Modelling accretion disk spectra from active galactic nuclei
by Manish Rawat
16 August, 2016
1 Active Galactic Nuclei (AGN)
1.1 Introduction
In some galaxies, it is observed that galaxy is overshadowed by violent and luminous activity in a very
concentrated volume at center of the galaxy. This violent and luminous activity is produced in the center
(nucleus) of such galaxies which are called active galactic nuclei (AGNs). AGNs are the center of the
galaxies which have higher luminosity than normal galaxies and this luminosity (i.e., excess emission)
is observed in almost over all of the electromagnetic spectrum. The galaxies which have such high
luminous centers are called Active Galaxies.
The luminosities of an AGN is such large that it outshines the entire host galaxy. Not only is the
luminosity is intense, but it usually fluctuates in intensity as well. At least 10 % of all known galaxies
have active cores, and in many instances they exhibit intense radio emission and other activity outside
their core in addition to their AGNs.
1.2 Types of Active Galaxies
Active galaxies can be broadly be classified as radio-loud and radio-quiet.
The active galaxies which have ratios of radio (5 GHz) to optical (B-band) flux, F FB 5 ≥ l0 are called
radio-loud, otherwise radio-quiet.
Roughly 15%-20% AGN are radio-loud active galaxies. Radio-loud galaxies have emission contribution from both jets and the lobes that the jets inflate. These emission contributions dominate the
luminosity of AGN at radio wavelengths and possibly at some other wavelengths.
Radio-quiet objects are simpler since jet and any jet-related emission can be neglected at all wavelengths.
Radio-Quiet Active Galaxies
1.2.1 Seyfert Galaxies
The Seyfert galaxies are named after Carl Seyfert, who discovered them in 1943. Their most important
characteristics are a bright, point like central nucleus and a spectrum showing broad emission lines.
The continuous spectrum has a non-thermal component, which is most prominent in the ultraviolet.The
emission lines are thought to be produced in gas clouds moving close to the nucleus with large velocities.
Host galaxies of Seyfert are spiral or irregular galaxies.
Seyfert galaxies can further be classified as Seyfert 1 and Seyfert 2 galaxies.
1Seyfert 1 galaxies have very broad emission lines that include both allowed lines (H I, He I, He
II) and narrower forbidden lines (such as [O III]). Seyfert 1 galaxies generally have “narrow” allowed
lines as well, although even the narrow lines are broad compared to the spectral lines exhibited by
normal galaxies. The width of the lines is attributed to Doppler broadening, indicating that the allowed
lines originate from sources with speeds typically between 1000 and 5000 km/sec, while forbidden lines
correspond to speeds of around 500 km/sec.
Figure 1: The visible spectrum of Mrk 1243, a Seyfert 1 galaxy. Here Mrk indicates an entry in the
galaxy catalog of E.B. Markarian (1913-1985), produced in 1968
Seyfert 2 galaxies have only narrow lines (both permitted and forbidden) with characteristic speeds
of about 500 km/sec. This implies that gas emitting the light is not moving so violently and these
galaxies are bright at infrared wavelengths. Seyfert 1 are also called NLS1 (Narrow Line Seyfert 1).
2Figure 2: The visible spectrum of Mrk 115, a Seyfert 2 galaxy
1.2.2 Radio-quiet Quasars
The radio-quiet quasars are high luminosity AGNs. These are same as Seyfert 1 galaxies except for their
high luminosity. The distinction between Seyfert 1 and radio-quiet quasars is arbitrary and is expressed
as in terms of limiting optical magnitude. Quasars were first seen as “quasi-stellar” (star-like) in optical
images as their optical luminosity is greater than their host galaxy. They are typically seen at greater
distances because of their relative rarity locally and thus rarely show an obvious galaxy surrounding the
bright central source.
They show strong optical and X-ray continuum emission and also they show broad and narrow
optical emission lines. As these are radio-quiet active galaxies, this class of AGN is called quasi-stellar
objects (QSOs). The term quasar is usually referred to radio-loud quasars. Although the term “quasar”
is now used commonly for both radio-quiet and radio-loud quasars.
