text:
New Villas
Tel: +34 952 838 705 Email: info@solsimpleproperties.com
The Hub - New Luxury Apartments - Malaga Capital €213,125
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The Hub - New luxury Apartments - Malaga Capital
Contact us today for further Information, price lists, plans and brochure
1 / 2 Beds Available
Located in the vibrant city of Malaga and 5 minutes walk to the beach, The HUB Malaga is a pioneer in our concept of Urban Living. 44 apartments of 1 and 2 bedrooms, the facilities include Co Working area, Skypool and chill-out, al fresco gourmet bar and boutique gym.
The best qualities are used in this full refurbishment project. Parking spaces available and surfaces from 53 to 89 sqm + terraces
Located in the heart of the Costa del Sol, in the vibrant city of Malaga, within the Huelin Neighborhood and just five minutes walk to the beach, you can find The Hub Malaga: one of the most 'transgressive' residential projects in the Malaga capital.Malaga has become a cosmopolitan and avant-garde city. The Huelin Neighborhood is being transformed into a new residential area with a marked emerging style that combines tradition, functionality and urban regeneration.The Hub Málaga is a new residential concept that will create a new vision for the capital. The project will be a boost for the Huelin Neighborhood, and an iconic building which will contribute to changing the architectural vision of Malaga. Home, social life, work space and leisure join together within a new concept of "Living Homes" thus creating an urban oasis.
The Hub was borne to establish new standards of lifestyle within urban environments. This new concept of residential living encapsulates impeccable design, services, state-of-the-art facilities, common areas and spaces adapted to new needs.The Hub will be made of buildings and homes far beyond an ordinary home or a second residence. The Hub is a building well above the expectations of a simple residence.The Hub philosophy encompasses apartments located in urban centers, built with the best qualities, extensive facilities (gym, common areas with barbecues and swimming pool) and services (co-working spaces and gastro bar).In this format, The Hub project uses the BIM (Building Information Modeling) methodology that guarantees, amongst other things the optimization of the running costs for the future conservation and maintenance of the building.
LET'S TALKabout the apartmentOF YOURDREAMS
The Hub Málaga, consists of two buildings - harmoniously integrated within its surroundings and which enjoys a privileged location. Located in the Huelin Neighborhood in the western part of the city and opposite the Princesa Metro stop. Perfectly located just five minutes walking distance to the beach of San Andrés and in front of what is considered to be 'the new lungs of the city,' the Park Europa.The building, completely renovated, has its own personality and the superior qualities, exterior and interior. With a total of 44 apartments consisting of 1 and 2 bedrooms, 56 garage spaces and storage rooms. This new concept of living extends to the outdoor areas with large premium common areas that invite you to enjoy the climate and Mediterranean life.The Hub Málaga has a swimming pool and chill-out area on one of the rooftop terraces overlooking the new Park Europa. Two barbecue areas and a Gastro Bar space will be located on the other terrace. In addition, the building includes a state-of-the-art gym and a co-working space with views.
On one of the terraces you can enjoy spectacular views of the new Parque Europa and Málaga Oeste from the Al Fresco Gourmet Bar. An unprecedented location with a barbecue area, external kitchen for personal use and a sofa and dining area. A perfect place for any type of event any time of year and of course, at any time of day, thanks to the infinity of possibilities it offers. In addition, you can make reservations directly from your APP
The Hub Malaga is characterized by its style and emphasizes a design that plays with open, wide, and bright spaces. The other rooftop that this property has is the ideal place to relax in its large modern design pool and enjoy the sensational views of the city. You will not be able to resist the magic of La Terraza Sky Pool & Chill-Out area!
Boutique Gym at The Hub Málaga is closer to a club than to a gym. Seen from the outside, it almost looks like a modern and elegant boutique, featuring a large, state-of-the-art fitness room with cardio areas (with treadmills, ellipticals and exercise bikes), weight machine area, free weight area and abdominal area.Once you enter, you find an elegant atmosphere and a perfect space to live a new premium fitness experience.Boutique Gym also stands out for its innovative technological standards with internet connection and televisions. Top of the range integration software and apps, as well as synchronization with smartwatches and techniques which are key elements in taking your workouts to a higher level.
The concept of The Hub never ceases to surprise, expanding its facilities and services to a cosy Co-Working space that implements all the services necessary to carry out business activities.The spectrum of services ranges from designer desks, tables, sofas, high-speed Wi-Fi, meeting areas and rest areas to guarantee all the necessary services for entrepreneurs and self-employed workers, with natural light and spectacular views guaranteed.
Disclaimer: The images are for guidance only, and are therefore susceptible to modifications for technical, legal or other reasons. The furniture reflected is purely decorative and does not constitute a contractual document. The equipment of the houses will be as per the corresponding quality specifications document
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Martha Marie Elizabeth Schultz, 99, of Hutchinson, died Sunday, Jan. 8, 2017, at Harmony River Living Center in Hutchinson.
Funeral services will be Thursday, Jan. 12, at 11 a.m., at Immanuel Lutheran Church in Brownton. The Rev. R. Allan Reed will officiate, and burial will follow in the Brownton City Cemetery in Brownton. Visitation will be Thursday from 9:30 a.m. to 11 a.m. at the church.
Judith "Judy" Ann Martineau, 74, died Wednesday, Jan. 4, 2017, at Abbott Northwestern Hospital in Minneapolis.
Marella F. Kemmer, 96, of Lakeville, formerly of Moorhead, died Friday, Jan. 6, 2017.
Funeral services will be held Thursday, Jan. 12, at 11 a.m., at St. John's Lutheran Church, 20165 Heath Ave., Lakeville, with interment in Moorhead Memorial Gardens. Friends may call at the church one hour prior to funeral services.
Casket bearers will be Dean Kemmer, David Kemmer, Jeff Kemmer, Bradley Kemmer, Jayson Stiefel and Christi Stiefel.
Arnold Leroy Hepner, 82, of Hackensack, died Dec. 28, 2016.
Services were held Jan. 6, 2017, at the Dobratz-Hantge Chapel in Hutchinson, with the Rev. Ronald Mathison officiating.
Urn bearers were Alisa Schmidt, Jonathan Larson, Joshua Larson and Joel Larson. Cody Schmidt was the honorary urn bearer. Military honors were by the Hutchinson Memorial Rifle Squad. Interment was in the Bohemian National Cemetery, rural Silver Lake.
Arnold "Arnie" D. Hanson, 80, of Faribault, died Tuesday, Jan. 3, 2017, at Abbott Northwestern Hospital in Minneapolis.
Funeral services were held Saturday, Jan. 7, at Trinity Lutheran Church in Faribault, with interment at Maple Lawn Cemetery, Faribault, with military rites by the Rice County Central Veterans Association.
E. Mae Barfknecht, 88, of Gaylord, died Thursday, Jan. 5, 2017, at Harmony River Living Center in Hutchinson.
Jon Barfknecht was the urn bearer. Interment was in the church cemetery.
Early in January, all the best seventh- to ninth-grade Knowledge Bowlers from Southwestern Minnesota converge on Marshall to compete. This year there were 48 teams from schools from Windom, Redwood Falls, and Minneota to Lester Prairie, Hutchinson, Winsted, and Glencoe. These teams all advanced from Region 6 and 8 meets in December.
There are a lot of dire predictions out there about our representative democracy. We're just past a presidential election campaign in which candidates complained about a rigged political system. Now, commentators worry about the imminent failure of the American experiment.
The inaugural 2017 meeting of the Glencoe City Council got off to a rocky start. Two new members joined the Council at its Jan. 3 meeting and promptly got into a procedural mess that looked clumsy at best.
It centered around the reorganization of the Council for the new year. Most agenda items were simply procedural and expected to take several minutes to complete. But not this year.


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Says Art 370 essential, unassailable
JAMMU & KASHMIR Saturday, July 20, 2019
Handwara, July 20 (KNO) : National Conference party president and Member of Parliament from Srinagar Dr Farooq Abdullah Saturday said annulling Art 370, Art 35-A will tantamount to a constitutional coup, saying Art 370 is essential and unassailable until the final resolution of the State was reached to. According to a statement issued to KNO, party president while addressing a mammoth convention of workers at environment park Handwara said that powers in New Delhi were frightened of National Conference and the very thought of NC coming back to power sends jitters through them. "They are well aware of the fact that until NC remains in the state it will painstakingly protect the interests of the state. They are frightened of us. Anxious and unnerved, New Delhi has employed various henchmen in the state to frustrate the efforts of NC towards protecting the special status of our state," he said. Party president while underscoring the need of rallying round the flag of the party said, "Today we see some parties coming up in every alley of Kashmir. You all have to remain very cautious about their nefarious designs. They are the agents and second in-command of New Delhi in the state. Their sole mission is to frustrate the efforts aimed at protecting the interests of the state. At present we see them tricking people through high-sounding statements. You all should remain cautious about their trickery. They are here to add colors to the nefarious designs of those forces as are inimical to our special status. If they have had a speck of concern for the interests of the state, they would have assisted National conference in its efforts to protect the special constitutional position of the state. However, they cannot. They are here to serve the interests of New Delhi only and not the state." Party president who was very vocal in rejecting the attacks on Art 370 said, "Not even a single day passes, when we don't hear our enemies cry about the temporality of the Art 370. Yes, the Art was deemed to be Temporary unless the issue of the state was not resolved. However, the issue is still lingering around; therefore the Article 370 will continue to remain around. The accession of the state took place in unusual circumstances. The Maharaja who was rather vacillating then did not want to accede to any of the two dominions. He wanted the erstwhile state to remain free. However that didn't happen, he had to make a choice in extra ordinary circumstances. Keeping in view the special circumstances, Art 370 was added to the constitution of India. The Art will remain as it is until the final resolution of the issue of Jammu and Kashmir is not reached to. That resolution hasn't been reached yet. So until the issue is solved, no power on earth can touch the article 370. The uneducated rants of those who want to obliterate Art 370 are far-fetched, fanciful and devoid of any genuineness," he said adding, "we are fighting for our rights. Our struggle is based on principles. We are seeking our rights. It is people of the state from whom we draw our energy, and god willing we will continue with our mission." Referring to Art 35-A party president said that the Art 35-A was the fall out of the Art 370 and fiddling with it will upset all the successive amendments to the presidential of 1954. "If Art 35-A is annulled, then all the successive presidential orders will get annulled automatically. It was the then Maharaja who gave the state subject rights to the people of state in 1927," he said adding, "it is the prime duty of the NC workers, functionaries to apprise the people living in towns and villages about the impending threats on our special position. The need of the hour calls for making people alert about the consequences of repealing Art 370, Art 35-A." Party president while highlighting the supreme sacrifices of the party workers said, "Our worker is the back bone of our party. It is the sheer resolve of our workers that keeps us going. I remember how one of our brave hearts, Ghulam Rasool Sahib laid his live for the ideals of the party. I pay earnest tributes to him. Our state is facing onslaught on its interests from all sides, the need of the hour calls for forging unity. We all have to unite our efforts to protect the special position of our state." Among others Haji Ali Muhammad Sagar, Muhammad Akbar lone, Hasnain Masoodi, Nasir Aslam Wani, Muhammad Ramzan Choudhary, Mir Saifullah, Qaisar Jamsheed lone, Sharief Ud Din Shariq, Farooq Shah, G R Naz, Shafkat Watali, Salam Ud Din Bajad, Rafiq Shah addressed the workers and impressed upon them to increase their engagement with the people. Earlier party president visited the Darbar e Aliya Yonisiya, Awoora, Kupwara following the three day annual urs observance. Party president paid obeisance at the shrine and prayed for the wellbeing of human kind. He was accompanied by party general secretary Haji Ali Muhammad Sagar, members of parliament Muhammad Akbar lone, Hassnain Masoodi; party senior leaders Mir Saifullah, Qaisar Jamsheed lone, and Farooq Ahmad Shah. Party president while addressing the devotees deliberated on the integration of different school of thoughts on one single platform. "Sufism stresses on universal brotherhood, humanism and peace." It is pertinent to mention here that more than 50000 people from across the country and state are attending the urs observance; the final Dua will be held on 21 July. Party president appreciated the efforts of Sajada Nisheen Mian Darbar e Aliya Younisia Alhaj Pir Mian Muhammad Maqbool for his service to the shrine and the people. Later a party convention was organised at Langate which was presided by General Secretary Haji Ali Mohammad Sagar. He stressed on workers to take party programme to every section of the society. Party Provincial President Nasir Aslam Wani, party functionaries also present.(KNO)

