}$ at 44 AU.

Quite interestingly, the value of $M_{lim}$
is not too far from $10$ M$_\oplus$ commonly accepted as the core
mass throughout the whole protoplanetary disk. This coincidence is
accidental since our estimate of $M_{lim}$ was derived
self-consistently rather than postulated in an ad hoc fashion.

\subsection{Limiting distance for core instability.}
\label{subsect:dist}

Given a constraint (\ref{eq:Sig_lim}) and having a particular
model of the radial distribution of $\Sigma_s$ one can determine
the maximal radial extent $a_{lim}$ of the region in protoplanetary disk
in which the CI can produce giant planets within the nebula life
time $\tau_{neb}$. Since $\Sigma_s$ is expected to be a decreasing
function of $a$ this would only be possible for $a<a_{lim}$.
Using our power-law parametrization (\ref{eq:Sig_s}) of $\Sigma_s$
we find that in a MMSN-like disk with $\alpha=3/2$
\begin{eqnarray}
a_{lim}^{MMSN}\approx 44~\mbox{AU}~\frac{\left(\chi\Sigma_1
\right)^{2/3}\mu_2^{0.35}}{\rho_{c,1}^{0.14}\kappa_{0.1}^{0.09}}
\left(\frac{\tau_{neb}}{3~\mbox{Myr}}\right)^{0.75}.
\label{eq:a_lim}
\end{eqnarray}
Accretion at $\dot M=\dot M_{max}$ corresponds to $\chi=1$
in this formula. The limiting distance found in equation
(\ref{eq:a_lim}) is similar to previous estimates
(e.g. Dodson-Robinson et al.\ \rm 2009) obtained for less vigorous
planetesimal accretion (i.e. for smaller $\dot M$) and fixed
$\dot M_{crit}=10$ M$_\oplus$ but this is just a coincidence.

Equation (\ref{eq:a_lim}) shows that as $\Sigma_s$ (or $\Sigma_1$)
increases, the extent of CI-capable part of the protoplanetary
disk also grows. But $\Sigma_s$ cannot be increased without limit ---
at some point the gaseous component of the protoplanetary nebula
 would become self-gravitating,
and, depending on its cooling time (Gammie 2000), would either
fragment or evolve quasi-viscously while maintaining the marginally
gravitationally unstable state (Rafikov 2009; Clarke 2009).
Thus the most extreme value of $a_{lim}$ can be obtained by taking
$\Sigma_s(a)=f_{dg}\Sigma_{Q=1}(a)$ in equation (\ref{eq:Sig_lim}),
where $f_{dg}$ is the dust to gas ratio, which we take to
be $10^{-2}$, and $\Sigma_{Q=1}(a)$ is the gas surface density
at which the disk is marginally gravitationally unstable, which
happens when the Toomre $Q\equiv\Omega c_{s,0}/(\pi G\Sigma_{Q=1})$
is of order unity (Safronov 1960; Toomre 1964). Assuming that the
disk is heated by a central star with luminosity 1
$L_\odot$ at normal incidence (i.e. neglecting complications related
to flaring geometry, Chiang \& Goldreich [1997]) we find that
\begin{eqnarray}
f_{dg}\Sigma_{Q=1}(a)\approx 20\,a_{10}^{-7/4}\,\mbox{g cm}^{-2}.
\label{eq:Sigma_sg}
\end{eqnarray}
Using this density profile in equation (\ref{eq:Sig_lim}) we find
\begin{eqnarray}
a_{lim}^{Q=1}\approx 200~\mbox{AU}~\frac{\chi^{0.57}
\mu_2^{0.3}}{\rho_{c,1}^{0.12}\kappa_{0.1}^{0.07}}
\left(\frac{\tau_{neb}}{3~\mbox{Myr}}\right)^{0.64}.
\label{eq:a_lim_sg}
\end{eqnarray}
In practice this limit will hardly apply to real systems
because self-gravitating disks with $Q\sim 1$ would
not persist for the lifetime of the nebula, as we already
mentioned.

