Accretion disc outbursts are reoccurring events observed in various astrophysical systems including Xray binaries and cataclysmic variables. These outbursts are characterized by a sudden increase in luminosity due to various instabilities in the accretion disc. We need to investigate the timedependent accretion flow models to understand the mechanisms driving these outbursts. Timedependent models incorporate the discs time evolution and can capture the buildup of instabilities. This review aims to give a basic overview of accretion disc outburst and stability analysis. The paper highlights the necessity of considering the hierarchy of different timescales dynamical viscous and thermal when investigating the instabilities occurring in the accretion disc. The importance and observational implications of studying these accretion disc outbursts are also discussed.
['Astrophysics - High Energy Astrophysical Phenomena']
Accretion Disc Outbursts and Stability Analysis
{'Accretion Disc Outbursts and Stability Analysis': "Liza Devi *; Asish Jyoti Boruah; Biplob Sarkar  \n* Corresponding author Department of Applied Sciences, Tezpur University, Napaam, Assam-784028, India E-mail of corresponding author: app23110@tezu.ac.in  \nAbstract - Accretion disc outbursts are re-occurring events observed  in  various  astrophysical systems, including  X-ray  binaries  and  cataclysmic  variables. These outbursts are characterized by a sudden increase in luminosity due to various instabilities in the accretion disc. We  need  to  investigate  the time-dependent accretion  flow  models  to  understand  the  mechanisms driving these outbursts. Time-dependent models incorporate the disc's time evolution and can capture the build-up of instabilities. This review aims to give a basic overview  of  accretion  disc  outburst  and  stability analysis. The paper highlights the necessity of considering the hierarchy of different timescales, dynamical, viscous, and thermal, when investigating the instabilities occurring in the accretion disc. The importance  and  observational  implications  of  studying these accretion disc outbursts are also discussed.  \nKeywords-accretion  disc  outburst,  instabilities,  dwarf novae, cataclysmic variable.", 'I. ACCRETION DISC OUTBURSTS: AN OVERVIEW': 'Accretion  discs  are  swirling  discs  of  gases  rotating differentially around a compact object, like a neutron star or a black hole. These discs are an important component of various binary systems. These systems are consist of a compact  object  and  a  companion  star,  including  X-ray binaries and cataclysmic variables [1,2,5,6,9]. The accretion  disc  is  fuelled  by  matter  drawn  from  the companion  star  and  accreted  onto  the  compact  object. Sometimes,  accretion  discs  experience  a  remarkable characteristic by a quick increase in brightness, followed by  a  slower  decline.  This  is  known  as  an  accretion  disc outburst  [1,2,8].  Due  to  the  sudden  increase  in  the accretion  rate,  outbursts  occur  and  result  in  a  dramatic increase in luminosity. This phenomenon  is  often observed  in  X-ray  binaries  and  cataclysmic  variables. After  an  outburst,  the  system  typically  enters  to  an extended  period  of  inactivity,  known  as  quiescence  [1]. This state can last for a significant amount of time before occurring  another  outburst.  The  period  and  recurrence time  of  outbursts  vary  from  system  to  system  ranging from minutes to months and even years.  \nThis  review  aims  to  provide  a  basic  overview  of accretion  disc  outbursts  and  stability  analysis  along  with highlighting  the  importance  of  considering  the  hierarchy of timescales involved.', 'II. TIME-DEPENDENT ACCRETION FLOW MODELS': "Time-dependent  accretion  flow  models  consider  the disc's  time  evolution,  which  helps  us  to  study  the generation  of  instabilities  that  lead  to  outbursts.  The magnitude  of  the  viscosity  governs  the  disc  flow's  time dependence  [8,11,40].  Thus,  one  of  the  few  sources  of  \nquantitative  data  regarding  disc  viscosity  is  provided  by observations  of  time-dependent  disc  models.  These models  helps  us  to  pinpoint  the  origins  of    various instabilities,  monitor  their  growth  rates  and    determine their    eventual  impacts  [2].  The  time  dependence  study mainly  deals  with  the  phenomenon  of  dwarf  nova outbursts and is a field of great interest on its own.  \nIn  traditional  steady-state  optically  thick  discs,  the viscosity  has  little  effect  on  the  observable  properties, which  is  fortunate  for  confirming  their  existence,  but  it also implies that it is unlikely that observations of steady discs  will  provide  much  information  about  the  viscosity [2,5,43]. Moreover these models failed to explain the time evolution  phenomenon  and  instabilities  occurring  in outbursting  candidates  in  binary  systems.  