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Part C Major Option Astrophysics
High-Energy AstrophysicsGarret Cotter
Office 756 DWB
Trinity Term 2014 Lecture 6Wed 14 May
Accretion discs and black holes in AGN continued
• Outline of the thin accretion disc
• The viscosity problem
• Spectrum of the this accretion disc
• Black Holes in AGN
Properties of the thin accretion disc
First of all, the disc must be in hydrostatic equilibrium. In the z direction perpen-dicular to the plane of the disc we must satisfy
dP
dz
= �⇢g
z
where g
z
is the vertical component of the gravitational acceleration due to thecentral object,
g
z
⇡GM
R
2
z
R
(using small-angle approximation). We can relate P and ⇢ via the sound speed inthe gas, dP = c
2
s
d⇢, and then integrate to find that the density in the disc falls off
exponentially with height
⇢(z) = ⇢
0
exp
✓�
z
2
2h
2
◆
with a height scale factor given by h
2
= c
2
s
R
2
/GM .
Next we consider the speed of rotation. The particles in the disc willhave orbits very close to Keplerian, so
v
2
rot
=
GM
R
.
The scale height can be re-written in terms of the rotational velocityvia
h
2
=
c
2
s
R
2
v
2
rot
so if we have R � h we must have v
rot
� c
2
s
and so the rotation
of the disc is highly supersonic. The same condition applies to keepcold gas in the plane of the Milky Way with a rotational velocity of⇡ 200 km s�1.
Viscosity in the disc
For the accreting material to fall into tighter orbits in the disc there must be anoutwards flow of angular momentum—a torque acting on the disc. Take the discviscosity to be ⌘ and consider a radius r in the disc with thickness t and angularvelocity !. The tangential force per unit area exerted by the disc inside r on thedisc outside r is given by
F = ⌘r
d!
dr
.
This force acts over an area 2⇡rt so the total torque between adjacent pieces ofthe disc is
� = 2⇡r
3
t⌘
d!
dr
.
Remember torque is rate of change of angular momentum!
Again taking the orbits to be Keplerian we have ! =
qGM
r
2
, so subbing into theprevious equation for the torque we have
dL
dt
= �3⇡⌘t(GMr)
1/2
which is the rate of change of angular momentum of the inner piece of the disc.This must equal the change of angular momentum duw to inflow of disc material,i.e.
dL
dt
= mr
2
! = m(GMr)
1/2
and so we have a relationship between the accretion rate and the disc viscosity,
m = 3⇡⌘t
The problem with viscosity... The problem arises when we consider the Reynolds
number of the material in the disc—a measure of how turbulent it is.
R ⇠V L
⌫
Here V and L are characteristic speed and length scales and ⌫ is the kinematicviscosity, ⌘/⇢. We find (see e.g. Longair pp 145–146) that R ⇠ 10
12. The flowis highly turbulent, and so standard kinetic theory dynamic visosity ⌘ =
1
3
⇢c� willmake a negligible contribution.
We do not yet understand the precise mechanism of viscosity in accretion discs.Highly turbulent flow helps, but precise calculations are difficult. Magnetic fieldswill be present and will certainly contribute. Much of the progress to date hascome from a neat side-step developed by Shakura and Sunyaev (1972). Theyinvented a parameter
↵ =
⌫
hc
s
which allows detailed models to be made without knowing the exact mechanismfor the viscosity.
Luminosity of a thin accretion disc
Neglecting the energy transport due to viscosity, we can calculate the rate atwhich accreting material in the disc must lose gravitational potential energy if it isto fall closer to the accreting object. For an annulus between r and r + dr, theenergy which must be dissipated will be
L(r) = �✓dE
dt
◆=
G
˙
MM
2r
2
dr
where ˙
M is the accretion rate and M is the mass of the central object. Includingviscous energy transport we gain a total luminosity three times this value (non-
examinable—see Longair pp 149-150).
Temperature structure of a physically thin, optically thick disc
If the disc is optically thick, each annulus between r and r + dr will radiate as ablackbody with the luminosity derived above. Hence via Stefan’s Law (remember-ing the disc has two surfaces), the annulus at at r will radiate with 2�T
4⇥2⇡rdr.Thus
�T
4
=
3G
˙
MM
8⇡r
3
and so
T (r) =
✓3G
˙
MM
8⇡r
3
�
◆1/4
Spectrum of the thin disc
We are now in a position to describe the form of the overall spectrum of the disc,i.e. the sum of all the black-body contributions at different radii.