1.2.3 Low Ionization Nuclear Emission-line Regions (LINERs)
The low ionization emission-line regions or the LINERs active galaxies show only weak nuclear emission
line regions. These galaxies have very low luminosities in their nuclei, but with fairly strong emission
lines of low-ionization species such as forbidden lines of [O I] and [N II]. The spectra of LINERs is
similar to the low-luminosity end of Seyfert 2 class, and LINER signatures are detected in many spiral
galaxies in high sensitivity studies. These low ionization lines are also detectable in starburst galaxies
and in H II regions. These galaxies do not show any other signs of a true AGN (powered by accretion
onto a supermassive black hole). So, it is unclear that these galaxies truly represent AGNs. If they
represent, then they are lowest luminosity classification of radio-quiet galaxies.
3Figure 3: Sombrero Galaxy (M104), a LINER galaxy. Credit: Hubble Space Telescope
Radio-Loud Active Galaxies
1.2.4 Radio Galaxies
Radio galaxies are powerful radio sources. They emit nuclear and extended radio emission. The radio emission of a radio galaxy is non-thermal synchrotron radiation. The radio luminosity of radio
galaxies is typically 1033 − 1038 W between 10 MHz and 100 GHz. The host galaxies of radio galaxies
are elliptical galaxies.
Like Seyfert galaxies, radio galaxies can also be divided into two classes by their emission lines:
broad-line radio galaxies (BLRGs, corresponding to Seyfert 1s) and narrow-line radio galaxies
(NLRGs, corresponding to Seyfert 2s).
BLRGs have bright, starlike nuclei surrounded by very faint, hazy envelopes. NLRGs are giant or
supergiant elliptical galaxies.
Despite their similarities, there are differences between Seyfert and Radio galaxies. Although Seyfert
nuclei emit some radio energy, they are relatively quiet at radio wavelengths compared with radio
galaxies. Also, while nearly all Seyferts are spiral galaxies, none of the strong radio galaxies are spirals.
1.2.5 Radio-loud Quasars
Radio loud quasars have very high luminosity and behave exactly the same way as the radio quiet
quasars with the addition of emission from a jet. They show strong optical continuum emission, broad
and narrow emission lines, and strong X-ray emission, together with nuclear and often extended radio
emission.
1.2.6 Blazars
The properties of rapid variability and a high degree of linear polarization at visible wavelengths define
the class of AGNs knows as blazars The most well-known object in this class is BL Lacertae, found in
4the northern constellation of Lacerta (the Lizard). BL Lac was originally classified as a variable star
because of its irregular variations in brightness; hence the variable star type of designation. In a week’s
time BL Lac would double its luminosity, and it would change by factor of 15 as the months passed.
But although BL Lac has a stellar appearance, its spectrum shows only a featureless continuum with
very weak emission and absorption lines. Careful observations reveal that the bright, starlike nucleus
of BL Lac is surrounded by a fuzzy halo that has a spectrum similar to that of an elliptical galaxy.
BL Lac objects are a subclass of blazars that are characterized by their rapid time-variability.
Their luminosities may change by upto 30% in just 24 hours and by a factor of 100 over a longer time
period. BL Lacs are also distinguished by their strongly polarized power-law continua (30-40% linear
polarization) that are nearly devoid of emission lines. However, observations of a few faint spectral lines
have revealed high redshifts, so that, like quasars, BL Lacs are at cosmological distances. Of those BL
Lacs that have been resolved, about 90% appear to reside in elliptical galaxies.
Another class of Blazars is optically violently variable quasars(OVVs). They are similar to
the BL Lacs except that they are typically much more luminous, and their spectra may display broad
emission lines.
2 Another Classification of Active Galaxies
This classification is based on according to the radio loudness and optical spectra of AGNs, i.e., whether
they have broad emission lines (Type 1), only narrow lines (Type 2), or weak or unusual line emission.
the following figure shows the principal classes of AGN (adapted from Lawrence 1987, 1993) based
on this classification. Within each of the groupings, different types of AGN are listed by increasing
luminosity.
With few exceptions, the optical and U.V. emission-line spectra and infrared to soft X-ray continuum
of most radio-loud and radio-quiet AGN are very similar and so must be produced in more or less the
same way.