redpajama_set_name: RedPajamaCommonCrawl


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AUC Chairperson Dr.
Home / Diplomacy / AUC Chairperson Dr.
AUC Chairperson Dr. Nkosazana Dlamini Zuma emphasises the importance to celebrate the determination of women around the world for equal rights and have their voices heard
Addis Ababa, Ethiopia, 08 March 2016: The African Union (AU) joins the international community to celebrate the social, economic, cultural and political achievements of women, while also reviewing past progress of women's development and looking ahead at future opportunities for women, especially in the African continent.
The celebration of the International Women's Day today Tuesday 8th of March 2016, at the AU headquarters featured a series of activities including an African Debate on the motion: "Electoral Laws should facilitate Gender Parity".
Holding under the theme "Agenda 2063: A Pledge for Gender Equality" the event at the African Union took place in the presence of H.R Dr. Nkosazana Dlamini Zuma, Chairperson of the AU Commission, H.E Ahmat Awad Sakine, Chair of the Permanent Representatives Committee (PRC) AND Ambassador of the Republic of Chad, the AUC Deputy Chairperson, Mr. Erastus Mwencha, the AU Commissioners, Members of the PRC, and invited guests among others.
Dr. Dlamini Zuma began by thanking the participants for the mass mobilization around the issues discussed during this celebration regarding women's development in the continent and the world at large. "I would like to thank you all for joining us and especially for standing as witnesses to all the pledges that have been made by the African Union Commission's Departments and other stakeholders committing to implementing at least two high impact actions that support gender equality and women's empowerment in furtherance of Agenda 2063……I can assure you that these are not empty pledges, as the AUC will develop a performance management plan that will track and report on progress, capture lessons learned and best practices as well as outcomes that will inform future decision-making on the implementation of gender equality and women's empowerment in the AU" Underscored the AUC Chairperson.
H.E. Dr. Nkosazana Dlamini-Zuma highlighted the importance of the role of women within the continent saying, women like men have equal rights and "these rights are entitlements not a privilege". She underscored the need for laws, noting that Africa has to move a step further to embedding champions that will convince law makers that laws are important and should be implemented. "If women are not active in political parties, gender parity will never be achieved and therefore women and progressive men must be active in political parties" said the AUC Chairperson. Dr. Dlamini Zuma further stressed on the need for African women to work hand-in-hand with progressive men and make the men understand that establishing gender parity is not a favor but it is a right which must be given.
Ambassador Ahmat Awad in his remarks supported the course and pledged to working together with stakeholders and partners to bring a stop to violence and all forms of discrimination which affects women and children in the continent. In addition, the Chairperson of the PRC appealed to the AUC Chairperson to have an active participation of the AU Heads of State in mainstreaming gender development in political parties and decision making organs at national, regional and continental levels. "Women should be policy-makers; they should be part of the legislative and constitution of their respective communities, what a man can do a woman can also do." said the Chair of the PRC.
Speaking earlier Mrs. Mahawa Kaba Wheeler, Director of Women Gender Development of the AUC recalled that IWD celebrations creates an opportunity for women and men to come together, network and organize themselves for social transformation, anchored on values and principles that ensure enjoyment of human rights and fundamental freedoms for all. The commemoration of this day, she said, accords nations the opportunity to marshal efforts to strengthen women's rights and their participation in social, economic and political development.
In order to improve the involvement and participation of women at all sphere of life, the Women, Gender and Development Directorate (WGDD) of the African Union Commission (AUC) organized a Zumba dance fitness activity geared towards the promotion of wellness as an ingredient to achieving the goals of Agenda 2063. The Zumba dance brought together all AU staff, Friends and well wishers on the 7th March 2016 an energetic momentum to kick-start the International Women's Day (IWD) .


text:
WILLIAMS TO BRING MAGIC TO EUSTIS
Compiled by Alan Schmadtke and Joe WilliamsTHE ORLANDO SENTINEL
Pat Williams, who is trying to bring an NBA franchise to Orlando, is scheduled to speak in Lake County on Sept. 26 at 12:15 p.m. at the Eustis Community Center. He has been invited as guest of the Rotary Club of Eustis, but the public is invited. The Orlando Magic, the name for the proposed team, has sold more than 6,000 reservations for season tickets at $100 a reservation. Williams has said that if the team receives at least 10,000 reservations before an NBA team owners meeting in Pheonix, Ariz., next month, that Orlando will be almost certain to recieve an expansion team. "Mr. Williams will discuss his group's efforts to land a National Basketball franchise for Orlando," said Wayne Bailey of the Rotary Club. "He will also talk about season ticket information." The fee to attend the luncheon meeting is $5 and reservations must be made through Bailey, 357-5903.


Abstract: Developing an effective methodology for obtaining optically pure compounds at a quantitative yield has been the main concern in the field of organic chemistry. In this chapter, Akai and Moustafa review the concept of dynamic kinetic resolution as a potential solution for overcoming this fundamental issue. The advantage of the use of an enzyme (such as lipase) and its combination with metal catalysts is demonstrated in this chapter.
Abstract: Fluorinated aromatic compounds are found in a variety of biologically active compounds, including clinical drugs and agrochemicals. Therefore, the synthesis of aryl fluorides is particularly important in the medicinal and process chemistry fields. In this paper, we report a method for the synthesis of aryl fluorides by benzyne fluorination under microflow conditions using the efficient Comet X-01 micromixer. In comparison to our previously reported method under ordinary batch conditions, this approach facilitates a significant reduction in reaction times, to ~10 s, as well as increases in the yields of fluoroarenes (by up to 51%).
Abstract: A reagent generated from TMSN3/KMnO4/BnEt3NCl was found to promote an SH2' radical azidation of a bromo silyl enol ether to furnish an azido silyl enol ether via olefin transposition. With the present azidation protocol, a new synthetic approach to agelastatin A, a potent antitumor marine alkaloid, has been established.
Abstract: A concise procedure to prepare optically active (3R)- and (3S)-(E)-1-iodohexa-1,5-dien-3-ol was developed. The racemic alcohol was prepared under flow and batch conditions via successive half reduction of the ester followed by Grignard reaction. Lipase catalyzed kinetic resolution under flow conditions afforded (3S)- alcohol and corresponding (3R)-acetate. The acetyl group was removed under flow conditions by using ion exchange resin.
Abstract: We have discovered that the racemization of configurationally stable, axially chiral 2,2'-dihydroxy-1,1'-biaryls proceeds with a catalytic amount of a cyclopentadienylruthenium(II) complex at 35–50 °C. Combining this racemization procedure with lipase-catalyzed kinetic resolution led to the first lipase/metal-combo-catalyzed dynamic kinetic resolution of racemic axially chiral biaryls. The method was applied to the synthesis of various enantio-enriched C1- and C2-symmetric biaryl diols in yields of up to 98% and enantiomeric excesses of up to 98%, which paves the way for new developments in the field of asymmetric synthesis.
Abstract: One-pot sequential reactions using the acyl moieties installed by enzymatic dynamic kinetic resolution of alcohols have been little investigated. In this work, the acryloyl moiety installed via the lipase/oxovanadium combo-catalyzed dynamic kinetic resolution of a racemic dienol [4-(cyclohex-1-en-1-yl)but-3-en-2-ol or 1-(cyclohex-1-en-1-yl)but-2-en-1-ol] with a (Z)-3-(phenylsulfonyl)acrylate underwent an intramolecular Diels–Alder reaction in a one-pot procedure to produce an optically active naphtho[2,3-c]furan-1(3H)-one derivative (98% ee). This method was successfully applied to the asymmetric total synthesis of (–)-himbacine.
Abstract: The dynamic kinetic resolution (DKR) has gained increasing attention because it is a simple method to obtain optically pure compounds from racemic substrates in quantitative yields. DKR is typically performed by a combination of kinetic resolution of racemates and in situ, continuous, and rapid interconversion between two enantiomers of the substrates, namely, racemization. We recently reported a new DKR method based on the combination of lipase-catalyzed kinetic resolution of racemic secondary alcohols and the V-MPS-catalyzed in situ racemization. In V-MPS, oxovanadium moieties were covalently bound to the inner surface of mesoporous silica (MPS) with a pore size of 2–4 nm. In this article, we introduce the background of why we used MPS, reaction characteristics of V-MPS, advantages of our DKR method, and its synthetic applications.
Abstract: A versatile synthesis of 1,4-benzodiazepine derivatives through the reaction of various 3-(trifluoromethanesulfonyloxy)benzynes with N-(p-toluenesulfonyl)imidazolidin-2-ones is reported. This reaction system provides a 1,4-benzodiazepine bearing a trifluoromethanesulfonyloxy group as a single regioisomer among the four possible regioisomers.
Abstract: We report a method for synthesizing optically active 2,3-disubstituted indolines by the cycloaddition reaction of benzynes with various 4-[(4-toluenesulfonyl)amino]-(E)-but-2-enenitriles, which are readily prepared from the corresponding ʟ-amino acid derivatives.


You know, that visual novel.
YU-NO: A Girl Who Chants Love at the Bound of this World is coming to PlayStation 4 and PC in North America and Europe in 2019, publisher Spike Chunsoft announced at Anime Expo 2018. It will feature Japanese audio and English text.
In Japan, YU-NO first launched for PlayStation 4 and PS Vita in March 2017. In March 2018, publisher 5pb. announced that it is also preparing Switch and PC versions of the visual novel. Spike Chunsoft did not announce the Switch or PS Vita versions for a western release.
Watch the announcement trailer and opening movie below.


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text:
\section{Introduction}
\label{Sec1}

In the last several years, there has been much interest in massive compact objects.
Sometimes the evidence is direct; however, it is indirect on some other occasions.
For instance, the~detection of gravitational wave (GW) event GW190814 \cite{Abbott2020} directly confirms the existence of a compact star with a mass of 2.5--2.67$M_\odot$. Thus far, the minimum mass of an astrophysical black hole is understood to be around
 $3M_\odot$ \cite{mac,pooley} (also see~\cite{ozel}) while the maximum mass of a neutron star (NS) is argued to be about $2.5M_\odot$ \cite{alsing}, leading to a so-called mass gap.
Therefore, the~above observation fills in the gap, arguing the compact
object to be a massive NS.
Note, however, that some researchers also claim against this mass gap.
There indeed can be a depression in mass distribution.
Nevertheless, there is other evidence for massive
NSs (although not as massive as that inferred from GW190814) based
on pulsar observations~\cite{Cromartie2020} with mass $>2M_\odot$.
Indeed, the statistics based on observation showed that while the averaged mass of NSs is about $1.4M_\odot$, the~accreted millisecond pulsars have masses, on~average, of $1.6M_\odot$ \cite{pul,miller}. Hence, there is no surprise with the discovery of more than two solar mass pulsars.

