Accretion in astrophysicsGas falls onto a star or a compact object (neutron star, white dwarf, black holes)Gravitational potential energy converted into thermal energyGas radiates as it heats up, produces a luminosity proportional to the loss of gravitational energy as it falls down

This “accretion luminosity” is enormous in the case of compact objectsbecause gas has to lose enormous amount of gravitational potential energy to fall onto them (E ~ GM2/r, r ~ 0 for a black hole!)

Accretion onto compact objects is most powerful energy source in the Universe,more powerful than nuclear reactions! Emission typically in X-rays bandIt produces active galactic nuclei (AGNS) and QUASARS, the highest luminosity objects that we know in the Universe (100 times more luminous than galaxies)!QUASARS are believed to be the result of accretion onto supermassiveblack holes (SMBHs) with masses 108-109 Mo ----> huge GM2/R

Galactic compact X-ray sources – from white dwarfs/neutron stars/stellar BHs,usually in binary system with main sequence star (< 100 Mo) Lx ~ 1037-1038 erg/sAGNs/ QUASARS - from SMBHs (> 106 Mo) Lx ~ 1042-1046 erg/sUltraluminous X-ray sources (ULXs)– from intermediate mass BHs (100-105 Mo)Lx ~ 1040 erg/s

Two types of accretion

-spherical---- gas has no angular momentum

-from a disk (accretion disk) --- gas has angular momentum

Accretion disks are the most common mode of accretion in astrophytsics.

Examples

Accretion disk around a protostar – this is essentially the protostellar/proto-planetary disk.

Accretion disk in a binary system made of a neutron star/white dwarf arounda main sequence star (or some other combination). Gas flows from the star to the compact object (gas infall produces cataclysmic variables or supernovae type II)

Accretion around a single compact object, e.g. neutron star, stellar black hole or supermassive black hole

A little digression - What is the “radius” of a black hole?

Stellar black holes form from collapse of very massive stars (M > 8 Mo)

Neither degenerate electron or degenerate neutron gas pressure canstop the collapse.Star collapses into a singularity, i.e. a region of space-time with infiniteDensity.Black holes are a prediction of General Relativity.

Black hole is “black” because not even photons can escape

If that is true it means GM/r > 1/2 ve2 = 1/2c2, and so there exists a minimum

radius rs at which photons can orbit the black hole while still being ableto escape

rs = 2GM/c2 = 3km x M/Mo (Schwarzschild radius) rs = 30 km for 10 Mo BH

From studying equation of motion of matter around a black hole in GeneralRelativity one finds that radius of last stable orbit is 3rs ---> this is the“radius” of a black hole relevant for accretion since inside this radius matter has no potential energy in the newtonian sense.

Accretion disks

Suppose matter (gas) moves in a disk around a star or compact object? Then it means matter is in centrifugal equilibrium.How can it fall onto the star or compact object?

Answer: there must be a non-conservative force that “extracts” angular momentum and rotational energy from some of the matter in the disk (canhappen even if the disk conserves angular momentum globally)

Assumption: viscous force (same meaning as friction in mechanics)Note: not the same as molecular viscosity because macroscopic force

Examples of viscosity

1- spiral waves in disk (gravitational disturbance)

2- turbulence in a clumpy medium = medium made of clouds andclouds collide anelastically transferring energy and angular momentum

2 – magnetic field can also extract energy and angular momentum from thegas. Important especially around compact objects because gas is very hotand ionized (so many charged particles, needed to maintain magnetic field)

Hard to know which mechanism produces viscosity in a given situation,observations of accretion disks are not detailed enough to study directly the role of turbulence or the interaction between gas and magnetic field

Simple “heuristic” model is the a-disk model (Shakura & Sunyaev 1977)

f ~vvisc

2 ~ P (last equality holds if vvisc ~aT)

f = viscous stress (=viscous force per unit area), enters both momentum and energy equation for disk fluid opposite

sign of thermal pressure force (pulls gas inward)

Since viscosity drives accretion ~ dM/dt - one determines by measuringthe accretion rate using observations of accretion disk luminosity/spectrum.It turns out that typically ~ 10-1– 1 for accretion disks around compact objects (e.g. black holes), 10-3 – 10-2 for protostellar disks.

viscous force removes angular momentum locally and-----> converts kinetic (rotational) into thermal energy

Viscosity does something to energy as well…

Converts rotational energy in to heat

Heat radiated away (important, otherwise rising pressure would stop accretion again!)

Energy being lost in heated gas ultimaltely depends on potential energy lost (Eth ~ T ~ h ~ Epot

~ Mbh) typically X-rays

Gas sinks deeper in the potential well if it cools efficiently

Viscosity

Gravitationalpotential energy

Radiation

Disk + Viscosity -> Accretion Disk

Accretion luminosity and accretion efficency

Efficiency = = L / (dM/dt x c2) = ½ GM / c2 rin dM/dt x c2 = maximum possible luminosity = power emitted if all mass converted into

energy

Neutron Star – rin ~ 10 km = 0.1

Black Hole - rin = 3rs ~ 0.08

But from GR different for non rotating (= 0.057) and rotating black holes ( = 0.42)

For nuclear reactions in stars ~ 0.007 , much smaller!!