I
⌫
/Z
r
outer
r
inner
2⇡rB
⌫
{T (r)}dr
where from lecture 1 we have
B
⌫
/ ⌫
3
⇣e
h⌫/kT � 1
⌘�1
From before we have T / r
�3/4 so dr / (1/T )
1/3
d(1/T ) and we can integratedT instead of dr. Including B
⌫
explicitly we therefore have
I
⌫
/Z
r
outer
r
inner
✓1
T
◆4/3
⌫
3
⇣e
h⌫/kT � 1
⌘�1
✓1
T
◆1/3
d
✓1
T
◆
We can proceed by changing variable x = (h⌫/kT )—recall from the second-yearthermo problem set where you used the same substitution to derive the functionalform u(T ) / T
4. This yields
I
⌫
/⌫
3
⌫
8/3
Zx
outer
x
inner
x
4/3
(
e
x � 1
)
�1
x
1/3
dx
The integral dx is just a numerical constant so we now have the shape of thespectrum over most of its range:
I
⌫
/ ⌫
1/3
.
Note, though, that the low- and high-frequency ends will have a different form:
• From the outer edge of the disc we will see the Rayleigh-Jeans tail of Touter
,I
⌫
/ ⌫
2.
• From the inner edge, an exponential cut-off I⌫
/�e
�h⌫/kT
inner
�.
Theoretical spectrum of thin accretion disc.
1994ApJS...95....1E
Total luminosity of the thin disc
Let’s now estimate ✏. If we approximate with a Newtonian potential, take theparticle to have started its trip at r = 1 and total energy zero, and calculate thetotal energy it has on the last stable orbit. The amount of GPE which must be lost(by radiation) is equivalent to 1/12 of the rest-mass energy of the particle.
For the best possible case—the closest orbit around a rapidly-rotating black hole—the efficiency rises to a whopping 0.42. Compare this to nuclear fusion in stars,which has an efficiency of only 0.7 percent!
Hence in practice, astronomers usually adopt an approximate value of ✏ = 0.1
for accretion onto black holes.
Example: estimating a quasar accretion rate
Suppose we observe a quasar to have a total power output of 1040 W. We arenow in a position to estimate the mass of the central black hole and the rate atwhich its mass is increasing.
First let us assume that the accretion is Eddington limited. From our equation forthe Eddington luminosity we have
L
Edd
=
4⇡GMcm
p
�
T
from whichM = 7⇥ 10
8
M�.
And from our Eddington accretion rate, using ✏ = 0.1 we have
˙
M
Edd
=
4⇡GMm
p
✏c�
T
⇡ 3M�yr�1
Getting round the Eddington limit
The accretion may not always be Eddington-limited. It is, for example, possible toachieve ˙
M much greater than would be inferred by using the Eddington luminositywith ✏ = 0.1, by making the disc physically thick, and very low density, so that it isoptically thin and matter doesn’t have time to radiate away so much energy beforeit falls over the horizon. This has the advantages of allowing black holes to growat a very high rate in the early Universe, and also of providing “funnels” whichcould be a mechanism for collimating outflows from accreting objects.
Unfortunately simple analytical models of these discs are unstable, but the ad-vantages of thick discs are so great that much effort is put into modelling themnumerically... including the effects of strong magnetic fields.
It is also possible for an object to have a luminosity significantly greater than theEddington luminosity:
• In supernovae (somewhat trivially!)
• Where spherical symmetry is broken, with extremely collimated radiation ina direction different to the accretion direction
• Where accretion is not steady, e.g. bursts or radiation emitted when discreteclouds of matter fall onto a neutron star or white dwarf.
Evidence for black holes in AGNThere are several canonical pieces of evidence that supermassive black holesreally are there at the heart of AGN. Among these are:
• Variability (in combination with Eddington luminosity).
• Stellar velocity dispersions.
• Rotation speeds inferred from emission lines.
• The controversial history of X-ray line profiles.