On the other hand, for~more than
15 years, at~least a dozen objects have been detected as peculiar over-luminous
Type Ia Supernovae (SNeIa), whose unusually high luminosity, along with their violation of
Khokhlov's limit~\cite{khok}, argue for progenitor masses that are as high as $2.8M_\odot$
\cite{Scalzo2010,Howell2006}.
This is evidence for a significant violation of the Chandrasekhar mass-limit of
$\sim$1.4$M_\odot$ of white dwarfs (WDs) and a super-Chandrasekhar~mass-limit.

An important question to ask is: how do we argue for the existence of such unconventional and massive compact
objects? Numerous investigations have been carried out by our group aimed at uncovering this mystery over the last one decade.
We have shown that strong magnetic field: its strength and anisotropic
effect~\cite{DM2012,DM2013,DM2013b,DM2014,DM2015,SM2015,MB2018,MB_euro2018,AG2020,MB2021,MB2022}, modified Einstein's gravity~\cite{Ban2017,EPL2019}, and matter encountering noncommutative physics at high density in the core of compact stars~\cite{IJMPD2020,IJMPD2021} can each independently lead to highly massive WDs and NSs.
After our initiation, several other groups independently started looking
at this issue based on other ideas, e.g., the~ungravity effect~\cite{BertM2016}, WDs
having net electric charge~\cite{Liu2014,Car2018}, lepton number violation~\cite{Belyaev2015} and anisotropic pressure~\cite{HB2013}.
Additionally, there appears to be a relation between equation of state (EoS)
and compact mass~\cite{glen}.

Magnetic fields in WDs and NSs, in~general, degenerate stars, have been
explored for a long time (see the review~\cite{cham} from three decades ago).
In general, it is not difficult to understand the surface field of WDs and
NSs to be $\sim$$10^9$ and $10^{15}$ G, respectively~\cite{cham,fw2005},
based on their different origins, e.g.,~fossil effect, dynamo, binary
evolution etc.~\cite{fw2000,fwk}. Spectropolarimetric observations suggest that the magnetic
flux of majority magnetic WDs is similar to that of main-sequence Ap-Bp
stars~\cite{fw2005}. This argues for an evolutionary link of
magnetism between the main sequence stars and WDs. A similar argument
can be made for isolated NSs if the underlying O-type progenitors have
effective dipolar fields $\sim$$10^3$ G~\cite{fw2005}. It has been statistically found
that magnetic WDs are, in general, massive, and their number with fields
at high as $\sim$$10^9$ G could even be $10\%$ of the total population~\cite{fwk}. However, magnetic field evolutions for isolated and accreting
WDs are different. This is no surprise because~in the outer layers
of WDs, the field structure is expected to be modified due to accretion,
particularly above a critical accretion rate \mbox{rate $\approx \text{(1--5)}\times
10^{-10} M_\odot {\rm yr}^{-1}$}. This results in a shorter Ohmic decay time,
leading to an apparent lack of field $\sim$$10^9$ G in observed accreting WDs~\cite{Cumming2002,zwf}.

Among other consequences of magnetized and modified gravity-induced WDs
are WD pulsars. These stars can also behave as soft gamma-ray repeaters (SGRs)
and anomalous X-ray pulsars (AXPs) \cite{MR2016,BM2016,BM2019}, generating significant gravitational
radiation that can be detected by space-based detectors~\cite{Kalita2019,Kalita2020,Kalita2021}.
Many of these transients can eventually turn out to be WDs with super-Chandrasekhar masses.
Observational data from the Sloan Digital Sky Survey (SDSS) suggest that
magnetic WDs can have larger masses compared to their non-magnetic counterparts, although~they span the same temperature range~\cite{Van2005,Ferrario2015}. Regardless of the rotation rate, strong magnetic fields have been shown to modify the EoS of electron degenerate matter and yield super-Chandrasekhar WDs with masses up to $\sim$$2.6\, M_{\odot}$ \cite{DM2012,DM2013,DM2013b,SM2015}.

The effect of strong magnetic fields on the mass-radius relation has been explored previously by our group for~both Newtonian~\cite{DM2012} and general relativistic~\cite{DM2014,SM2015,DM2015,MB2021} formalisms. These studies were performed for various different field configurations and were in good agreement with the results from independent studies~\cite{Bos2013,FS2015,Car2018}. All the above ventures, including the underlying observations, by~the aforementioned groups have brought super-Chandrasekhar WDs into the limelight in recent times. Magnetized WDs (B-WDs) have many important implications other than their apparent link to peculiar SNeIa, and, therefore, their other properties should also be explored~\cite{MR2016,BM2016}. Strong magnetic fields can influence the thermal properties of WDs, namely their luminosity, temperature gradient and cooling rate~\cite{MB2018,AG2020, MB2021,MB2022}.
Moreover, the~magnetic field of WD has a crucial role in their binary
evolution as progenitors of SNeIa or accretion-induced collapse~\cite{imin1,imin2,imin3}. Additionally, a possible merger through
binary evolution leading to a peculiar SNeIa could be an
alternate explanation for super-Chandrasekhar progenitors~\cite{imin4}.

In this review paper, we discuss the broad implications of these compact stars as well as the current status of observations. We aim at gathering all the
underlying results obtained in last decade or so and try to assess the
overall state of the art of the subject as it stands now, in~particular regarding the effects of magnetic
fields in NS and WD. This paper is organised as follows. In~Section~\ref{Sec2}, we discuss the origin and evolution of strong magnetic fields inside WD stars. In~Section~\ref{Sec3}, we simulate the rotating and magnetized B-WDs based
on their structural stability for toroidal and poloidal magnetic field geometries. In~Section~\ref{Sec4}, we compute the mass-radius relation for non-rotating B-WDs, assuming a spherical geometry and radially varying field strength. In~Section~\ref{Sec5}, we explore the effect of cooling as well as magnetic field dissipation on the modified mass limit and the suppressed luminosities of B-WDs. In~Section~\ref{Sec6}, we briefly describe the quantisation of energy states of
matter in B-WDs and derive their modified mass limit, assuming both constant and varying magnetic fields. In~Section~\ref{Sec7}, we investigate the GW emission originating from WDs with misaligned rotation and magnetic axes, along with their detection prospects.  Finally, we discuss the effect of anisotropy of dense matter in~the presence of strong magnetic fields on~the properties of B-WDs in Section~\ref{Sec8} and conclude in Section~\ref{Sec9}.

\section{Origin and Evolution of Magnetic Fields inside White~Dwarfs}
\label{Sec2}
Here, we discuss the origin of strong fields inside magnetised WDs and their evolution at the end of the main sequence during~the asymptotic giant phase. We present both numerical results from the Cambridge stellar evolution code STARS and analytical estimates from virial theorem that provide a relation between various energy components of the~WD.

\subsection{Origin of Magnetic~Field}
Magnetic fields in stars exhibit complex behaviour with structure on both small and large scales. The~origin of large-scale fields is debated. They can be fossil fields (e.g.,~\cite{DM2013b,fw2005}) that can be dated back to the formation of the star or~fields generated by a dynamic effect. Additionally, the field can be generated due to WD mergers (e.g.
~\cite{fw2005}). As large-scale magnetic fields in stars can be unstable, dynamo effects are interesting because they can regenerate these fields.
It is well known that purely poloidal or toroidal fields in stars are both structurally unstable~\cite{MT1973,Tayler1973}. However, magnetized stars and B-WDs with a toroidally dominated mixed field configuration, along with a small poloidal component, are among the most plausible cases~\cite{Wick2013} and also have an approximately spherical shape~\cite{SM2015}. Unlike the surface magnetic field, which can be observationally inferred, fields in the interior of the WD cannot be directly constrained. However, there is a sufficient evidence that stars exhibit dipolar fields in their outer regions and are expected to
have stronger toroidally dominated fields in their interior. Numerical simulations have already shown that the central fields of B-WDs can be several orders of magnitude higher than those at the surface~\cite{SM2015,BM2017,QT2018}.

\subsubsection{Dynamo-Model Based STARS~Simulations}
The evolution of magnetic field components along with the angular momentum
during stellar evolution has been modelled recently with the Cambridge stellar evolution code STARS~\cite{QT2018} using advection-diffusion equations coupled to the structural and compositional equations. It has been shown that the magnetic field is likely to be dipolar in nature, decaying as an inverse square law for most of the star. The~simulation results also suggest that, at~the end of main sequence, stars have toroidally dominated magnetic fields.
The authors reported \cite{QT2018} the evolution of toroidal field in the interior of the WD as a function of mass coordinate at various times in the end of main sequence after the exhaustion of helium nuclei in the core, i.e. during the asymptotic giant phase, as displayed in Figure \ref{fig1} for completeness.

\begin{figure}[H]%
\begin{center}
	\parbox{2.75in}{\includegraphics[width=2.75in]{Fig1_right}}

	\hspace*{1pt}

  \caption{
	Toroidal field inside the star is shown as a function of the mass coordinate at various times after
	the helium exhaustion in the core, during the asymptotic giant phase.
	See \cite{QT2018} for details.}%
 \label{fig1}
 \end{center}
\end{figure}

Large-scale magnetic fields can be present in the degenerate core of magnetised WDs or B-WDs; even during the late stages of stellar evolution~\cite{QT2018} very high fields can develop based on the conservation of magnetic flux, as well as~from the dynamo mechanism. Therefore, strong fields inside magnetized WDs can also be of fossil origin. While the mass of the WD increases due to accretion, the magnetic field is advected into its interior. Consequently, the~gravitational power dominates over the degeneracy pressure, which leads to the contraction of the star. Hence, the~initial seed magnetic field is amplified as the total magnetic flux remains~conserved.

For magnetic field $B \sim 10^8\, {\rm G}$ within a star of size $R \sim 10^6\, {\rm km}$, the resultant flux will be $\sim$$10^{20}\, {\rm G\, km^2}$. From~flux freezing, for~a 1000 km size B-WD, the~magnetic field can then grow up to $\sim$$10^{14}\, {\rm G}$. Once the field increases, the~total outward force further builds up to balance the inward gravitational force, and the whole cycle is repeated multiple times. Therefore, the~magnetic fields of B-WDs are likely to be fossil remnants from their main-sequence progenitor~stars.

\subsubsection{Modified Virial~Theorem}
Virial theorem relates the integrated gravitational potential, thermal, kinetic and magnetic energies of a physical system to provide insight into its equilibrium configuration. By~invoking magnetic flux conservation and based on the
variation of the internal magnetic field with the matter density as a power law, the~modified virial theorem can be derived using the equation of magnetostatic equilibrium~\cite{AS2021}. The~well-known form of the virial theorem is
\begin{equation}
2T + W + 3\Pi + \mu = 0,
\end{equation}
where $T$, $W$, $\Pi$ and $\mu$ are the kinetic, gravitational, thermal and magnetic energies, respectively. For~the case of a static non-rotating WD, we have $T=0$. Assuming that a polytropic EoS is satisfied through the entire star such
that $P = K\rho^\Gamma$, where $K$ and $\Gamma$ are polytropic constants, and~$M=(4\pi/3)R^3 \rho$ with $\rho$ is the mean density, the~scalar virial theorem reduces to
\begin{equation}
-\alpha \frac{GM^2}{R} + \beta \frac{M^\Gamma}{R^{3(\Gamma-1)}} + \gamma \frac{\Phi_M^2}{R} = 0.
\end{equation}

After rearranging the terms, we obtain
\begin{equation}
M = \sqrt{\frac{\gamma \Phi_M^2}{\alpha G \left(1 - \frac{\beta M^{\Gamma-2}}{\alpha G R^{3\Gamma-4}}\right)}}
\end{equation}
for any $\Gamma$. For~$\Gamma = 4/3$, appropriate for extremely relativistic non-magnetic degenerate electrons, we obtain a mass which is independent of radius for a fixed magnetic flux, as~expected from Chandrasekhar's theory.  We evaluate the coefficients $\alpha$, $\beta$ and $\gamma$ to establish the modified virial theorem for a high magnetic~field.