This  is  where the need of time-dependent accretion flow models comes into picture. Instabilities in accretion discs are caused by a variety of physical mechanisms.", 'III. THE HIERARCHY OF TIMESCALES': "When  studying  instabilities  in  accretion  discs,  it's important to examine the hierarchy of timescales. The three main timescales are:", 'A. Dynamical Timescale': "The  shortest  timescale  in  an  accretion  disc.  This timescale  characterizes  the  disc  material's  orbital  period around the central object. In other words, it is the amount of time, the disc particles require to move around the accreting object  in  a  single  Kepler  orbit.  In  addition,  it  is  the characteristic  period  for  restoring  hydrostatic  equilibrium perpendicular to the disc plane [1,2,9,28].", 'B. Thermal Timescale': 'The  timescale  on  which  the  disc  can  adjust  its temperature  in  response  to  external  perturbations.  The thermal time scale indicates the duration needed for viscous forces  to  produce  the  thermal  energy  of  a  specific  disc annulus. It is calculated by dividing the total thermal energy present by the local energy dissipation rate [1,2,9,28].', 'C. Viscous Timescale': "The  viscous  time  or  radial  drift  timescale  refers  to  the period required for the fluid in a disc to move substantially in a radial direction. We can also say that it is the period needed for  matter  to  slowly  move  inward  due  to  viscosity.  The viscous  timescale  is  substantially  longer  than  the  thermal timescale  and  the  dynamical  timescale  is  shorter  than  the later [1,2,9,28].  \nThe  accretion  disc's  stability  depends  on  all  three timescales.  Radiative  processes  on  the  thermal  timescale alter the disc's vertical structure, while viscous forces on the viscous timescale drive angular momentum  transfer [7,10,40]. A comprehensive stability analysis must  \nencompass  all  three  timescales  in  a  coherent  framework  to accurately  understand  the  origin  of  possible  instabilities resulting in outbursts [1,2,39].", 'IV. INSTABILITIES IN THE ACCRETION DISC': 'Various instabilities can arise in accretion discs due to the interaction between  these  timescales. Some  common instabilities include:', 'A. Thermal instability': 'It  is  caused by temperature fluctuations within the disc. In a disc annulus, when rate of heating is out of step with the rate  of  cooling,  then  a  non-equilibrium  situation  arises.  In this  case  it  is  said  that  the  disc  is  subjected  to  thermal instability.  This  instability  grows  in  thermal  timescale [1,2,7,30].', 'B. Viscous instability': 'Viscous instabilities occur due to the buildup of viscous stress  in  the  disc  (resulting  from  changes  in  viscosity  or angular  momentum  transfer).  We  can  explain  viscous instability  by  applying  a  perturbation  in  the  form  of  extra mass  to  a  disc  ring.  If  the  extra  mass  we  added  as  a disturbance, diffuses  or  drifts  away  from  the  disc  ring  and and  the  disc  ring  gradually  returns  to  its  original  surface density,  the  disc  ring  is  said  to  be  viscously  stable. Otherwise, it is considered viscously unstable [1,2,30].', 'C. Thermal-Viscous instability': "Examining how an accretion disc reacts to perturbations, which might be local or global, is necessary to determine the stability  of  the  disc.  The  most  well-known  and  researched instability  is  the  thermal-viscous  instability,  which  is responsible  for  some  X-ray  binary  outbursts  as  well  as dwarf  nova  outbursts  [29].  When  angular  momentum  is transported  viscously  and  there  is  a  non-linear  feedback between the heating and cooling processes, thermal-viscous instability results. Under some circumstances, a local region in the disc may change from a cool, low-viscosity state to a hot, high-viscosity one. The mass accretion rate significantly increases because of the change in the viscous regime, which in turn impacts the disc's heating and cooling processes.  The  hot  state's  temperature  increase  causes  a higher equilibrium mass accretion rate, which enhances the outburst [26].  \nThe basic principles of thermal-viscous instability can be explained by the S-shaped equilibrium curve [7] (Fig.1.). In an  accretion  disc,  hydrogen  is  the  most  abundant  element. The  degree  of  ionization  of  hydrogen  determines  the stability of the disc [38,39,40].  The hydrogen fully ionized state  is  known  as  the  high  state  or  HII  state.  