The controversial story of the iron K-↵ lineor
“The Good, The Bad and The Ugly”
442 L. Miller et al.: The X-ray spectral variability of MCG–6-30-15
Fig. 2. Principal component spectra for (top) Suzaku data and (bottom) XMM-Newton data. Left-hand panels show limits on the o�set component,right-hand panels show eigenvector one. Spectral points in the range 0.4�11 keV are from the��or���instruments, those in the range 15�45 keVare from the���. See text for further details.
of photon index can produce similar absorption spectra but witha shift in the e�ective ionisation parameter. The gas density wasassumed to be 1010 cm�3 for all zones except the highest ion-isation zone, where a value 108 cm�3 was assumed. Althoughthis parameter a�ects the absorption spectral shape, this againis largely degenerate with the ionisation parameter, although ifthe assumed absorbing shell is su�ciently thick (as may occurfor the combination of high � and low density values) then therecan be significant variation in � through the zone arising frominverse-square-law dilution, thus leading to a broader range ofionisation states than might be the case in a denser, thinner shellof gas. All the absorption models are a�ected by such uncertain-ties. We adopt units of erg cm s�1 for �.
4.2.2. Joint fit to eigenvector one and offset components
Qualitatively, the PCA “eigenvector one” in both the Suzaku andthe XMM-Newton data has the appearance of a powerlaw af-fected by ionised absorption. In reality the absorbing zones arecomplex, and their physical parameters are unlikely to be unam-biguously measurable in data with CCD resolution. However,we know a priori of the existence of ionised absorbing zonesfrom the previous analyses of Chandra and XMM-Newton high-resolution grating data, so we can start by seeing whether amodel that includes those zones can explain the CCD spectrawe observe.
We first create a model based on the simplest interpreta-tion of the PCA, namely that the eigenvector one represents avariable-amplitude powerlaw, with ionised absorption, and that
the o�set component arises from distant reflection, with lighttravel-time erasing any reflected amplitude variations. The pri-mary absorbing layers that have already been identified in thegrating data comprise:
Zone 1 with log � � 2 (Lee et al. 2001);
Zone 2 with log � � 0.5 (Lee et al. 2001);
Zone 3, a highly ionised, log � >� 3.5, outflow at line-of-sight ve-locity v � 1800 km s�1 (Young et al. 2005). The 6.7 and 6.97 keVlines only appear in the o�set component in the PCA, not oneigenvector one, so for now we assume that this zone is onlyassociated with the o�set component (when fitting to the datalater, we allow zone 3 to absorb all components). This zonealso produces velocity-shifted lines of 2.0 keV Si���Ly� and2.62 keV S���Ly� which were observed by Young et al. (2005)in the Chandra���data. In this absorption layer the ratio ofthe 6.7 and 6.97 keV lines depends on both the ionisation and onthe microturbulent velocity dispersion of the gas, since the linesare easily saturated, and the relatively high equivalent width ofthe lines is most easily achieved by models with line broad-ening. Hence the absorption model is broadened by a velocitydispersion of 500 km s�1, consistent with the findings of Younget al. (2005) (see Sect. 5), and we fix the ionisation parameter atlog � = 3.85 as described later in Sect. 5.1.
Fe I edge at 0.707 keV. A further feature apparent in the high-resolution data, but not in data of CCD resolution, is the com-plex absorption edge structure at �0.7 keV, discussed exten-sively by Branduardi-Raymont et al. (2001), Sako et al. (2003),
444 L. Miller et al.: The X-ray spectral variability of MCG–6-30-15
Fig. 3. The fit of model B to the principal component spectra: (left) Suzaku 0.5�45 keV, (right) XMM-Newton 0.4�10 keV; (top) eigenvector one,(bottom) o�set component. The model is shown in units of Ef(E), points with error bars show the “unfolded” component spectrum values.
Table 5. Fit parameters and statistics for the joint fit of Model B toeigenvector one and the o�set components, for each dataset, table en-tries as in Table 4.