In the presence of a strong magnetic field, the~contribution of the magnetic pressure to the magnetostatic balance cannot be neglected. The~new momentum balance condition, neglecting the effect of magnetic tension, is then given by
\begin{equation}
\frac{1}{\rho}\left(\frac{dP}{dr} + \frac{dP_B}{dr}\right) = - \frac{Gm(r)}{r^2}
\end{equation}
at an arbitrary radius $r$ with mass enclosed at that radius $m(r)$, where $\rho$ includes the contribution from magnetic field and $P_B$ is the pressure due to the magnetic field of the star. We consider two different approaches to compute the modified virial theorem (see~\cite{AS2021}): (i) invoke flux conservation (freezing), which is quite common in stars when conductivity is high, and~(ii) assume
the magnetic field varies as a power law with respect to density, just as the EoS of thermal pressure, throughout.

For the case (i), the~coefficients of the modified virial theorem are obtained as
\begin{equation}
\alpha = \frac{3(\Gamma-1)}{5\Gamma-6},\ \beta = \frac{3^\Gamma K}{(4\pi)^{\Gamma-1}},\ \gamma = \frac{2(n-1)}{5\Gamma-6} + \frac{4n-3}{6}.
\end{equation}
For $n=1$, the situation simplifies to that of a non-magnetized or weakly magnetized WD or a B-WD with constant magnetic field and hence constant $P_B$ throughout. For~power-law fields of case (ii), corresponding coefficients are
\begin{equation}
\alpha = \frac{3(\Gamma_1-1)}{5\Gamma_1-6},\ \beta = \left(1 + \frac{\Gamma - \Gamma_1}{(5\Gamma_1 - 6)(\Gamma-1)}\right)\frac{3^\Gamma K}{(4\pi)^{\Gamma-1}},\ \gamma = \frac{1}{6},
\end{equation}
where $\Gamma_1$ is the constant from the relation $P_B = K_1 \rho^{\Gamma_1}$. It is important to note that $\alpha$ is related to the scaling of $B$ with $\rho$. Therefore, the~presence of magnetic pressure allows either a more massive or smaller star. For~$\Gamma=\Gamma_1$, the~result reduces to that of the non-magnetic case with a redefined $K$. Figure~\ref{fig2} shows the variation of radius with indices $n$ and $\Gamma_1$ for the conserved flux and power-law field models, respectively, with~various $\Gamma$ and total~masses.

\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.56in]{Fig2_left}
&\includegraphics[width=2.56in]{Fig2_right}\\
\end{tabular}
  \caption{
	{\it Left panel:} Variation of radius with $n$ for the conserved flux model with various $\Gamma$ and total masses.
	{\it Right panel:} Variation of radius with $\Gamma_1$ for the power-law field model with various $\Gamma$ and total masses.
	In each case, $\Gamma = 4/3$ corresponds to $\overline{B} = 10^{14}\ {\rm G}$, and $\Gamma= 2$ corresponds to $\overline{B} = 10^{16}\ {\rm G}$, with
	$\overline{B}$ being the average magnetic field. Parameter sets from
	top to bottom mentioned in the figures correspond to the lines
	sequentially from bottom to top in the low $n$ and $\Gamma_1$ regimes. See~\cite{AS2021}.}%
 \label{fig2}
\end{figure}

\subsection{Evolution of Magnetic~Field}
Highly magnetised WDs can possibly result from repeated episodes of accretion and spin-down~\cite{BM2017}. The~entire evolution of the B-WD can be classified in two phases: accretion-powered and rotation-powered. The~accretion-powered phase is governed by three conservation laws: linear and angular momenta conservation and conservation of magnetic flux around the stellar surface, given as
\begin{equation}
l\Omega(t)^2 R(t) = \frac{GM(t)}{R(t)^2},\ I(t)\Omega(t) = {\rm constant},\ B_s(t)R(t)^2 = {\rm constant},
\end{equation}
where $l>1$ accounts for the dominance of gravitational force over the centrifugal force, $I$ is the stellar moment of inertia and $\Omega$ is the angular velocity of the star that also includes the contribution acquired due to accretion. Here, we neglect any possible field decay and assume a specific field
geometry. Solving the above equations simultaneously gives the time evolutions of the radius, magnetic field and angular velocity during the accretion phase. Accretion discontinues when
\begin{equation}
-\frac{GM}{R^2} = \frac{1}{\rho} \frac{d}{dr} \left(\frac{B^2}{8\pi}\right)_{r=R} \sim -\frac{B_s^2}{8\pi R\rho},
\end{equation}
where $\rho$ is the density of the inner disk~edge.

\vspace{-3pt}
\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.35in]{MRB17_Fig2a}
&\includegraphics[width=2.35in]{MRB17_Fig2b}\\
\end{tabular}
	\caption{Time
 evolution of angular velocity (left panel) and magnetic field (right panel) as functions of mass. The~solid curves correspond to the case with $n=3$, $m=2.7$, $\rho=0.05\, {\rm g\, cm^{-3}}$, $l=1.5$ and dotted curves correspond to the case with $n=3$, $m=2$, $\rho=0.1\, {\rm g\, cm^{-3}}$, and $l=2.5$. The~other parameters are fixed with $k=10^{-14}$, $\dot{M}=10^{-8}\, M_{\odot} {\rm yr}^{-1}$, $\chi=10^{\circ}$ and $R=10^4\, {\rm km}$ at $t=0$. See~\cite{BM2017}.}%
 \label{fig3}
\end{figure}

If the magnetic field is dipolar, $\dot{\Omega} \propto \Omega^3$ for a fixed magnetic field. Generalizing it to $\dot{\Omega} = k\Omega^n$ with constant $k$ giving for the spin-powered phase, we obtain
\begin{eqnarray}
	\Omega &=& [\Omega_0^{1-n} - k(1-n)(t-t_0)]^{1/1-n}, \\
	{\rm and}\,\,\,B_s &=& \sqrt{\frac{5c^3 I k \Omega^{n-m}}{R^6 {\rm sin}^2 \chi}}.
\end{eqnarray}

Here, $\Omega_0$ is the initial angular velocity for the spin-powered phase (once accretion stops) at time $t=t_0$, and $\chi$ is the angle between the rotation
and magnetic axes. The value of $k$ is fixed such that $B_s$ can be constrained at $t=t_0$, which is known from the field evolution in the preceding accretion-powered phase. Here, $n=m=3$ corresponds to the dipole field configuration; therefore $m$ represents the deviation from the dipolar field, especially for $n=3$. Figure~\ref{fig3} shows the sample evolutions of angular velocity and magnetic field as functions of stellar mass.
The initial angular velocity and particularly the magnetic field are chosen in such a way that they do not affect the stellar structure with~respect to when they are zero. In both cases, initially larger $\Omega$ with accretion drops significantly during the spin-powered phase, followed by a phase of its increasing trend. At~the end of the evolution, the~star can be left either as a super-Chandrasekhar WD and/or an SGR/AXP candidate with a higher spin frequency. Other initial conditions do not produce qualitatively different results.

\section{Simulating Magnetized and Rotating White Dwarfs and Their~Stability}
\label{Sec3}
In nature, WDs are expected to consist of mixed field geometry, including both toroidal and poloidal magnetic field components. However, here we consider toroidally dominated magnetic fields as they ensure the stability of these stars.
Note, in fact, that the core of WDs may contain a stronger toroidal field,
which contributes to the stellar structure, mostly including mass, with~a weaker
surface poloidal field. We simulate WDs and B-WDs based on the Einstein equation solver, \emph{XNS} code~\cite{Pili2014}.
It has been shown that toroidally dominated and purely toroidal fields not only make the star slightly prolate but also increases its equatorial radius~\cite{KS2004,FR2012,SM2015}. It is seen that the deformation at the core is more prominent than the outer region. Nevertheless, the rotation of a star makes it oblate, and hence, there is always a competition between these two opposing effects to decide whether the star will be an overall oblate or~prolate.

The combined effect of toroidal field and rotation (uniform and
differential) was already explored in detail earlier \cite{Kalita2019}.
It was shown that as the angular frequency is small, it does not affect the
star considerably and results in a marginally prolate star due to magnetic
effects. However, for high angular velocity, the low density region is affected more due to rotation than the high density central region, resulting in an
oblate shaped star.
For both the explorations, the~magnetic to gravitational energies ratio (ME/GE) as well as kinetic to gravitational energies ratio (KE/GE) are chosen to be $\lesssim 0.1$ to maintain stable equilibrium~\cite{ChF1953,Komatsu1989,Braithwaite2009}. The authors found that for the central density $\rho_c \sim 2.2\times10^{10}\, {\rm g\ cm^{-3}}$, the magnetic field at the interior (close to the center) of the WD is $B_{max} \sim 2.7\times10^{14}\, {\rm G}$. Several WDs are observed with the surface magnetic field $\sim 10^9\, {\rm G}$
\cite{Heyl2000,Van2005,Brink2013}; hence, the central field might be much larger than $10^9\, {\rm G}$. In~fact, it has already been argued in the literature that the central field could be as large as $10^{14}\text{--}10^{15}\, {\rm G}$ \cite{FS2015,SS2017}.

For differentially rotating B-WDs~\cite{SM2015}, the~angular velocity profile is specified as~\cite{Ster2003,BDZ2011}
\begin{equation}
F(\Omega) = A^2 (\Omega_c - \Omega) = \frac{R^2 (\Omega - \omega)}{\alpha^2 - R^2 (\Omega- \omega)^2},
\end{equation}
which is implemented in the \emph{XNS} code used to simulate them,
where $A$ is a constant that indicates the extent of differential rotation, $R = \psi^2 r {\rm sin}\theta$, $\omega = -\beta^{\phi}$, $\Omega_c$ is the angular velocity at the center and $\omega$ is the angular velocity at radius $r$.
Previous works (e.g. \cite{DM2015,SM2015,Kalita2019}) reported in detail the density isocontours
of differentially rotating B-WDs for toroidal and poloidal fields.
It was shown that `polar hollow' structure can form with differential rotation regardless of the specific geometry of the magnetic field.

\section{Mass-Limit and Luminosity Suppression in Non-Rotating White~Dwarfs}
\label{Sec4}
Apart from increasing the limiting mass of WDs, strong magnetic fields can also influence the thermal properties of the star, such as its luminosity, temperature gradient and cooling rate~\cite{MB2018,MB_euro2018,AG2020,MB2021,MB2022}. To~model such a WD, the~total pressure inside the star is modelled by including the contributions from the degenerate electron gas, ideal gas and magnetic pressures. The~interface is at the radius where the contributions from electron degenerate core and outer ideal gas pressures are the same. The~presence of strong fields gives rise to additional pressure $P_B = B^2/8\pi$ and density $\rho_B = B^2/8\pi c^2$ inside the magnetized WDs~\cite{Sinha2013} when $c$ is the speed of~light.