Based on characteristics of their optical and U.V. spectra, AGN can be separated into three types:
1. AGNs having bright continua and broad emission lines from hot, high velocity gas, presumably
located deep in gravitational well of central black hole are called Type 1 AGN.
In radio-quiet, these include Seyfert 1 galaxies, which have relatively low-luminosities and are only
seen nearby and high-luminosity radio-quiet quasars (QSQs), seen at greater distances.
5Radio loud type 1 AGN called broad-line radio galaxies (BLRG) at low luminosities and radio-loud
quasar at high luminosity, either Steep spectrum radio quasars (SSRQ) or Flat spectrum radio quasars
(FSRQ) depending on radio continuum shape, with dividing line set at αr = 0:5
2. Type 2 AGN have weak continua and only narrow emission lines, meaning either that they
have no high velocity as or, line of sight to such gas is obscured by a thick wall of absorbing material.
Radio-quiet type 2 AGN includes Seyfert 2 galaxies at low-luminosities, as well as narrow emission-line
X-ray galaxies.
Radio loud type 2, also called Narrow-line radio galaxies (NLRGs), is of two types:
Low-luminosity Fanaroff-Riley type I (FR I) radio galaxies which have often symmetric radio
jets whose intensity falls away from nucleus and high luminosity Fanaroff-Riley type II (FR II)
radio galaxies which have more highly collimated jets leading to well defined lobes with prominent hot
spots.
3. A small no. of AGN have very unusual spectral characteristics and called Type 0 AGN and
speculate that they are related by a small angle to the line of sight (“near 0 degrees”). These include BL
Lacertae (BL Lac) objects, which are radio loud AGN that lack strong emission or absorption features.
10% of radio-quiet have unusually broad absorption features in their optical and U.V. spectra and are
called as BAL (Broad Absorption Line).
If BAL special features are caused by polar outflows at small angles to line of sight, then they are
also type 0 AGN; alternatively they may have edge-on disks with winds instead. These are no known
radio-quiet BL Lacs.
3 Cause of Activity in Galaxies
Any successful model for active galactic nuclei must explain how such a small central region can emit
so much energy over such a broad range of wavelengths. Astronomers then hypothesize that the core
must contain something very unusual. No ordinary single star could be so luminous. No ordinary group
of stars cloud be packed into so small in a region.
One kind of object that is very small but that can also emit intense radiation is the accretion disk
around a black hole. When matter falls toward a black hole, it generally does not fall straight in,
but ends up orbiting just outside the black hole, where it collides with other matter and emits intense
radiation.
4 Accretion Disk and Its Basic Physics
An accretion disk is a rotating disk of matter around a central massive object such as a star or a black
hole. Gravity of central object causes the matter to slowly spiral inwards toward the central object.
Then in orbit it continuously looses angular momentum and falls to the surface of central object.
For a body of mass M and radius R∗, gravitational potential energy released by accretion of mass m
onto its surface is:
∆E
acc =
GMm
R∗
(1)
If accreting body is a neutron star with radius R∗ ≈ 10km and mass M approax to one solar mass,
then yield ∆Eacc is about 1013 J per accreted kg, which is huge.
6Now, the AGNs have supermassive black holes at the centre of the galaxy and their mass is million
to billion times of solar mass. Hence, the energy released when gases present in accretion disk losses
its energy as it spirals towards black hole is very large as compared to above example of neutron star,
which is the source of intense radiation from AGNs.
Thus we see larger the ratio R M ∗, greater is the energy released from accretion disk.
5 Methods of Black Hole Mass Determination
5.1 Using Kepler’s and Newton’s laws
This technique is old and simply uses the Kepler’s and Newton’s laws of gravitation.However, not every
black hole can be measured like that. The method best applies to stellar black holes that are part of
a binary system and live together with a companion star. It can sometimes be used for suppermassive
black hole founds in the centres of galaxies. In galaxy, majority of stars are part of binary systems or
higher systems. Solitary stars such as our sun make only 20%. So, it is usually excepted that most of
the stellar black holes are part of the binary systems. Practically, it is not possible to find a lonely black
hole as there is nothing to reveal its presence in its vicinity.