Model Bparameter Suzaku XMM-Newton
� 2.23 2.21log �REFLIONX 2.04 1.94Galactic NH 0.051 0.045Chandra�����& XMM-Newton���zoneszone 1 log � 2.36 2.14
NH 0.45 0.41zone 2 log � 0.22 �0.42
NH 0.07 0.04zone 3 log � (3.85) (3.85)
NH 2.60 2.90additional o�set component absorption zoneszone 4 log � 1.95 2.04
NH 34.0 28.1zone 5 log � 1.35 1.89
NH 3.59 9.81
�2/d.o.f. 261/332 257/309
5. Results – model fits to data
5.1. Fitting methodology and initial constraints
Fitting to the PCA components should only be considered asyielding an indication of the model components that may be
required, for two reasons. First, although the o�set componentand eigenvector one alone provide a good description of thevariable X-ray spectrum at energies above 2 keV (Tables 2, 3),more complex source behaviour is implied at softer energies.Second, there is no unique interpretation of the o�set compo-nent: this component essentially describes the appearance of thesource around its lowest possible flux state, and the o�set com-ponent spectrum we deduce may either be pure reflection orpure absorption, for a source with a variable absorption cover-ing fraction, or some combination of the two, as indicated bythe PCA. Hence we need to test the model against the actualdata. This is done in this section. Model B is summarised byFig. 4 which shows the model fitted to the mean Suzaku spec-trum as described below and showing the three emission compo-nents of the model (“direct” power-law, with ionised absorption;“partially-covered” power-law, with higher opacity absorptionfrom zone 5; and low-ionisation reflection with absorptionfrom zone 4). We also show the component of cosmic X-raybackground emission included in the fit to������data.
The full dataset that we investigate here comprises obser-vations taken at a number of epochs with a variety of instru-ments. In fitting to the data we require the model to fit simultane-ously any data that were obtained simultaneously (e.g.���and��data for XMM-Newton or���and���data for Suzaku ). Wedo allow variations in model parameter values between datasetstaken at di�erent epochs, although the model components arenot changed.
Accretion history and remnant black holes today
From the known lifetimes of radiosources and the Eddingtonmass-doubling time for SMBH accretion, we expect that anindividual AGN episode is only a short fraction of the lifetime of anindividual galaxy. Hence we now have a picture in which mostmassive galaxies underwent an AGN phase early in their history.This implies that there are relic quiescent SMBH’s in the centre ofmost galaxies today.
Accretion history contd.
That the quiescent black holes in today’s galaxies are the relics ofAGN activity in the past can be shown by considering the diffuse
high-energy background as measured by satellite, at energiesranging from UV upwards.
The background high-energy flux which we observe across the skyis dominated by the individual contributions from AGN emitting atthe epoch of their peak activity, at around z ⇡ 2–3.
Accretion history contd.
If we integrate under the X-ray background spectrum, we find thatthere is a mean energy density in the local Universe✏ ⇡ 10�16Jm�3.
Let us approximate that this energy was all released at z = 2.Radiation energy density scales as (1 + z)4 so this implies anenergy density at the peak of the quasar era of 81✏.
Using our standard accretion efficiency of 10%, this implies thatthere was a mass density in recently-fed black holes of 810✏/c2 atz = 2. Now, as the Universe expands mass density scales as(1 + z)3, which means that there will be a mass density in blackholes at the present day equal to 30✏/c2.
Accretion history contd.
Now let us use our numbers: ✏ ⇡ 10�16Jm�3.
This gives us a present-day mass density in relic black holes ofabout 3⇥ 10�32 kg m�3.
If we take the mean number density of galaxies today (0.01Mpc�3) and allow one black hole to inhabit each galaxy, we find amean black hole mass per galaxy of 5⇥ 107M�.
Can we see why this is an underestimate?
The radio-loud/radio-quiet dichotomy
Although we now know that even low-luminosity AGN such asSeyferts have jets, the extraordinarily powerful jets of the mostluminous sources gives rise to a traditional breakdown of AGN into“radio loud” and “radio quiet” categories.
The classical radiogalaxies and quasars are generally taken to fallinto the radio-loud category, and lower luminosity objects such asthe Seyferts fall into the radio-quiet class.
However the a complication arises in that there seem to be manyobjects with the same optical/UV/X-ray properties as the(radio-loud) quasars, but which lack strong radio jets. To avoidconfusion we shall refer to these as radio-quiet quasars.