For an approximately spherical B-WD, the~model equations for magnetostatic equilibrium, photon diffusion and mass conservation can be written within a Newtonian framework as
\begin{eqnarray}
&\frac{{\rm d}}{{\rm d}r}(P_{\rm deg}+P_{\rm ig}+P_{B}) = -\frac{Gm(r)}{r^2}(\rho + \rho_B), \nonumber \\
&\frac{{\rm d}T}{{\rm d}r} = -{\rm max}\left[\frac{3}{4ac} \frac{\kappa \rho}{T^3} \frac{L_r}{4\pi r^2}, \left(1 - \frac{1}{\gamma}\right)\frac{T}{P} \frac{{\rm d}P}{{\rm d}r}\right], \vspace{0.1in} \\
&\frac{{\rm d}m}{{\rm d}r} = 4\pi r^2 (\rho + \rho_B), \nonumber
\end{eqnarray}
where we have ignored the magnetic tension term for radially varying magnetic field strength. In~these equations, $P_{\rm deg}$ and $P_{\rm ig} = \rho kT/\mu m_p$ are the electron degeneracy pressure and the ideal gas pressure, $\rho$ is the matter density, $T$ is the temperature, $m(r)$ is the mass enclosed within radius $r$, $\kappa$ is the radiative opacity, $L_r$ is the luminosity at radius $r$, and~$\gamma$ is the adiabatic index of the~gas.

The opacity in the surface layers of non-magnetised WD is approximated with Kramers' formula, $\kappa = \kappa_0 \rho T^{-3.5}$, where $\kappa_0 = 4.34\times10^{24}Z(1+X)\ {\rm cm^2\ g^{-1}}$, and~$X$ and $Z$ are the mass fractions of hydrogen and heavy elements (other than hydrogen and helium) in the stellar interior, respectively. Assuming helium WDs for our study here, we set the helium mass fraction to $Y = 0.9$ and $Z = 0.1$. The~radiative opacity in the surface layers is mainly due to the bound-free and free-free transitions of electrons~\cite{ST1983}.
For the radial variation of the field magnitude within the B-WD, we adopt a profile used extensively to model magnetized NSs and B-WDs~\cite{Band1997,DM2014},
given by
\begin{equation}
B\left(\frac{\rho}{\rho_0}\right) = B_{\rm s} + B_0 \left[1 - {\rm exp}\left(-\eta \left(\frac{\rho}{\rho_0}\right)^{\gamma}\right)\right],
\label{Bprof}
\end{equation}
where $B_s$ is the surface magnetic field, $B_0$ is a fiducial magnetic field, and $\eta$ and $\gamma$ are dimensionless parameters along with $\rho_0=10^9\, {\rm g/cm^3}$, which determine how the magnetic field decays from the degenerate core to the surface. We set $\eta = 0.8$ and $\gamma = 0.9$ for all~calculations, unless stated otherwise.

Radial luminosity is assumed to be constant, $L_r = L$, as~there is no hydrogen burning or other nuclear fusion reactions that take place within the WD core. The~differential equations are solved by providing the matter density at the surface, total WD mass and surface temperature as the boundary conditions. For~strong magnetic fields, the~variation of radiative opacity with $B$ can be modelled as $\kappa = \kappa_B \approx 5.5\times10^{31}\rho T^{-1.5} B^{-2} {\rm cm^{2}\ g^{-1}}$~\cite{PY2001,VP2001}. The~field-dependent Potekhin's opacity is used instead of Kramers' opacity if $B/10^{12}\ {\rm G} \geq T/10^6\ {\rm K}$, which is relevant for the strong fields that we~consider.

The left panel of Figure~\ref{fig6} shows the effect of luminosity on the mass-radius relation for non-magnetized WDs compared to Chandrasekhar's results~\cite{Ch1935}. Although~the increase in $L$ leads to progressively higher masses for larger WDs, Chandrasekhar mass limit is retained irrespective of the luminosity. The~right panel of Figure~\ref{fig6} shows the effect of magnetic field on the mass-radius relation for B-WDs with $L=10^{-4}\, L_{\odot}$ and compares them with the non-magnetic Chandrasekhar results. It can be seen that the magnetic field affects the mass-radius relation in a manner analogous to increasing $L$ by~shifting the curve towards higher masses for these WDs having larger radii. The~mass-radius curves for $B_0 \lesssim 10^{13}\, {\rm G}$ practically overlap with each other in the smaller radius regime and retain the Chandrasekhar mass limit. However, for~strong central fields with $B_0 \sim 10^{14}\, {\rm G}$, super-Chandrasekhar WDs are obtained, with masses as high as $\sim$$1.9\, M_{\odot}$.

\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.6in]{MB21_Fig1a}
&\includegraphics[width=2.6in]{MB21_Fig3}\\
\end{tabular}
  \caption{\emph{Left panel:} The effect of $L$ on the mass--radius relation of non-magnetised WDs is shown for $L=10^{-4}\ L_{\odot}$ (blue diamonds), $L=10^{-3}\ L_{\odot}$ (green circles) and $L=10^{-2}\ L_{\odot}$ (red triangles), along with the Chandrasekhar result (black squares). \emph{Right panel:} The effect of field strength on the mass--radius relation of B-WDs is shown for
	$B=(B_s,B_0)=(0,0)$ (blue diamonds), $B=(10^9,10^{13})\ {\rm G}$ (orange crosses), $B=(10^7,10^{14})\ {\rm G}$ (green circles) and $B=(10^9,10^{14})\ {\rm G}$ (red pluses), along with the Chandrasekhar result (black squares) for $L=10^{-4}\ L_{\odot}$. See~\cite{MB2022}.}%
\label{fig6}
\end{figure}

To ensure structural stability of B-WD, an~increase in magnetic energy density has to be compensated by a corresponding decrease in the thermal energy and hence the luminosity. This effect is especially prominent for B-WDs with larger radii where the magnetic, thermal and gravitational energies are comparable with each other. We find that in the presence of stronger field, a~slight decrease in the luminosity for $R \gtrsim 12000\, {\rm km}$ WDs leads to masses that are similar to their non-magnetic counterparts. However, the~smaller radius B-WDs require a substantial drop in their luminosity (well outside the observable range) and still do not really achieve masses that are similar to the non-magnetized WDs.
As a result, for~stars with $2000 \leq R/{\rm km} \leq$~10,000, even if the luminosity decreases substantially with $10^{-16} \leq L/L \leq 10^{-12}$, the~resulting mass of the B-WD remains larger than its non-magnetic counterpart. This leads to an extended branch in the mass-radius~relation.

\section{Effect of Magnetic Field Dissipation and Cooling~Evolution}
\label{Sec5}
Magnetic fields inside WD undergo decay by Ohmic dissipation and Hall drift processes with timescales given by~\cite{HK1998,Cumming2002}, respectively,
\begin{eqnarray}
t_{\rm Ohm} = (7\times10^{10}\ {\rm yr})\, \rho_{c,6}^{1/3} R_{4}^{1/2} (\rho_{\rm avg}/\rho_{\rm c}),\\
t_{\rm Hall} = (5\times10^{10}\, {\rm yr})\ l_8^2 B_{0,14}^{-1} T_{\rm c,7}^{2} \rho_{\rm c,10},
\label{tOhm_tHall}
\end{eqnarray}
where $\rho_{\rm c,n} = \rho_c/10^n\, {\rm g\, cm^{-3}}$, $R_4 = R/10^4\, {\rm km}$, $T_{c,7}=T_c/10^7\, {\rm K}$, $B_{0,14}=B_0/10^{14}\, {\rm G}$ and $l = l_8 \times 10^8\, {\rm cm}$ is the characteristic length scale of the flux loops through the outer core of WD. Ohmic decay is the dominant field dissipation process for $B \lesssim 10^{12}\ {\rm G}$, while for $10^{12} \leq B/{\rm G} \leq 10^{14}$, the decay occurs via Hall drift, and for $B \gtrsim 10^{14}\ {\rm G}$, the~principal decay mechanism is likely to be ambipolar diffusion~\cite{HK1998}.

The field decay can be solved using
\begin{equation}
\frac{{\rm d}B}{{\rm d}t} = -B\left(\frac{1}{t_{\rm Ohm}} + \frac{1}{t_{\rm Amb}} + \frac{1}{t_{\rm Hall}}\right),
\end{equation}
where $t_{\rm Amb}$ denotes the ambipolar diffusion time scale. We consider two separate cases: (a) when only Ohmic dissipation occurs for both the surface and central magnetic fields, and (b) while $B_s$ continues to evolve over $t_{\rm Ohm}$, Hall drift determines the $B_0$ evolution until the central field drops to about $10^{12}\, {\rm G}$,
below which Ohmic dissipation sets~in.

The thermal energy is radiated away gradually over time in the observed luminosity from the surface layers as the star evolves. Because~most of the electrons occupy the lowest energy states in a degenerate gas, the~thermal energy of ions is the only significant energy source that can be radiated. The~rate at which thermal energy of ions is transported to surface and radiated depends on the specific heat, given by
\begin{equation}
L = -\frac{d}{dt}\int c_{v} dT = (2\times10^6\ {\rm erg/s)}\, \frac{Am_{\mu}}{M_{\odot}}\left(\frac{T}{K}\right)^{7/2},
\end{equation}
where $c_v \approx 3k_B/2$ is the specific heat at constant volume. Given an initial $L$ and temperature $T_0$ at time $t_0$, final temperature after cooling is given by $(T/{\rm K})^{-5/2} - (T_0/{\rm K})^{-5/2} = 2.406\times10^{-34}\, \tau/{\rm s}$, where $\tau = t-t_0$ is the WD age.  It is important to note that for simplicity in calculations, we have assumed self-similarity of the cooling process over the entire~evolution.

Table~\ref{Table1} lists the luminosities and masses for WDs with radii $2000 \lesssim R/{\rm km} \lesssim$~20,000 and initial $B=(10^9,10^{14})\, {\rm G}$ at~time $t=0$ and $10\, {\rm Gyr}$. The~fraction of time when the Hall drift dominates, i.e.,~$t_{\rm Hall}/\tau$, falls significantly with increasing stellar radius. Therefore, the~magnetic field decays considerably more because the faster Ohmic dissipation process turns out to be critical for much of the cooling evolution. The~mass-radius relations merge for $R \gtrsim 6000\, {\rm km}$ WDs. The~inferred luminosities are also much less suppressed for the intermediate radius WDs with $6000 \lesssim R/{\rm km} \lesssim 12000$. However, for~low radius B-WDs,
field decay affects mass and luminosity significantly. As~it is the high
magnetic pressure which helps to hold more mass, the~field decay significantly
sheds off mass in its new equilibrium if the radius is fixed. The~limiting
mass for small radius B-WDs with a fixed $R \approx 2000\, {\rm km}$ drops to about
$1.5\ M_{\odot}$ due to field decay compared to $1.9\ M_{\odot}$ without field evolution.
The majority of these small radius B-WDs still remain practically hidden throughout their cooling evolution because of their strong fields and correspondingly low luminosity.