The method is as follows:
It is seen that the stars near the galactic nuclei, i.e., near the center of galaxy orbits faster to the
center in a very compact region. It is suggested that there must be a black hole present for this rapid
movements of stars in such a compact region.
Let M be the mass of the central black hole and m be that of any one of the star orbiting it. Then
according to the Newton’s law of gravitation, gravitational attraction between the star and black hole
is
Fg
=
GMm
r2 (2)
in radially inward direction toward centre of black hole.
The centripetal force due to gravitational attraction is:
Fc
= mr!2 (3)
where ! is angular velocity of star’s orbit.
Then from F = ma, we have equation of motion
m
d2r
dt2 = −
GMm
r2 + mr2!
or,
d2r
dt2 − mr2! = −GM r2 (4)
Now, as angular momentum remains constant under inverse square law, then L = mr2!
Replacing ! in equation (4) by L, we have:
d2r
dt2 = −
GM
r2 +
L2
mr3 (5)
7To solve equation (5), let u = 1 r.
Now, L = mr2! = mr2 dθ dt
so, dt d = Lu m2 dθ d
As, r = 1
u
, then dr dt = dt d ( u 1) = − u 12 dt d = − m L dθ d
Similarly, d dt 22 r = − L m 2u 22 d dθ 2u 2
Using these values in equation (5), we get
d2u
dθ2 + u =
GMm2
L2 (6)
On solving this differential equation, we get it’s solution as:
u =
1 r
=
GMm2
L2 + Acosθ (7)
where A is a constant of integration, determined by initial conditions.
On comparing equation (7) with standard polar form (r,θ) of an ellipse
a(1 − e2)
r
= 1 + cosθ (8)
we get,
a(1 − e2) = L2
GMm2 (9)
where a is semi-major axis of elliptical orbit.
If b is semi-minor axis of elliptical orbit then, we know
b2 = a2(1 − e2)
Now, we know area of ellipse Πab and from Kepler’s second law, rate of sweeping this area is 2L m,
Then if T is time period for one complete elliptical orbit, we have
T = πab
L=2m (10)
On squaring equation (10),
T 2 = 4π2a2b2m2
L2
8or,
T 2 = 4π2a2b2m2
a(1 − e2)GMm2
or,
T 2 = 4π2a2b2
aGM(b2=a2)
or,
T 2 = 4π2
GM ! a3 (11)
or,
GM
4π2 =
a3
T 2 ) M = 4G π2 ! T a3 2 (12)
Equation (11) is Kepler’s third law, and, shows that mass of central object (here black hole) is
proportional to the ratio of the cube of semi-major axis and time period of elliptical orbit of an orbiting
body to the central object and equation (12) can be used to find mass of the black hole if we observe
any orbiting star to it and measure the parameters of elliptical orbit ,i.e., semi-major axis and time
period of elliptical orbit.
The most well-determined mass with help of studying the motions of individual stars around supper
massive black hole is that of Sgr A∗, which is supper massive black hole at center of our Milky Way
galaxy. Two decades of observations of the proper motions and radial velocities of individual stars (e.g.,
Genzel, Eisenhauer, and Gillessen 2010; Meyer et al. 2012) enabled by the combination of advanced
infrared detectors and adaptive optics on large telescopes, has led to a measurement of a black hole
mass of 4.1(±0:4) × 106 M
5.2 By Reverbation Mapping of AGNs
It is known that continuum emission from AGNs varies on short-time scales since the discovery of
quasars. It was also figured out that emission-line varies too, which were in each case quite extreme
changes. With advancement of sensitive electronic detectors in 1980s, a strong connection between continuum and emission-line variability was established. An older idea (Bahcall, Kozlovsky, and Salpeter
1972) about how emission-line region structure could be probed by variability was cast into a mathematical formalism by Blandford and McKee (1982) and given the name reverberation mapping.
Theory of Reverberation Mapping
5.2.1 Assumptions
From the observations of AGNs, following assumptions can be made:
91. The emission lines respond rapidly to continuum changes, showing that the BLR (Broad Line
Region) is small (because the light-travel time is short) and the gas density in the BLR is high (so the
recombination time is much shorter than the light-travel time). It is also noted that the dynamical
timescale (of order RBLR=∆V ) of the BLR is much longer than the reverberation timescale (of order
RBLR=c), so the BLR is essentially stationary over a reverberation monitoring program.