\subsection{Sensitivity to planetesimal accretion efficiency.}
\label{subsect:acc_efficiency}

Through our calculations we have retained in all formulae the
parameter $\chi$ defined in equation (\ref{eq:mdot}), which
characterizes the efficiency of planetesimal accretion. This
allows us to see how the variation of planetesimal $\dot M$
with respect to $\dot M_{max}$ affects the
planet formation by the CI. From equation (\ref{eq:Sig_lim})
we see that smaller $\chi$ (corresponding to less efficient
accretion) results in {\it higher} $\Sigma_s$ and {\it smaller}
$a_{lim}$, see equations (\ref{eq:a_lim}), (\ref{eq:a_lim_sg}).

\begin{figure}
\plotone{f1.eps}
\caption{
Plot of the limiting semi-major axis $a_{lim}$ beyond which core
instability cannot produce planets within 3 Myr as a function of
$\Sigma_1$ --- surface density of solids at 10 AU in a MMSN-like
protoplanetary disk with $\Sigma_s\propto a^{-3/2}$
(see Eq. [\ref{eq:Sig_s}]). Solid curve is for planetesimal accretion
proceeding at maximum possible rate $\dot M_{max}$, dotted curve
is for $\dot M=\dot M_{tr}$ (accretion of planetesimals with
$\sigma\sim \Omega R_H$, see Eq. [\ref{eq:M_tr}]). Dashed lines
correspond to $a_{lim}$ in a
marginally gravitationally unstable disk, both for
$\dot M=\dot M_{max}$ (upper line) and
$\dot M=\dot M_{tr}$ (lower line). See text for details.
\label{fig:f1}}
\end{figure}

As we emphasized in \S \ref{sect:atm}, this result is not trivial
since less efficient accretion implies lower planetesimal
luminosity, bigger $M_{atm}$ for the same $M_c$, and lower
$M_{crit}$, making CI easier to get going. However, it turns
out that the competing effect
of being able to grow the massive core faster at higher $\dot M$
is more important for the CI to commence {\it within the
limited amount of time}. For this reason the largest extent of
the region in which the CI can happen in time $\tau_{neb}$ is
reached when the core is able to accrete at the highest possible
rate, namely at $\dot M_{max}$ (we discuss whether it is potentially
possible to {\it exceed} $\dot M_{max}$ in
\S\ref{subsect:strengthen}). This makes our numerical estimates
in equations (\ref{eq:a_lim}), (\ref{eq:a_lim_sg}) very robust.

Just for illustration let us also consider a situation in which
the core accretes planetesimals at the rate $\dot M_{tr}$ given
by equation (\ref{eq:M_tr}), and determine $a_{lim}$ in this case.
Comparing expressions (\ref{eq:max_rate}) and (\ref{eq:M_tr}) one
can easily see that we can do this by simply setting
$\chi=p^{1/2}$ ($p(a)$ is defined by equation [\ref{eq:p}]) either
in equation  (\ref{eq:Sig_lim}) or in equation (\ref{eq:a_lim})
and determining  $a_{lim}$ from the resulting expression:
\begin{eqnarray}
a_{lim}^{tr}\approx 5~\mbox{AU}~\frac{\Sigma_1^{0.5}
\mu_2^{0.27}M_{\star,1}^{0.08}}{\rho_{c,1}^{0.19}\kappa_{0.1}^{0.07}}
\left(\frac{\tau_{neb}}{3~\mbox{Myr}}\right)^{0.57},
\label{eq:a_lim_tr}
\end{eqnarray}
where we assumed MMSN-like disk properties to provide direct
comparison with $a_{lim}^{MMSN}$ given by equation (\ref{eq:a_lim}).
If, as we did in deriving equation (\ref{eq:a_lim_sg}), we assume
that the disk is marginally gravitationally unstable we would
find $a_{lim}^{Q=1,tr}\approx 20$ AU for our standard choice of
parameters. In Figure \ref{fig:f1}
we display different expressions for $a_{lim}$
as functions of $\Sigma_1$ --- value of $\Sigma_s$ at 10 AU ---
obtained for $\kappa_{0.1}=1$, $\rho_{c,1}=1$, $\mu_{2}=1$,
$\tau_{neb}=3$ Myr under a variety of assumptions regarding $\Sigma_s$
and $\chi$.