\begin{table}[H]
\caption{The effect of magnetic field on luminosity when the initial field is fixed at $B=(10^9,10^{14})\ {\rm G}$ for all the radii. The topmost entry for each radius is the initial time, whereas the bottom two entries list corresponding parameters for $t=\tau=10\ {\rm Gyr}$. We evaluate fields assuming Ohmic dissipation is the dominant process for the top entries of $\tau=10\ {\rm Gyr}$, and~for the bottom entries, we assume Hall drift is the primary process until the field parameter $B_0$ decays to $\sim$$10^{12}\ {\rm G}$, below~which Ohmic dissipation dominates.
}
\label{Table1}
\bgroup
\def1.2{1.2}
\begin{tabular}{c c c c c c c c }
\hline
\hline
\centering
\textbf{\emph{R}/1000 km} & \boldmath{$t/{\rm Gyr}$} & \boldmath{$t_{\rm Hall}/\tau$ }
& \boldmath{$B_{\rm s}/{\rm G}$} & \boldmath{$B_0/{\rm G}$} & \boldmath{$L/L_{\odot}$ }& \boldmath{$M_{B=0}/M_{\odot}$ }& \boldmath{$M/M_{\odot}$} \\ \midrule
2.0 & 0 & & $10^9$ & $10^{14}$ & $10^{-16}$ & 1.378 & 1.865 \\ \noalign{\smallskip} \cmidrule{2-8} \noalign{\smallskip}
    & 10 & 0 & $4.58\times10^8$ & $4.58\times10^{13}$ & $10^{-16}$ & 1.377 & 1.478 \\
    & & 1 & & $5.83\times10^{13}$ & $10^{-16}$ & & 1.542 \\ \midrule
8.0 & 0 & & $10^9$ & $10^{14}$ & $10^{-12}$ & 0.709 & 0.762 \\ \noalign{\smallskip} \cmidrule{2-8} \noalign{\smallskip}
    & 10 & 0 & $9.86\times10^7$ & $9.86\times10^{12}$ & $2\times10^{-6}$ & 0.699 & 0.699 \\
    & & $2.28\times10^{-2}$ & & $1.04\times10^{11}$ & $7\times10^{-6}$ & & 0.699 \\ \midrule
14.0 & 0 & & $10^9$ & $10^{14}$ & $2\times10^{-6}$ & 0.286 & 0.286 \\ \noalign{\smallskip} \cmidrule{2-8} \noalign{\smallskip}
    & 10 & 0 & $3.06\times10^7$ & $3.06\times10^{12}$ & $10^{-5}$ & 0.262 & 0.262 \\
    & & $2.22\times10^{-3}$ & & $3.08\times10^{10}$ & $10^{-5}$ & & 0.262 \\ \midrule
20.0 & 0 & & $10^9$ & $10^{14}$ & $7\times10^{-6}$ & 0.164 & 0.164 \\ \noalign{\smallskip} \cmidrule{2-8} \noalign{\smallskip}
    & 10 & 0 & $2.97\times10^7$ & $2.97\times10^{12}$ & $10^{-5}$ & 0.138 & 0.138 \\
    & & $3.70\times10^{-4}$ & & $2.98\times10^{10}$ & $10^{-5}$ & & 0.138 \\
			\bottomrule
\end{tabular}
\egroup
\end{table}

The left panel of Figure~\ref{fig7} shows the effect of B-WD evolution on their mass-radius relations, including both magnetic field decay and thermal cooling effects but neglecting neutrino cooling.
The luminosities are varied with field strength such that the B-WD masses can match those obtained for the non-magnetized WDs. For~$B=(0,0)\, {\rm G}$, the~mass-radius relation is shifted more towards the Chandrasekhar result as a result of cooling, and the mass limit remains unaltered. However, for~$B=(10^9,10^{14})\, {\rm G}$, although~the maximum mass $\sim$$1.9\, M_{\odot}$ at a small radius turns out to be much larger than the Chandrasekhar limit, we find that it is lowered considerably to $\sim$$1.5\, M_{\odot}$ primarily as a result of magnetic field decay and also thermal cooling over $t=10\, {\rm Gyr}$. The~right panel of Figure~\ref{fig7} shows the radial variation of matter density for the same cases. Matter density at the core is slightly suppressed in the presence of strong field and also as a result of the evolution. As~the total stellar energy is conserved, an~increase in the magnetic energy has to be compensated by a similar decrease in gravitational energy, and, hence, the central density. Once the field decays and luminosity drops due to cooling, the~central density adjusts itself to be slightly lower to balance the loss of magnetic and thermal energies with~time.

We also use the STARS stellar evolution code to qualitatively investigate the B-WD mass-radius relationship at different field strengths, with~the objective of numerically validating our semi-analytical models. We find that the numerical results are in good agreement with our analytical formalism, and the magnitude of $B_0$ dictates the shape of the mass-radius curve. In~validation of our analytical approach, we have found that the limiting mass $\sim$$1.8703 M_{\odot}$ obtained with the STARS numerical models is in very good agreement with $M \approx 1.87 M_{\odot}$, which is inferred from the semi-analytical calculations for B-WDs with strong fields $B=(10^{\text{6--9}},10^{14})\, {\rm G}$ \cite{MB2022} for a given magnetic field profile.

We argue that the young super-Chandrasekhar B-WDs only sustain their large masses up to $\sim$$10^{5}\text{--}10^6\, {\rm yr}$ since their formation, and this essentially explains their apparent scarcity even without the difficulty of detection owing to their suppressed luminosities. We plan to explore this
issue in detail in the future, particularly the timescale of their formation
considering simultaneously the growth (e.g., by accretion~\cite{BM2017}) and
decay of fields.
It is important to note that if matter accretes at a higher
rate such that the total mass accreted exceeds $\sim$0.1--0.2$M_\odot$ before
the field diffuses,
the field may be restructured~\cite{zwf}, leading to a reduced polar field
independent of the initial field. We also plan to rigorously explore the
fate of B-WDs once the fields decay, if~they actually collapse, in~place of
keeping the radius~fixed.

\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.6in]{MB21_Fig5}
&\includegraphics[width=2.6in]{MB21_Fig6a}\\
\end{tabular}
	\caption{\emph{Left panel:} Effect of magnetic field on $L$ set to match the non-magnetised mass-radius relation. Results are shown for $B=(0,0)$ at $t=0$ (green circles), $B=(0,0)$ at $t=10\ {\rm Gyr}$ (blue diamonds), $B=(10^9,10^{14})\, {\rm G}$ at $t=0$ (orange triangles) and $B=(10^9,10^{14})\, {\rm G}$ at $t=10\ {\rm Gyr}$ (magenta crosses). \emph{Right panel:} The matter density profiles for the same cases are shown for \emph{R} = 10,000 km. See~\cite{MB2022}.}%
\label{fig7}
\end{figure}

\section{Mass Limit Estimate Including Quantum~Effects}
\label{Sec6}

\textls[-15]{Several magnetized WDs have been discovered with surface fields as high as $\sim$$10^5\text{--}10^9\, {\rm G}$}. It is likely that stronger fields ($\sim$$10^{12}\text{--}10^{14}\, {\rm G}$) exist at their interiors. Energy states of a free electron in a uniform strong magnetic field are quantized into Landau orbitals, which defines the motion of the electron in a plane perpendicular to the field. The maximum number of Landau levels occupied by cold electrons in a magnetic field is given by $\nu_m = \frac{(E_{Fmax}/m_e c^2)^2 - 1}{2 B_D}$, where $m_e$ is the rest mass of the electron, $E_{Fmax}$ is the maximum Fermi energy of the system and $B_D = B/B_c$, where $B_c = 4.414\times10^{13}\, {\rm G}$, is the critical field strength. High magnetic field strength therefore modifies the equation of state (EoS) of degenerate matter by causing Landau quantization of electrons~\cite{LS1991}. The larger the magnetic field, the smaller is the number of Landau levels occupied~\cite{DM2013}. This results in a significant modification of the mass-radius relation of the underlying WD, particularly for $B_D\gtrsim 100$.

To obtain the revised mass-radius relation for a strongly magnetized WD, the~modified EoS has to be combined with the condition of magnetostatic equilibrium. If~the B-WD is approximated to be spherical, then its mass is obtained from
\begin{equation}
\frac{1}{\rho+\rho_B}\frac{d}{dr}\left(P+\frac{B^2}{8\pi}\right) = -\frac{GM}{r^2} + \left[\frac{\vec{B}\ . \nabla \vec{B}}{4\pi (\rho+\rho_B)}\right]_{r}, \ \frac{dM}{dr} = 4\pi r^2 (\rho + \rho_B).
\end{equation}

If the field is uniform or highly fluctuating, the~magnetic terms can be neglected in the above equations, and~following Lane--Emden formalism~\cite{ARC2010}, the~scalings of mass and radius with central density are obtained as
\begin{equation}
M \propto K_m^{3/2}\rho_c^{(3-n)/2n}, \ R \propto K_m^{1/2} \rho_c^{(1-n)/2n}, \ K_m = K\rho_c^{-2/3}.
\end{equation}

Here, $n=1$ ($\Gamma=2$), which is the case for a high $B_D$ as shown in
Figure~\ref{fig8}, corresponds to central density independent mass, unlike Chandrasekhar's case when $K_m$ is independent of the field strength, and limiting mass corresponds to $n=3$.

\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.45in]{BM13_Fig1}
&\includegraphics[width=2.69in]{BM13_Fig2}\\
\end{tabular}
	\caption{\emph{Left panel:} EoS for different cases: solid line represents Chandrasekhar's EoS, and the~dotted and dashed lines represent the 5-level (corresponding to very strong $B$) and 1-level (corresponding to ultimate EoS for extreme $B$) systems of Landau quantization, respectively. \emph{Right panel:} Mass-radius relations: pure solid line represents Chandrasekhar's result, and the one marked with filled circles represents the evolutionary track of WD with an increase in $B$. The~dot-dashed, dotted and dashed lines represent WDs with 50124-level, 200-level and 1-level systems of Landau quantization, respectively (corresponding to increasing $B$). We set $E_{Fmax}=200 m_e c^2$ for both panels. See~\cite{DM2013,DM2013b}.}%
\label{fig8}
\end{figure}

Substituting the proportionality constants appropriately, the~limiting mass is obtained~as
\begin{equation}
M_{lim} = \left(\frac{hc}{2G}\right)^{3/2} \frac{1}{(\mu_e m_H)^2} \approx \frac{10.312}{\mu_e^2}M_{\odot},
\end{equation}
when the limiting radius $R_{lim} \rightarrow 0$. For~$\mu_e=2$, which is the case for a carbon-oxygen WD, we obtain $M_{lim} = 2.58 M_{\odot}$. For~a finite but high-density and magnetic field, e.g.,~$\rho_c = 2\times10^{10}\, {\rm g\ cm^{-3}}$ and $B=8.8\times10^{15}\, {\rm G}$ when $E_{Fmax}=20m_e c^2$, $M_{lim}=2.44 M_{\odot}$ and $R \approx 650\, {\rm km}$. It should be noted that these $\rho_c$ and $B$ are similar or below their respective upper limits set by the instabilities of pycnonuclear fusion, inverse-$\beta$ decay and general relativistic effects~\cite{DM2014}. Note, however, that pycnonuclear reaction rates are quite
uncertain and not well constrained. Moreover,~slightly lower $\rho_c$
and $B$ still would lead to B-WD mass well above the Chandrasekhar~mass-limit.

The left panel of Figure~\ref{fig8} shows how the EoS of electron degenerate matter is modified as a result of magnetic field. With~increasing field, the~inward gravitational force is balanced by the outward force due to modified matter
pressure, and a quasi-equilibrium state is attained. Consequently, with~
decreasing stellar radius due to a higher gravitational field, a~very high magnetic field is generated, which prevents the WD from collapsing, thus making its
equilibrium configuration more massive. Subsequently, with~the continuation of accretion the WD approaches the new mass limit $M_{lim} \sim 2.58M_{\odot}$.
This new limiting mass WD plausibly sparks off a violent thermonuclear reaction with further accretion, as is the case with the idea of the $1.4M_\odot$ limit of nonmagnetic WDs,
thus exploding it and giving rise to an over-luminous type Ia~supernova.

The evolution of the mass-radius relationship with the evolution of magnetic field
for a super-Chandrasekhar WD of maximum possible mass is shown in the right panel of Figure~\ref{fig8}, along with a few typical mass-radius relations for different fixed magnetic field strengths describing possible stars in intermediate equilibrium states. The~ultimate WD, corresponding to the maximum mass $M_{lim} \sim 2.58M_{\odot}$, lies on the mass-radius relation for a one-Landau-level system, but~the intermediate WDs having weaker magnetic fields correspond to multilevel systems. The~said one-Landau-level system corresponds to the hypothetical high central magnetic field $8.8\times10^{17}\, {\rm G}$. This is in the spirit of a Chandrasekhar mass-limit for a non-magnetic WD, which also
corresponds to the hypothetical high (even infinity) central density. However, the~mass for
$B_D\sim 30-100$ still appears to be significantly super-Chandrasekhar. The~intermediate systems of 200-level and 50124-level systems correspond to central magnetic fields of $4.4\times10^{15}\, {\rm G}$ and $1.7\times10^{13}\, {\rm G}$, respectively.