Plus all mass participates to accretion in disk, only a fraction to nuclear burning in stars

Total accretion luminosity does not depend on viscosity or details of radiation physics it is simply (as for protostellar disks L = G x M x dM/dt /rin

Eddington Limit

Radiation coming from the disk produces radiation pressure the higher theaccretion flow the hotter the disk and the stronger the effect of radiation pressure

Radiation pressure is felt by accreting matter -- eventually radiation pressure becomes higher than gravitational pull of compact object/star and accretion stops.

Radiation pressure force will be proportional to luminosity (more photons=moreradiation pressure) and luminosity is proportional to accretion rate.

The limiting luminosity at which an object can accrete in “steady state” is:

4cGMmp

Derived for spherical accretion but approximately correct also for accretion disk(photons emitted mostly perpendicular from the disk)

L > Le still possible (e.g. supernovae type Ia and novae) but only transient andoutflow occurs!

T

Ledd == Thomson cross section

Energy of typical photon = pc (gas hot and ionized free electrons)

The number of photons crossing unit area in unit time at radius r is: L/4r2pc

Number of collisions per electron per unit time= L T/4r2pc

Force per electron = rate at which momentum is deposited per unit time

= Frad = LT/4r2pc X p = LT/4r2c

For accretion to occur it must be Frad < Fgrav

Fgrav (gravitational force per electron) = GMmp/r2

(protons and electrons coupled by Coulomb interaction so gravitational force communicated via protons)

Obtain Ledd by setting Fgrav=Frad.

Unique phenomena produced by accretion in binaries

Nova – white dwarf + main sequencestar

outbursts of luminosity produced bythermonuclear burning of hydrogenrich accreted material

systems brightens for about a month

with L >> Ledd

Enova ~ 1046 erg

Supernova Type Ia

White dwarf + main sequence star but much stronger outburst because 1 Mo of helium/carbon is ignited and synthesized into iron group elements

Esup ~ 1051 – 1052 erg.

Small range of luminosities, standard candles important for cosmology!

How do we know that black holes exist?

How can we prove existence? Example: measure velocity of gas or stars on the last stable orbit because GR makes accurate predictions on the equations of motions that are valid only for black holes

Unfortunately no instrument has enough resolution to take measurements so close to a BH.

In general, we think that black holes exist because gas accretion onto black holesis the only way to explain X-ray luminosity of the most powerful sources that we see in the Universe, from some Galactic X-ray sources to AGNs and QUASARS in distant galaxies

AGNs and QUASARS like powered by supermassive black holes (SMBHs). These were probably born in the early Universe from collapse of Supermassive Stars (> 100 Mo) and then accrete gas from the galaxy in which they were born: connection between galaxy formation and supermassive black holes, hot topic of current research!

Active Galactic Nuclei (AGNs) are some of the most powerful energy outbursts in the Universe (X-ray, radio, optical)The most powerful AGNs, distant QUASARS, have X-ray luminosities up to 1046 erg/s (>100 times brighter than our Milky Way)Associated with galaxies, powered by a central SMBHRadio jets produced by electrons accelerated by strong magnetic fieldproduced in accretion disk (synchrotron radiation)

FIRST QUASAR DISCOVERED in 1964, 3C 273

AGNs as indirect evidence for SMBHs

HOST GALAXY OF 3C273

A nearby QUASAR M87

Magnetic field entangles andaccelerates part of the infallinggas into powerful jets

Accretion disk

~0.01 pc

AGNs: indirect evidence for SMBHs

Now do the SMBHs feed?With large reservoirs of gas in galaxies at kpc scales(108-109 Mo, same mass as SMBHs!)How does the gas get to the SMBHs that sit at the center?

Merging galaxies are often associated with AGNs……

X - rays

Evolution of the gas component in major merger

Accretion disks: structure equations

One can solve the a system of structure equations for accretion disks around compact object assuming (1) steady state, (2) neglecting gas infall (no protostellar envelope in this case although gas inflow may occur as gas comes from the donor star in a binary system), (3) thin axisymmetric and (4) viscosity is the only source of heating in the disk - determines the disk temperature together with cooling processes and pressure law.Self-gravity neglected for AGN/X-ray binaries accretion discs, not necessarilyfor protostellar disks (self-gravity can be coupled with viscosity there).

Equations to solve; momentum (Euler + viscosity), continuity, equation of state(e.g. polytropic), energy equation (gives luminosity, source is viscosity rather than nuclear reactions as in the case of stars), energy transport equation (e.g.diffusion equation in optically thick regions).In addition auxiliary equations for viscosity law and for opacity law.

Final result; M (mass of compact object) and dM/dt (accretion rate of gas from disk to star, related to viscous mass transport, constant by assumption of steady-state) determine completely disk structure (in the case of stars was just M).