Clearly, less efficient accretion results in significantly
more compact region of the disk where the CI can occur. In fact,
equation (\ref{eq:a_lim_tr}) implies that the formation of Saturn
by the CI at $10$ AU in our Solar System would only be possible
if the surface density in the proto-Solar nebula was at least 4
times higher than in the MMSN or if the nebula dissipation
timescale was at least 10 Myr, in agreement with existing
studies which assume planetesimal accretion at
$\dot M\sim \dot M_{tr}$ (Ida \& Lin 2004).

\subsection{Sensitivity to opacity variations.}
\label{subsect:opacity}

Rather interestingly, our calculations find very weak dependence of
both $\Sigma_{lim}$ and $a_{lim}$ on $\kappa_0$. Although the
reduction of $\kappa_0$ does help to reduce $M_{crit}$ and
$M_{lim}$, the values of $\Sigma_{lim}$ and $a_{lim}$ remain
virtually unaffected. Given the large uncertainly in the
value of $\kappa_0$ this property further strengthens our
estimates of $\Sigma_{lim}$ and $a_{lim}$.

Recently Hori \& Ikoma (2010) calculated
$M_{crit}$ as a function of $\dot M$ for protoplanets with
dust-free (possible if dust grains sediment from the outer
layers of the protoplanetary atmosphere) and metal-free
(i.e. containing only H and He) atmospheres and found
values of $M_{crit}$ lower by up to an order of magnitude
(as low as $2$ M$_\odot$ in the metal-free case for
$\dot M=10^{-6}$ M$_\odot$ yr$^{-1}$) than in the dusty
case. It would certainly be interesting to
repeat calculations done in \S\S\ref{sect:mcrit},
\ref{sect:constr} and Appendix \ref{app:zeta} for the case of
dust-free atmosphere to see how this extreme reduction of opacity
would extend the radial range available for CI.

In practice, we cannot do such calculation at present
since it is not possible to calibrate $M_{atm}$ against the results
of Hori \& Ikoma (2010) as we did in equations
(\ref{eq:Matm})-(\ref{eq:zeta}) and Appendix \ref{app:zeta}.
This is because in the absence of dust $\kappa$ is
a function not only of gas temperature but also of gas
density. As demonstrated in R06 in this case $M_{crit}$ is no
longer independent of the ambient temperature $T_0$ and density
$\rho_0$ of the nebula as the original analyses of Mizuno (1980)
and Stevenson (1982) suggest. Instead one finds that
$M_{atm}\propto (\rho_0/T_0^3)^{q/(1+q)}$, where
$q\equiv \partial\ln\kappa/\partial\ln\rho$ (R06). Obviously, $M_{crit}$
then also depend on $\rho_0$ and $T_0$, and the knowledge of
this dependence is very important for obtaining $\Sigma_{lim}$
and $a_{lim}$ in the dust-free case. Unfortunately, we do not
possess this knowledge from first principles as opacity
calculations are rather complicated, and in any case
we cannot currently calibrate $M_{crit}$ as functions of $M_c$,
$T_0$ and $\rho_0$ against numerical results because calculations of
Hori \& Ikoma (2010) were done for a single value of the planetary
semi-major axis (meaning fixed values of $T_0$ and
$\rho_0$), while $\dot M$ was varied.
The scaling of $M_{atm}$ and $M_{crit}$ with $T_0$ and $\rho_0$
and its implication for the possibility of the CI thus remain worthwhile
issues for future investigation.