2. The continuum-emitting region is so small compared to the BLR, it can be considered to be a point
source. It does not have to be assumed that the continuum emits isotropically, though that is often a
useful starting point.
3. There is a simple, though not necessarily linear or instantaneous, relationship between variations of
the ionizing continuum (at < 912 A) and the observed continuum (typically at 5100 A). The fact
that the reverberation works at all, justifies this at some level of confidence.
5.2.2 The Transfer Equation
During reverberation monitoring program, the continuum behaviour over time can be written as C(t) =<
C > +∆C(t) and the emission-line response as a function of line of sight velocity VLOS is L(VLOS; t) =
< L(VLOS) > +∆L(VLOS; t) where < C > and < L(VLOS) > represent mean values. The continuum
and emission line variability are small on reverberation time scale. So, the relationship between them
can be modelled as linear on short timescales.
Then the relation between the two is:
∆L(VLOS; t) = Z Ψ(VLOS; τ)∆C(t − τ)dτ (13)
which is called transfer equation. and Ψ(VLOS; t) is the transfer function. This equation shows
that Ψ(VLOS; τ) is the observed response to a δ-function continuum outburst.
The transfer function is the six dimensional phase space of BLR projected into the observable
coordinates, line of sight velocity (Doppler effect) and time delay relative to the continuum variations.
Then, transfer function can be constructed geometrically. It is therefore common to refer to the transfer
function Ψ(VLOS; τ) as a velocity delay map. It should be clear that each emission line has a different
velocitydelay map because the combination of emissivity and responsivity is optimized at different
locations of the BLR for different lines. To map out all of the BLR gas would require velocity delay
maps for multiple emission lines with different response timescales.
5.2.3 Construction of a Velocity Delay Map
Consider a simple BLR model, in which a circular ring of gas orbiting counterclockwise at speed vorb
around the central source at distance R. Let a distant observer sees the system edge-on (Figure 4), defines
a polar coordinate system centered on the continuum source at an angle θ measured from observer’s
line of sight. Two clouds are shown at (R,±θ).
Photons from a δ-function continuum outburst travel toward observer along −xaxis. The dotted
line shows the path taken by an ionizing photon from same outburst will reach BLR cloud after a travel
time R
c ; an emission-line photon produced in response by the cloud and if directed toward the distant
observer travels an additional distance R cos θ
c . After travelling additional distance, this photon will be
10at the same distance from the observer as the continuum source. Hence, these emission line photons
are delayed by sum of two dotted segments,i.e, by
τ = (1 + cos θ)R=c (14)
The locus of points that all have the same time delay to the observer is labeled as an isodelay surface
in the top part of the figure. This isodelay surface is a paraboloid.
The corresponding Doppler shifts of the clouds at coordinate (R; ±θ) are ∓vorb sin θ. These transformations are general, and a ring of radius R and orbital speed vorb projects in velocity delay space
to an ellipse with axes 2 vorb centered on VLOS = 0 and 2R/c centered on R c . Here the ring is pictured
edge-on, at inclination 90; at any other inclination i, the projected axes of the ellipse in velocity delay
space are correspondingly reduced to 2 vorb sini and 2Rsini/c. Thus, a face-on (i = 0) disk projects
in velocity delay space to a single point at (0, (R=c); all of the ring responds simultaneously and no
Doppler shift is detected.
Figure 4: Top: A notional BLR comprised of clouds orbiting the central source counterclockwise in a
circular orbit of radius R. Bottom: the same circular BLR projected into the observable quantities of
Doppler velocity and time-delay; this is a very simple velocity delay map.