The full solution of the disk structure equation shows the the accretion disk canbe divided into three regions;

(1)an outer region, a radius r >> rI, rI = inner radius of the disk, wheregas pressure dominates over radiation pressure and in which the opacity is controlledby free-free absorption (inverse brehmsstrahlung)

(2) a middle region, at small r, in which gas pressure dominates radiation pressurebut the opacity is mainly due to electron (Thomson) scattering

(3) an inner region, at very small r, r ~ rI, at which radiation pressure dominatesgas pressure and electron scattering dominates absorption in the opacity

Important: at the inner radius of the disk rI , i.e. closest to the compact object, most of the gravitational energy is released -- most of the viscous heating is generated - most of the radiation is emitted

Therefore the inner region is what one needs to study in order to understandthe observed spectrum of an accretion disk. For this the steady state solution gives the disk interior temperature as (note the dependence on); T = (5 x 107 K)(M) -1/4 r – 3/8

Spectrum of accretion diskTo compute the spectrum (power/luminosity per frequency) one needs to take into account that the different regions of the accretion disk will produce a different specific flux F depending on the local properties, i.e.;

(1)Optical depth – optically thick vs. optically thin regions

(2) Source of opacity (in optically thick regions). Scattering or absorption, which scattering or absorption process?

In optically thick regions (no matter the opacity source) one can use the usualdiffusion approximation for vertical radiation transport to calculate F(r,z).replacing differentials with finite differences and integrating on z one obtainsflux at the surface, i.e F(r,z = h) = F(r)

F(r) ~ acT4/ ~ acT4/<>

At sufficiently high altitude above the disk midplane, quite soon if the disk is thin, the disk will become optically thin. In a thin disk the transition will be sharp - the surface of the disk emits as a blackbody. So the emitted flux Fe will be:

Fe = aTs4, where Ts (surface temperature)= [4F(r)/a]1/4

In optically thin regions ( < 1 in an entire column above the midplane –this happens at disk inner and outer edge for example)the emitted flux will be equal to the emergent flux and will be equal to;

F(r) = Fe ~ h(r,T) (r, T) average photon emissivity, depends on specific radiative process (erg s-1 cm-3)

But middle and inner region of the disk belongs to a third regime; disk is optically thick (so diffusion equation ok for radiation transport) but opacity is mostly due to scattering of photons rather than absorption -- cannot assume blackbody for emergent flux, valid only when absorption dominates!

In this case, the emergent flux is that of a “modified blackbody”. Scatteringnear the surface increases absorption probability before photon can escapeat the surface so that the intensity goes down compared to blackbody case.The specific intensity is given by:

I ~ jff/ff(ff/es) ~ B(Ts)(ff/es)1/2

Note that absorption opacity is due to free-free (and emission as well) in this region (high midplane temperature, gas is ionized -> see expression for Ts).Scattering is due to electron scattering instead

The total emitted flux follows from I and is given by; (*) Fe ~ (6.2 x 1019 erg cm2 s -1)1/2Ts

9/4 rather than Fe ~ aTs4 (blackbody)

(<ff> << es ) (<ff> >>es)

<ff> = Rosseland mean absorption opacity

Using Rosseland mean opacities and Fe = Teff4 (Stefan-Boltzmann’s law)

one obtains the relation between the emission temperature and theblackbody effective temperature.

Teff ~ Ts(<ff>/es)1/8 rather than Teff ~ Ts

Ts = surface temperature, characterizes the energy of emitted photons

The effect of scattering is thus to increase the mean energy of the emergentphotons, kBTs, above the value it would have been if the radiation occurred inthermodynamical equilibrium (i.e. the blackbody case).

One can then use equation (*) in combination with the structure equation that relates the emergent flux to the mass (M) and mass accretion flow (dM/dt) to express the surface temperature Ts as a function of M and dM/dt;

Ts = (2 x 109 K) 2/9(M/Mo)-10/9( (dM/dt)17)8/9(r/rs)-17/9 f8/9

rs = Schwarzschild radius

(dM/dt)17= mass accretion rate measured in units of 1017 g s-1 ~ 10-9 Mo/yr

f = 1 – (6/r)1/2

For a blackbody (see 14.5.38) the temperature constant would be much lower ~5 x 107 K --- the photons emitted have higher energy (“harder”) thanif the disk radiated as a true blackbody. Photons are “hard” X rays, i.e. X-rayswith very high energies (10-100 keV, more for SMBHs).This is a very good feature because it allows to distinguish emission by accretiondisks around compact objects from other astrophysical objects that producelower energy “softer” X-rays (e.g. galaxy clusters, protostellar outflows, emissionat 0.1-1 keV)

The total spectrum of the accretion disk emission is the superposition of the flux coming from the different regions of the disk but, as anticipated, the highest flux (so most of the luminosity) is produced by the middle/inner region that gives rise to the modified blackbody shape (scattering dominates).The outer region is optically thick and absorption dominated and is well described by a blackbody spectrum.

The innermost region is optically thin and the emission is dominated by free-free emission and inverse Compton scattering – inverse Compton is the process by which photons gain energy by scattering off electrons at very high speed and produces the high energy tail in the spectrum (hottest region).