However,  in low  state  or  HI  state,  hydrogen  is  neutral.  Both  these branches  are  stable  (thermally  and  viscously)  [30,31]. Partially ionized hydrogen occurs in the intermediate state. In the accretion disc's outer regions, when the temperatures vary  between  log10T  =  3.5-4  K,  the  partial  ionization  of hydrogen  occurs  [27,28].  During  partial  ionization  of hydrogen, the disc may become unstable due to changes in opacity. In the context of viscous and thermal instabilities, this  state  is  unstable  [30].  Three  solutions  are  found  for  a specific  range  of  surface  density  Σ  (ΣB  <  Σ  <ΣA)  for  an  \naccretion  disc  annulus  [30].  Let  us  consider  a  disc  ring where Ṁout and Ṁin are mass outflow and inflow rate from and into the ring respectively. In the HI state, where Ṁin > Ṁout,  surface  density  profile  as  well  as  disc  mass  will increase, until a maximum density ΣA is reached and jumps to HII state [30]. In HII state, Ṁin < Ṁout satisfies. This as a result decreases the disc's mass there. This continues until a minimum surface density ΣB is reached and the disc annulus jumps to HI state. The outbursts phenomena result from the constant oscillations between these two states.  \nFig.1. Schematic diagram of S-shaped equilibrium curve.  \n<!-- image -->", 'D. Radiation Pressure Instability': "When the pressure of radiation surpasses the pressure of gas,  the  black  hole  accretion  disc,  which  has  a  classical heating component  proportional  to  the  pressure  and viscosity  parameter  α  (a  parameter  that  describes  the efficiency  of  angular  momentum  transport  in  an  accretion disc)  [20,42],  can  experience  both  thermal  and  viscous instabilities. This  happens  within  the  accretion  disc's innermost  radii  that  encircle  a  compact  object  (in  case  of black holes and neutron stars) [20,21]. Shakura and Sunyaev (1973)  identified  radiation  pressure  instability  in  their traditional α-models early on [22,43], which they thoroughly examined  in  1976.  According  to  Taam  et  al.  [24]  and Deegan  et  al.  [23],  the  microquasar  GRS  1915+105's recurrent  outbursts  lasting  hundreds  of  seconds  could  be explained by radiation pressure instability.  \nThe magnetorotational instability (MRI), which is caused by the interplay between weak magnetic fields and the disc's differential  rotation,  is  another  significant  instability  [39]. Viscosity can be efficiently enhanced, and angular momentum can be transported within the disc via an MRI. On  the  other  hand,  stability  analysis  of  discs  under  MRI influence  is  a  more  complex  field  that  is  still  being investigated.", 'V. STABILITY ANALYSIS': 'To analyze the stability of an accretion disc, we require knowledge  on  how  the  perturbations  or  disruption  to  its  \nstructure  grows  and  evolves  over  time.  Linear  stability analysis  is  commonly  used  to  examine  how  a  system responds to small perturbations [1,42]. After the addition of perturbation,  if  the  perturbation  grows,  it  may  indicate  an instability that could lead to sudden and intense outburst of energy.', 'A. Linear Stability Analysis': "Linear  stability  analysis  involves  simplifying  the  basic governing equations of the flow adding small perturbations to  the  disc's  equilibrium  state  to  examine  their  stability [1,19,34].  By  solving  the  simplified  equations,  we  can determine  the  conditions  which  give  rise  to  instabilities. Fujimoto  and  Arai  [18],  investigated  the  stability  of  an optically  thick,  slim  accretion  disc  around  a  black  hole. They  obtained  a  dispersion  relation  of  the  fourth  order within the context of the study of linear stability. Also, for the  cataclysmic  variable  disc  models  and  their  boundary layer (BL), Collins et al. [19] reported on the findings of a linearized  perturbation  analysis  using  local,  linear  stability analysis.", 'B. Non-Linear stability  analysis': 'Although linear stability  analysis  helps  us  to know how perturbations first develop, nonlinear analysis and numerical simulations are mostly required to fully understand occurrence of instabilities. Non-linear effects can lead to a range  of  outcomes,  which  includes  the  formation  of turbulent  patterns  [16,17],  interactions  between  various unstable modes [14] and saturation of large-scale instabilities [15]. Balbus et al. [16] used a mix of analytical and numerical approach to explore the hydrodynamical nonlinear  stability  of  fluid  flows  that  rotate  differentially  at different distances from the center (differential rotation) and flows that slide past each other (pure shear flows) in threedimensions.', 'C. Numerical Simulations': 'Numerical simulations work as the most effective way to investigate  the  broad  range  dynamics  of  accretion  discs [10,12,13,35].  