Let us assume that line-radiation emitted by individual clouds is isotropic,i.e., Ψ(θ) = , a constant
To transform this to the observable velocity time delay coordinates, we have:
11Ψ(τ)dτ = Ψ(θ) dθ
dτ dτ
From equation (14),
dτ
dθ = −
R c
sin θ
Then, using these equations, we have
Ψ(τ)dτ = c
R(2cτ =R)1=2(1 − cτ =2R)1=2 dτ (15)
and the mean response time is
< τ >=
R τΨ(τ)dτ
R Ψ(τ)dτ =
R c
(16)
A more realistic assumption is that much of the line emission is directed back toward the ionizing
source because the BLR clouds are very optically thick even in the lines. A simple parametrization is
that Ψ(θ) = (1 + A cos θ)=2. Isotropy is the case A = 0 and complete anisotropy (which is, incidentally,
perhaps appropriate for Lyαλ1215) corresponds to A = 1 (Ferland et al. 1992).
A useful measure of the line width is the line dispersion σline; for a ring, the line dispersion is:
σline = (< VLOS 2 > − < VLOS >2)1=2 = vorb
21=2 (17)
For comparison, for such a ring, FWHM=2vorb
A velocity delay map for any other geometry and velocity field can be constructed similarly.
5.2.4 Reverberation Mapping Results: Lags
The primary goal of most reverberation monitoring campaigns that have been undertaken to date has
been to determine the mean response time of the integrated emission line, i.e., to measure the lag
between continuum and emission line variations. The most up-to-date methodology for making these
measurements is that described by Zu, Kochanek, and Peterson (2011). The first high-sampling rate
multi wavelength reverberation monitoring program was undertaken in 1988 89 by the International
AGN Watch (Clavel et al. 1991; Peterso et al. 1991; Maoz et al. 1993; Dietrich et al. 1993; Alloin et
al. 1994).
Programs to measure the C iv λ1549 lag in a very low-luminosity AGN (Peterson et al. 2005) and
a very high-luminosity AGN (Kaspi et al. 2007) have also been reported.
These studies have led to several important findings:
1. Within a given AGN, the higher-ionization lines have smaller lags than the lower ionization lags,
demonstrating ionization stratification of the BLR. This also shows that the BLR gas is distributed over
a range of radii from the central source, and the lag for a particular emission line represents the radius
at which the combination of emissivity and responsivity is optimized for that particular emission line.
2. The variable part of the emission line can be isolated by constructing the rms residual spectrum
from all monitoring data. In the rms residual spectrum, the higher ionization lines are broader than
the low ionization lines, in such a way that the the product ∆V 2τ is constant within a given source,
suggesting that the BLR is virialized.
3. There is a relationship between the size of the BLR R and the AGN luminosity L of the approximate form R
125.2.5 Reverberation-Based Black Hole Masses
Virial Mass Estimates
For every AGN for which emission-line lags and line widths have been measured, consistency with the
virial relationship is found. This also appears to be true when the lag and line width are measured
for the same emission line when the AGN is in very different luminosity states. This strongly suggests
that the BLR dynamics are dominated by the central mass, which is then
MBH = f(∆V 2R
G ) (18)
where ∆V is the line width and R is the reverberation radius cτ . The quantity in parentheses that
contains the two directly observable parameters has units of mass and is sometimes referred to as the
virial product. The effects of everything unknown the BLR geometry, kinematics, and inclination are
then subsumed into the dimensionless factor f, which will be different for each AGN, but is expected to
be of order unity. Presumably, individual values of f can be determined if there is some other way of
determining the black hole mass. In the absence of a second direct measurement, it has been common
practice to use the MBH∗ relationship for this purpose. By assuming that the MBH∗ relationship is
the same in quiescent and active galaxies, it becomes possible to compute a mean value for the scaling
factor, which turns out to be < f > ∼ 5. Figure 5 shows the MBH∗ relationship for quiescent galaxies
and AGNs using the assumption that <f> = 5.25. The scatter around this relationship amounts to about
∼ 0.4 dex, which is a reasonable estimate of the accuracy of the virial method of estimating black hole
masses.
Figure 5: The MBH − σ∗ relationship. Quiescent galaxies are shown in black and AGNs are shown in
blue. Woo et al. (2010)
135.3 By Modelling Accretion Disk Spectra
5.3.1 The Spectra of Active Galactic Nuclei
Figure 6: A Sketch of continuum observed for many types of AGNs
The figure 6 shows a rough schematic of the continuum observed for many types of AGNs. The notable
feature of this spectral energy distribution (SED), is its persistence over some 10 orders of magnitude in frequency. This wide spectrum is different from thermal (blackbody) spectrum of a star or the
combined spectra of a galaxy of stars.