We can still get a qualitative idea of how $a_{lim}$ changes in
the dust-free case by setting opacity in equation (\ref{eq:a_lim})
at the very low level consistent with pure gas opacity, e.g.
$\kappa=10^{-4}$ cm$^2$ g$^{-1}$. We then find
$a_{lim}^{MMSN}\approx 80$ AU compared to $44$ AU that equation
(\ref{eq:a_lim}) predicts for $\kappa=0.1$ cm$^2$ g$^{-1}$.
Thus, opacity reduction due to sedimentation and coagulation
of dust grains in the protoplanetary atmosphere may
help in extending the range of distances in which the CI is possible.

\section{Discussion.}
\label{sect:disc}

Despite the robustness of our arguments
it is not inconceivable that some additional factors can weaken
them and make giant planet formation by the CI possible
even beyond the limits represented by equations (\ref{eq:Sig_lim})
and (\ref{eq:a_lim}). Alternatively, it is quite possible that
some of the assumptions used in deriving these results are too
extreme and one can get even better constraints by focussing on
less dramatic assumptions. Below we review factors that
can work one way or another.

\subsection{Extending CI to larger radii.}
\label{subsect:weaken}

One possible way to  facilitate CI and increase $a_{lim}$ is to
consider possibility of planetesimal accretion at rates {\it exceeding}
$\dot M_{max}$. This is very difficult (since there are many factors
that tend to reduce $\dot M$ compared to $\dot M_{max}$, see
\S\ref{subsect:strengthen}) but may be possible if e.g. one takes into
account the increase of planetesimal capture cross-section by
the core caused by its extended, dense atmosphere. This effect has
been previously investigated by Inaba \& Ikoma (2003) who
demonstrated that an increase of $\dot M$ by a factor
of $\sim 10$ compared to the value computed without atmosphere
is possible. According to equation (\ref{eq:a_lim}) such an
enhancement of $\dot M$ (incorporated by increasing $\chi$) would boost
$a_{lim}^{MMSN}$ by a factor of $\sim 4$.

Our present calculations assume that the core is accreting planetesimals
continuously until the protoplanetary nebula dissipates --- this is
important at large $a$ since massive core requires long time to be built.
But one may wonder if building smaller core in shorter time and then
cutting off subsequent planetesimal accretion (and energy release
at the core surface, which supports atmosphere against going unstable)
completely may still lead to the CI and potentially extend it to
larger semi-major axes. Such accretion scenario has been adopted
by e.g. Pollack et al.\ \rm (1996). The problem in this case is that
even if $\dot M=0$ it still takes long time for the atmosphere around
the core to grow to the mass comparable to $M_c$. INE0 show that this
process occurs on thermal timescale of the atmosphere and typically
takes millions of years.

Similar problem is also encountered in a scenario where the core
grows rapidly by planetesimal accretion in the inner regions of
protoplanetary disk and then gets scattered out to large radii by
some massive perturber. One might expect that after the orbit of the scattered
core circularizes by dynamical friction the core would gradually accrete
massive atmosphere and undergo CI at some point. Given that both the
orbit circularization and envelope accretion are likely to take long
time it is not at all obvious whether the CI could be achieved in
this scenario within several Myrs.

\subsection{Limiting CI to smaller radii.}
\label{subsect:strengthen}

There are many factors that can potentially reduce $a_{lim}$
compared to 44 AU estimated in equation (\ref{eq:a_lim}). For example,
there are several reasons why it may not be possible for
$\dot M$ to reach the maximum rate $\dot M_{max}$.

First, the growing core
may clear out a gap in planetesimal disk around its orbit, thus
significantly
reducing $\dot M$ (Tanaka \& Ida 1997; Rafikov 2001; 2003a).
In our previous calculations we implicitly assumed
this not to happen e.g. because of the core migration through
the disk, which allows fresh planetesimal material to be constantly
supplied for core accretion (Alibert et al.\ \rm 2005).