Simulations  can  help  us  to  understand  how the interplay between thermal, viscous, and magnetorotational  instabilities  lead  to  disc  outbursts  and fluctuations [36,37,38]. Bergaulinger et al. [13] used semiglobal  simulations  to  explore  the  growth  and  saturation  of the  MRI  in  core  collapse  supernovae  by  studying  its evolution.  The  accretion-ejection  instability  in  magnetized accretion  discs  is  simulated  numerically  by  Caunt  and Tagger [12].', 'VI. OBSERVATIONAL IMPLICATIONS OF OUTBURSTS': 'Accretion disc outbursts can be observed in a variety of astrophysical systems, including cataclysmic variables (dwarf novae) and X-ray binaries.', 'A. Cataclysmic variables': 'This  kind  of  binary  system  consists  of  a  white  dwarf which  draws  matter  from  a  companion  star  [1,4,29]. Primarily, the companion star is a main sequence star or red dwarf [27]. Of all known cataclysmic variables, nearly fifty percent belong to the dwarf novae class, which has a known orbital  period  [1,41].  The  outbursts  last  a  few  days,  then happen  again  in  a  few  weeks  or  months,  and  have  an  \namplitude  ranging  from  two  to  five  magnitudes  [1,25,43]. The  recurrence  time  and  the  form  of  the  outburst  light curves are not exactly periodic. The most studied example is the  SS  Cygni  system  [6].  The  spectral  characteristics  of dwarf novae in outbursts are very similar to those of novalike  variables  that  are  continuously  bright,  or  cataclysmic variables without dwarf nova outbursts.  It is assumed that dwarf  novae  also  feature  a  quasi-steady  state  disc  during outbursts  since  nova-likes  are  believed  to  have  bright, steady-state accretion discs [1].', 'B. X-ray Binaries': 'A  compact  object  (a  neutron  star  or  black  hole)  that  is accumulating mass from a companion star is what makes up an X-ray binary [1,2,4,5]. These systems are highly variable, exhibiting outbursts capable of several orders of magnitude increase  in  X-ray  luminosity.  Soft  X-ray  transients  are  a kind of low mass X-ray binaries that exhibit outbursts that are similar to dwarf novae outbursts, but with a considerably longer timescale and amplitude [1,3,32]. Compared to dwarf novae,  the  form  of  the  light  curve  is  significantly  more variable. They have a typical shape which rises quickly over a  few  days  and  then  slowly  decays  exponentially  over several  months.  While  there  is  concurrent  brightening  in other  wavebands,  particularly  in  the  optical  range,  the outburst is most noticeable in the X-ray regime. There has only  been  one  observation  of  several  soft  X-ray  transient outbursts  [1].  This  suggests  that  the  recurrence  period  is extremely long (decades, centuries, or even longer). Certain systems are seen repeating every year or every few years.  \nStudying  accretion  disc  outbursts  may  enhance  our understanding  of  other  astrophysical  systems,  including active galactic nuclei (AGN) [20,21] and gamma-ray bursts (GRB)  [33].  These  systems  show  rapid  and  extreme brightness increases, possibly due to similar instabilities in their accretion discs.', 'VII. CONCLUSION': 'This  paper  presents  a  summary  of  current  state  of knowledge  about  stability  analysis  and  accretion  disc outbursts.  It  is  necessary  to  use  time-dependent  models which  use  hierarchy  of  dynamical,  thermal,  and  viscous timescales  to  explain  accretion  disc  outbursts.  The  various instabilities which lead to outbursts are also reviewed here. We  have  also  discussed  linear,  non-linear  and  numerical stability  analysis  which  are  essential  for  determining  the factors that contribute to outbursts. Investigation of accretion  disc  outbursts  can  help  us  to  understand  the importance of different complex binary systems along with other astrophysical systems which display similar behaviors. Further  research  in  this  area  is  necessary  for  fully understanding the process that drives outbursts in accretion discs.', 'ACKNOWLEDGMENT': 'The  authors  LD,  AJB  and  BS  acknowledge  the reviewer  for  thoroughly  reviewing  the  manuscript  and providing beneficial comments. The authors also acknowledge the use of Meta AI and Google Gemini AI to improve the readability of the text.', 'REFERENCES': "- [1] Kolb, U., Extreme Environment Astrophysics . 2010.\n- [2] Frank,  J.,  King,  A.,  and  Raine,  D.  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