\section{Detectability of Gravitational Waves from Magnetised White~Dwarfs}
\label{Sec7}
One question that remains to be answered is how can B-WDs be detected directly.
Continuous gravitational waves can be among the alternate ways to detect super-Chandrasekhar WD candidates. If~B-WDs are rotating with certain angular frequency, then they can efficiently emit gravitational radiation, provided that their magnetic field and rotation axes are not aligned~\cite{BG1996}, and~these gravitational waves can be detected by upcoming instruments.
The dimensionless amplitudes of the two polarizations of a gravitational wave (GW) at a time $t$ are given by~\cite{BG1996,ZS1979}
\begin{eqnarray}
h_+ = h_0 {\rm sin}\,\chi \, \left[\frac{1}{2}{\rm cos}\,i\, {\rm sin}\,i \,{\rm cos}\,\chi \, {\rm cos}\,\Omega t - \frac{1+{\rm cos}^2 i}{2}{{\rm sin}\,\chi\,}{\rm cos}\,2\Omega t\right], \nonumber \\
h_{\times} = h_0 {\rm sin}\,\chi \left[\frac{1}{2}{\rm sin}\,i\, {\rm cos}\,\chi\, {\rm sin}\,\Omega t - {\rm cos}\,i\, {\rm sin}\,\chi\, {\rm sin}\,2\Omega t \right],
\end{eqnarray}
with $h_0 = (-6G/c^4)Q_{z^{\prime}z^{\prime}}(\Omega^2/d)$, where $Q_{z^{\prime}z^{\prime}}$ is the quadrupole moment of the distorted star, $\chi$ is the angle between the rotation axis $z^{\prime}$ and the body's third principal axis $z$, and $i$ is the angle between the rotation axis of the object and our line of sight. The~left panel of Figure~\ref{fig9} shows a schematic diagram of a pulsar with $z^{\prime}$ being the rotational axis and $z$ the magnetic field axis, where the angle between these two axes is $\chi$. The~GW amplitude is
\begin{equation}
h_0 = \frac{4G}{c^4} \frac{\Omega^2 \epsilon I_{xx}}{d},
\end{equation}
where $\epsilon = (I_{zz} - I_{xx})/I_{xx}$ is the ellipticity of the body, and $I_{xx}$, $I_{yy}$, $I_{zz}$ are the principal moments of inertia. We have used \emph{XNS} code~\cite{Pili2014} to simulate the underlying axisymmetric equilibrium configuration of B-WDs in general relativity. Moreover, we assume the distance between the WD and the detector to be 100~pc.

\begin{figure}[H]%
\begin{tabular}{cc}
\hspace{-30pt}\includegraphics[width=2.2in]{KM20_fig1}
&\includegraphics[width=2.8in]{KM20_fig3}\\
\end{tabular}
	\caption{\emph{Left panel:} Schematic diagram of a B-WD with $z^{\prime}$ being the rotational axis and $z$ as the magnetic axis. \emph{Right panel:} The dimensionless GW amplitudes for WDs are shown as functions of frequency, along with the sensitivity curves of various detectors. Optimum $i$ is chosen for $\chi$ at $t=0$. See~\cite{Kalita2020}.}%
\label{fig9}
\end{figure}

Since a pulsating WD can emit both dipole and GW radiations simultaneously, it is associated with both dipole and quadrupolar luminosities. Dipole luminosity for an axisymmetric WD is~\cite{Melatos2000}
\begin{equation}
L_D = \frac{B_p^2 R_p^6 \Omega^4}{2c^3} {\rm sin}^2 \, \chi \, F(x_0),
\end{equation}
where $x_0 = R_0 \Omega/c$, $B_p$ is the magnetic field strength at the pole, $R_p$ is the radius of the pole and $R_0$ is the average WD radius. The~function $F(x_0)$ is defined as
\begin{equation}
F(x_0) = \frac{x_0^4}{5(x_0^6 - 3x_0^4 + 36)} + \frac{1}{3(x_0^2 + 1)}.
\end{equation}

Similarly, the~quadrupolar GW luminosity is given by~\cite{ZS1979}
\begin{equation}
L_{GW} = \frac{2G}{5c^5}(I_{zz} - I_{xx})^2 \Omega^6 {\rm sin}^2\, \chi \, (1+15{\rm sin}^2 \, \chi).
\end{equation}

It should be noted that this formula is valid if $\chi$ is very small. The~total luminosity is due to both dipole and gravitational radiations. Therefore,
$\Omega$ and $\chi$ decay with time due to both $L_D$ and $L_{GW}$, given by~\cite{Melatos2000}
\begin{eqnarray}
\nonumber
\frac{d(\Omega I_{z^{\prime}z^{\prime}})}{dt} = -\frac{2G}{5c^5}(I_{zz}-I_{xx})^2 \Omega^5 {\rm sin}^2\, \chi \, (1+15\, {\rm sin}^2\, \chi) - \frac{B_p^2 R_p^6 \Omega^3}{2c^3}{\rm sin}^2\, \chi \, F(x_0), \\
\\
\nonumber
I_{z^{\prime}z^{\prime}}\frac{d\chi}{dt} = -\frac{12G}{5c^5}(I_{zz} - I_{xx})^2 \Omega^4 {\rm sin}^3\, \chi\, {\rm cos}\, \chi - \frac{B_p^2 R_p^6 \Omega^2}{2c^3}{\rm sin}\, \chi \, {\rm cos}\, \chi \, F(x_0),\\
\end{eqnarray}
where $I_{z^{\prime}z^{\prime}}$ is the moment of inertia about the $z^{\prime}$-axis. Equations~(26) and (27) need to be solved simultaneously to obtain the timescale over which a WD can~radiate.

The right panel of Figure~\ref{fig9} shows the dimensionless GW amplitudes for WDs as functions of their frequencies, along with the sensitivity curves of various detectors. We find that the isolated WDs may not be detected directly by LISA but~can be detected after integrating the signal-to-noise ratio (SNR) for 1 year. As~WDs are larger in size compared to NS, they cannot rotate as fast as NS and hence ground-based GW detectors such as LIGO, Virgo and KAGRA are not expected to detect the isolated WDs.
These isolated WDs are also free from the noise due to the galactic binaries as well as from the extreme mass ratio inspirals (EMRIs).

A pulsar-like object radiates GWs at two frequencies. When we observe such a GW signal whose strength remains unchanged during the observation time $T$, the~corresponding detector's cumulative SNR is given by~\cite{Jara1998,Bennett2010}
\begin{equation}
	{\rm SNR} = \sqrt{S/N_{\Omega}^2 + S/N_{2\Omega}^2}
\end{equation}
where
\begin{equation}
\langle S/N_{\Omega}^2 \rangle = \frac{{\rm sin}^2 \zeta}{100} \frac{h_0^2 T {\rm sin}^2 2\chi}{S_n(f)}, \langle S/N_{2\Omega}^2 \rangle = \frac{4{\rm sin}^2 \zeta}{25} \frac{h_0^2 T {\rm sin}^4 \chi}{S_n(2f)}
\end{equation}
where $\zeta$ is the angle between the interferometer arms and $S_n(f)$ is the detector's power spectral density (PSD) at the frequency $f$ with $\Omega = 2\pi f$. As~we mostly deal with space-based interferometers such as
LISA, we assume $\zeta = 60^{\circ}$. The~average is over all possible angles, including $i$, which determine the object's orientation with respect to the celestial sphere reference~frame.

Figure~\ref{fig10} shows the SNR as a function of time for toroidal field-dominated WDs with different field strengths. We assume that these toroidal-dominated WDs have a poloidal surface field which is nearly four orders of magnitude smaller than the maximum toroidal field $B_{\rm max}$ inside the WD. Of~course, such a poloidal field cannot change the shape and size of the WD, as does the toroidal field. The~surface field strength is relatively very small (as is the dipole luminosity), and so it hardly changes $\Omega$ and $\chi$ within a 1 yr period. The~left panel of Figure~\ref{fig10} shows the SNR for a B-WD with $B_{\rm max} = 2.6\times10^{14}\, {\rm G}$ with mass $1.7 M_{\odot}$. All the GW detectors except \emph{LISA} can easily detect such a WD almost immediately, and \emph{LISA} can detect it in 5 months of integration. In~contrast, when the field strength decreases ($B_{\rm max} \approx 10^{14}\, {\rm G}$) the SNR decreases, and \emph{LISA} and \emph{TianQin} can no longer detect them, as shown in the right panel. However, they can still be detected by \emph{ALIA}, \emph{BBO} and \emph{DECIGO} within 1 yr of integration~time.

\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.55in]{Fig10_left}
&\includegraphics[width=2.55in]{Fig10_right}\\
\end{tabular}
	\caption{SNR as a function of integration time for a toroidal field-dominated WD with central density $\rho_c = 2\times10^{10}\, {\rm g\ cm^{-3}}$, spin period $P=2\, {\rm s}$, $\chi = 30^{\circ}$ and $d=100\, {\rm pc}$. The green (top) line represents \emph{ALIA}, the red (middle) line represents \emph{TianQin} and blue (bottom) line represents \emph{LISA}. The solid black line corresponds to $\langle S/N \rangle \approx 5$. See~\cite{Kalita2021}.}%
\label{fig10}
\end{figure}

Figure~\ref{fig11} shows the SNR as a function of integration time for XTE J1810-197, assuming $\chi=45^{\circ}$. We choose the maximum radius of this source to be 3000 km instead of 4000 km if it is a WD because its spin is fast and the \emph{XNS} code does not run for 4000 km with such high rotation frequency. The~left panel of Figure~\ref{fig11} shows that \emph{BBO} and \emph{DECIGO} would be able to detect it within 20 d and 100 d, respectively, if~it is a 3000 km poloidal field-dominated WD. If~it is a toroidally dominated WD, \emph{BBO} and \emph{DECIGO} could immediately detect it, and \emph{ALIA} would be able to detect it within 5 months only if it is a 3000 km toroidally dominated WD (see right panel).

There are many other plausible detectabilities, e.g.,~via their activity in
binary systems, which will be explored in the future.

\begin{figure}[H]
\begin{tabular}{cc}
\includegraphics[width=2.55in]{Fig11_left}
&\includegraphics[width=2.55in]{Fig11_right}\\
\end{tabular}
	\caption{SNR as a function of integration time for XTE J1810-197 assuming $\chi=45^{\circ}$.  The first, second and fifth lines from top correspond to a WD with radius 3000 km for \emph{BBO}, \emph{DECIGO} and \emph{ALIA} respectively, while third, fourth and sixth lines from top represent a WD with radius 1000 km for
	\emph{BBO}, \emph{DECIGO} and \emph{ALIA} respectively. The solid black line corresponds to $\langle S/N \rangle \approx 5$. See~\cite{Kalita2021}.}%
\label{fig11}
\end{figure}

\section{Matter Anisotropy Effects in Highly Magnetised White~Dwarfs}
\label{Sec8}
It has been shown that the presence of a strong magnetic field, the~anisotropy of dense matter, and~the orientation of a magnetic field can significantly influence the properties of neutron and quark stars~\cite{Deb2021}. The stability of them
and B-WDs is not achieved unless one considers the anisotropy of the system arising from the combined effects of~\cite{Deb2021,Deb2022}: (i)~anisotropy due to strong magnetic fields and (ii) anisotropy of the system~fluid.