Second, as we mentioned in
\S\ref{sect:accr}, a known pathway to $\dot M\approx\dot M_{max}$ is
via the growth of the core to the size at which it starts dominating
dynamical evolution of nearby planetesimals and triggers
their efficient collisional fragmentation (Rafikov 2004).
However, there is a strong implicit assumption in this scenario ---
that the core can reach this critical size within the nebula lifetime.
Rafikov (2003b) has shown that at $a\sim 30-40$ AU a dynamically dominant
core would need to have mass of order $10^{24}$ g and would require on
the order of $10-100$ Myr to grow in the MMSN. This time scale is
apparently in conflict with the typical dissipation times of
protoplanetary disks. Thus, one may need to either require a more
massive disk at these radii or to find other pathways for accretion at
$\dot M_{max}$. Formation of massive solid bodies by direct gravitational
instability facilitated by various streaming instabilities
(Johansen et al.\ \rm 2009) may be quite relevant for the latter option.
Regardless of this (arguably rather serious) issue our estimate
(\ref{eq:a_lim}) still remains a useful upper limit on $a_{lim}$.

Among other factors hindering the onset of CI and reducing
$a_{lim}$ we should mention the possibility of high opacity in the
protoplanetary atmosphere. It was suggested in \S\ref{subsect:opacity}
that $\kappa$ may be very low because of the dust sedimentation
and growth. However, infalling planetesimals which feed core accretion
likely get partly disrupted in the atmosphere leaving behind large amount
of refractory material. This may actually {\it increase} $\kappa$ compared
to the value of $0.1$ cm$^2$ g$^{-1}$ assumed in
equation (\ref{eq:a_lim}). Nevertheless, given the weak sensitivity of
$a_{lim}$ to $\kappa$ the potential increase of atmospheric opacity
is unlikely to have huge effect on $a_{lim}$.

\section{Summary.}
\label{sect:sum}

We studied the formation of giant planets by core instability at
different locations in the protoplanetary disk with the goal of
determining the range of radii where the CI is feasible
within the several Myr lifetime of the protoplanetary
disk. We demonstrate that this range is determined by two key
factors:
\begin{itemize}

\item The high planetesimal accretion rate is necessary to build
the solid core as rapidly as possible at large separations
from the star.

\item Intense energy release at the core surface caused by
planetesimal accretion increases the critical core mass and
delays the CI.

\end{itemize}
The first factor turns out to be more important and the largest
distance at which CI can happen, around 40-50 AU, is obtained when
$\dot M$ is at its highest possible value corresponding to
accretion of dynamically cold planetesimals. The core mass
corresponding to this case is around $15$ M$_\oplus$, likely
compatible with the isolation mass at this distance.

Our approach is quite different from other similar studies which
often assume that (1) accretion proceeds at much slower rate $\dot M_{tr}$
defined by equation (\ref{eq:M_tr}) corresponding to accretion of
planetesimals moving with random velocities at the level of $\Omega R_H$
and/or (2) the CI commences after the core has reached a fixed mass of
around 10 M$_\oplus$ irrespective of the planetesimal accretion rate
or the location in the disk. Relaxing these two arbitrary assumptions we
are able to obtain a significantly more robust and self-consistent
limit on the CI operation which can be represented as a lower
bound on the solid surface density ($\gtrsim 0.1$ g cm$^{-2}$)
or an upper bound on the size of the region where
the CI can get going within several Myr timescale. These limits
are insensitive to the mass of the central star and depend only
weakly on the opacity in the core atmosphere. Our predictions are
relevant for interpreting the results of current and future direct
imaging surveys (Marois et al.\ \rm 2008) designed to uncover and
characterize the population of giant planets at large separations
from their parent stars.

\acknowledgements

Author is grateful to Ruth Murray-Clay for useful discussion and
the suggestion of the limiting radius calculation for the marginally
gravitationally unstable disk. The financial support for this work
is provided by the Sloan Foundation and NASA via grant NNX08AH87G.

redpajama_set_name: RedPajamaArXiv


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