The effective contributions from the matter and magnetic field lead to the system density given by
\begin{equation}
\tilde{\rho} = \rho + \frac{B^2}{8\pi}.
\end{equation}

The system pressure along the direction of magnetic field is represented as parallel pressure and takes the form based on magnetic field orientations as
\begin{equation}
p_{||} = \begin{cases}
p_r - \frac{B^2}{8\pi}, {\rm \ for\ radial\ orientation}\\
p_t - \frac{B^2}{8\pi}, {\rm \ for\ transverse\ orientation}.
\end{cases}
\end{equation}

Similarly, the~system pressure that aligns perpendicular to the magnetic fields is defined as transverse pressure and is based on the magnetic field orientations
\begin{equation}
p_{\perp} = \begin{cases}
p_t + \frac{B^2}{8\pi}, {\rm \ for\ radial\ orientation}\\
p_r + \frac{B^2}{8\pi}, {\rm \ for\ transverse\ orientation}.
\end{cases}
\end{equation}

The essential magnetostatic stellar equations which describe static, spherically symmetric B-WDs are
\begin{equation}
\frac{dm}{dr} = 4\pi \left(\rho + \frac{B^2}{8\pi}\right)r^2,
\end{equation}
and for radial orientation (RO)
\begin{equation}
\frac{dp_r}{dr} = \frac{-(\rho+p_r)\frac{4\pi r^3 \left(p_r - \frac{B^2}{8\pi}\right)+m}{r(r-2m)} + \frac{2}{r}\Delta}{\left[1- \frac{d}{d\rho}\left(\frac{B^2}{8\pi}\right)\frac{d\rho}{dp_r}\right]},
\end{equation}
whereas for transverse orientation (TO)
\begin{equation}
\frac{dp_r}{dr} = \frac{-\left(\rho+p_r+\frac{B^2}{4\pi}\right)\frac{4\pi r^3 \left(p_r + \frac{B^2}{8\pi}\right)+m}{r(r-2m)} + \frac{2}{r}\Delta}{\left[1+ \frac{d}{d\rho}\left(\frac{B^2}{8\pi}\right)\frac{d\rho}{dp_r}\right]}.
\end{equation}

We describe the effective anisotropy of the stars with $\Delta$, which depends on the magnetic field orientations given by $p_t - p_r + \frac{B^2}{4\pi}$ in case of RO and $p_t - p_r - \frac{B^2}{4\pi}$ for TO. We have for~RO
\begin{equation}
\Delta = \kappa \frac{(\rho+\rho_r)\left(\rho+3p_r - \frac{B^2}{4\pi}\right)}{\left(1 - \frac{2m}{r}\right)}r^2,
\end{equation}
and for TO
\begin{equation}
\Delta = \kappa \frac{(\rho+\rho_r+\frac{B^2}{4\pi})\left(\rho+3p_r + \frac{B^2}{2\pi}\right)}{\left(1 - \frac{2m}{r}\right)}r^2,
\end{equation}
where $\kappa$ is a dimensionless constant that describes the strength of anisotropy within the stellar structure. Note that $\kappa = 0$ implies the anisotropy effects that arise due to matter properties
and magnetic field both vanish. However, the~case of $B = 0$ but $\kappa \neq 0$ implies that only the anisotropy due to magnetic field vanishes. We have shown that highly magnetized WD models, which do not account for the combined anisotropic effects of the fluid and field, are eliminated, as they suffer from an
instability at the stellar~center.

To solve the magneto-hydrostatic stellar structure equations from the stellar center to surface, it is needed to supply an EoS along with a functional form of $\Delta$. Here, we consider the EoS proposed by Chandrasekhar to describe
degenrate electrons of WDs as
\begin{equation}
p_r = \frac{\pi m_e^4 c^5}{3h^3} \left[x(2x^2 - 3) \sqrt{x^2 + 1} + 3 {\rm sinh}^{-1} x\right],\ \rho = \frac{8\pi \mu_e m_H (m_e c)^3}{3h^3} x^3,
\end{equation}
where $m_e$ is the mass of an electron, $m_H$ is the mass of a hydrogen atom, $h$ is Planck's constant, $\mu_e$ is the mean molecular weight per electron, and~$x = p_F/m_{e}c$ with $p_F$ is the Fermi momentum. For~the carbon-oxygen WDs, $\mu_e=2$.

We show the mass-radius relations of B-WDs for different $B_0$, $\kappa$ and $\gamma$ in Figure~\ref{fig12}. For~TO fields with $B_0 = 3.79\times10^{14}\, {\rm G}$, a~maximum mass B-WD of $2.8 M_{\odot}$ is obtained, whose radius is 1457.67 km. For an~RO field with $B_0 = 1.2\times10^{14}\, {\rm G}$, the~maximum mass drops to $1.62 M_{\odot}$, and the radius of the B-WD is 454.67 km. For~$B_{0,TO} = 3.79\times10^{14}\, {\rm G}$, the~maximum mass and the corresponding radius of B-WDs increase by $\sim 70\%$ and $\sim 57\%$, respectively, compared to the non-magnetized but anisotropic case. For~$B_{0,RO} = 1.2\times10^{14}\, {\rm G}$, the~maximum mass and the corresponding radius decrease by $\sim 2\%$ and $\sim 52\%$, respectively, compared to the values
of non-magnetized but anisotropic WDs. Even without considering the magnetic field and incorporating the effects of local anisotropy due to the fluid, it is possible to push the maximum mass of WDs beyond the Chandrasekhar mass limit. For~example, by~considering $\kappa = 2/3$, we obtain a maximum mass for a non-magnetized but anisotropic WD of $1.81 M_{\odot}$. The~corresponding radius is 956.08 km. These values are $\sim 29\%$ and $\sim 8\%$, respectively, higher than the respective values of WDs at the Chandrasekhar mass limit. Moreover, with~increasing $\eta$ and $\gamma$ for the TO field case, the~mass of anisotropic B-WDs changes significantly, as~can be seen in the right panel of Figure~\ref{fig12}. It shows that the maximum mass of a B-WD with $\gamma = 0.9$ and $\eta = 0.2$ increases by $\sim 46\%$ compared to $\gamma = 0.6$ and $\eta = 0.1$.

\vspace{-5pt}
\begin{figure}[H]
\centering
\begin{tabular}{ccc}
\includegraphics[width=2.2in]{Deb21_Fig5a}
&\includegraphics[width=2.28in]{Deb21_Fig5b}
&\includegraphics[width=2.2in]{Deb21_Fig5c}\\
({\bf a})&({\bf b})&({\bf c})\\
\end{tabular}
	\caption{Stellar radius ($R$) as a function of gravitational mass ($M/M_{\odot}$) for different (\textbf{a}) $B_0$ (left~panel), (\textbf{b}) $\kappa$ (middle panel), and~(\textbf{c}) $\eta$ and $\gamma$ (right~panel). Solid circles represent the stars with the maximum-possible masses. See~\cite{Deb2022}.}%
\label{fig12}
\end{figure}

For RO fields, we can explain sub-, standard- and super-Chandrasekhar B-WDs by making appropriate choices of $B_{0,RO}$ and $\kappa$ (see Figure~\ref{fig13}). By~changing both $B_{0,RO}$ and $\kappa$, as~shown in the left panel of Figure~\ref{fig13}, we successfully explain both (i) the sub- and standard-Chandrasekhar progenitor B-WDs and (ii) the standard and super-Chandrasekhar progenitor
B-WDs by~a single mass-radius curve for the respective cases. On~the other hand, through changes of only $B_{0,RO}$ or $\kappa$, sub-, standard- and super-Chandrasekhar B-WDs line up in a series of mass-radius curves; see, e.g.,~the middle and right panels of Figure~\ref{fig13}. This leads to a complete explanation of under-, regular- and over-luminous SNeIa in a single~theory.

\vspace{-5pt}
\begin{figure}[H]
\centering
\begin{tabular}{ccc}
\includegraphics[width=2.232in]{Deb21_Fig6a}
&\includegraphics[width=2.23in]{Deb21_Fig6b}
&\includegraphics[width=2.23in]{Deb21_Fig6c}\\
({\bf a})&({\bf b})&({\bf c})\\
\end{tabular}
	\caption{Stellar radius ($R$) as a function of gravitational mass ($M/M_{\odot}$) for varying (\textbf{a}) $B_{0,RO}$ and $\kappa$ (left panel), (\textbf{b}) $B_{0,RO}$ (middle panel), and~(\textbf{c}) $\kappa$ (right panel). Solid circles represent the stars with the maximum possible masses. See~\cite{Deb2022}.}%
\label{fig13}
\end{figure}

\section{Conclusions}
\label{Sec9}
Reviewing the work in last decade or so, we have confirmed that, at~least theoretically, massive NSs and WDs, heavier
than their conventional counterparts as observed/inferred from recent data,
are possibly highly magnetized, rotating stable compact stars. They have
multiple implications including enigmatic peculiar over-luminous SNeIa.
Numerical simulations based on Cambridge stellar evolution code STARS argue
B-WDs to be toroidally (centrally) dominated with lower, maybe dipole,
surface magnetic fields. The~new generic mass-limit of WDs seems to be
well above $2M_\odot$, around $2.8M_\odot$, depending on the magnetic field
profile/geometry and rotational~frequency.

However, these massive compact objects, particularly WDs, have hardly been
observed directly due to their apparently very low luminosity. On~the other
hand, the~presence of magnetic fields and rotation by definition brings in
anisotropy in these compact objects, leading them to be triaxial when the
magnetic and rotation axes are misaligned. Therefore, they are expected to
emit continuous GW. Hence, B-WDs can be detected directly by the missions
LISA (in one year integration time), DECIGO/BBO and the magnetised NSs
can be detected by aLIGO, aVIRGO, Einstein~Telescope.

Nevertheless, the~magnetic fields therein start decaying beyond million years,
hence these massive compact objects may not survive beyond this time.
Moreover, due to their electromagnetic and gravitational radiation, the~angular velocity and the misalignment angle between magnetic and
rotation axes decrease, leading them to lose their pulsar nature and detection
possibility. This is another reason why they are so rare and/or hard to
detect unless targeted at an appropriate time. Therefore, appropriate missions
in GW astronomy and otherwise, e.g.,~radio astronomy, should be planned to
probe~them.

\vspace{6pt}

\authorcontributions{\textls[-20]{Both the authors share an equal authorship.
However, various figures presented in this manuscript are based on previous
works by B.M. and his collaborators.
The work was conceptualized by B.M.;
the methodology was initiated by B.M. and further was carried out
by B.M. and M.B.; investigation was initiated by B.M. and was further
taken over by M.B.; the first draft was prepared by M.B. and further was
revised and edited by B.M. and M.B.}}

\funding{M.B. would like to acknowledge support from the Eberly Research Fellowship at the Pennsylvania State University.}

\acknowledgments{B.M. thanks the organizers of ``The Modern Physics of Compact Stars and Relativistic Gravity 2021" held in September 27-30, 2021, in
Yerevan, Armenia, to invite for a talk, based on which the present paper
is composed. }

  \conflictsofinterest{The authors declare no conflict of interest.}

redpajama_set_name: RedPajamaArXiv


Thank you for selecting me. Let's get onto health, wealth and immortality.
Dear mentor Mark Hamilton, it has been brought to my understanding that I posses a rare trait that not everyone has. I am pure in heart. I want to help others dreams come true. Why dont people understand what you are saying when it's clear that what is not known is our higher self.
The library is closing to night. I'll get back to you very soon. My computer is not set up at home to attend the meetings yet.


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