When AGNs first studied, it was thought that their spectra were quiet flat. Accordingly, a power
law of the following form:
Fν
/ ν−α (19)
was used to describe monochromatic energy flux Fν (energy per unit area per second on a detector
aimed at source). α is called spectral index
The power received within any frequency interval between ν1 and ν2 is:
Linterval / Zν1 ν2 Fνdν = Zν1 ν2 νFν dν ν = ln 10 Zν1 ν2 νFνd log10 ν (20)
14so that equal areas under a graph of νFν vs. log10 ν corresponds to equal amounts of energy.
The continuous spectra of AGNs are known to be more complicated, involving a mix of thermal and
non-thermal emission. However, equation (1) is still used to parametrize the continuum. The spectral
index typically has a value between 0.5 and 2 that usually increases with increasing frequency, so the
curve of log10 νFν vs. log10ν in figure 6. The thermal component appears as the Big Blue Bump
(BBB) which can contain an appreciable amount of bolometric luminosity (luminosity over entire
wavelength range) of source. The emission from BBB is due to an optically thick accretion disk. There
is a therma infrared bump (IR Bump) to the left of the BBB. It is due to emission from torus.
The BBB is caused by the accretion disk and is a multi-temperature blackbody spectrum. We will
model the accretion disk spectra and with help of it will find black hole mass and accretion rate.
The figure below is the blackbody spectrum of the Planckian cure of single bodybody temperature
of 6000 K.
0
5×1012
1×1013
1.5×1013
2×1013
2.5×1013
3×1013
3.5×1013
0 5000 10000 15000 20000 25000 30000
Spectral Radiance (W (sr)-1 (m)-3)
Wavelength (in Angestrom)
Dependence of Spectral Radiance of Blackbody on Wavelength
Spectral Radiance
Figure 7: Blackbody spectrum for a temperature of 6000 K
5.3.2 The Sample
We have used 3 samples from the Yuan et al. (2008). The SDSS uses five filters namely ultraviolet
(u), green (g), red (r), infrared (i), infrared (z). The GALEX has two filters, near UV (NUV), far UV
(FUV). The WISE has four filters, W1, W2, W3, W4 (all infrared). The following table shows the
magnitude of the sources in different filters mentioned. The SDSS and GALEX uses the AB magnitude
system and WISE uses the Vega system.
The Source u filter g filter r filter i filter z filter NUV FUV W1 W2 W3 W4
SDSS J081432.11+560956.6 18.33 18.06 18.11 17.99 18.03 18.76 19.12 14.221 13.139 10.411 8.428
SDSS J085001.17+462600.5 19.82 19.12 18.82 18.47 18.43 21.17 21.75 14.622 13.808 11.504 8.235
SDSS J095317.09+283601.5 19.28 18.99 18.97 18.77 18.73 19.74 20.27 14.809 13.924 11.335 8.310
15The following table shows the frequencies used for various filters:
Filter Frequency (in Hz)
u 8 × 1014
g 6:4 × 1014
r 4:8 × 1014
i 4 × 1014
z 3 × 1014
NUV 1:32 × 1015
FUV 1:98 × 1015
W1 8:82 × 1013
W2 6:52 × 1013
W3 2:5 × 1013
W4 1:36 × 1013
Now, we change the the magnitudes of the SDSS, GALEX and WISE into flux (erg cm−2sec−1Hz−1)
The SDSS and GALEX magnitudes are in AB magnitude system. To convert these magnitudes into
flux, we use following formula:
mAB = −2:5 log10fν(erg cm−2 sec−1 Hz−1) − 48:6 (21)
where fν is the flux.
The WISE magnitudes are in Vega system. We first convert these into AB system and then will
convert these into flux by using equation (21). The conversion formula is:
mAB = mV ega + ∆m (22)
The following table gives value of the ∆m for various filters of the WISE:
Band Magnitude Offset(∆m)
W1 2.683
W2 3.319
W3 5.242
W4 6.604
Now, the luminosity distance (in Mpc) as given on NED server (NASA/IPAC Extragalactic Database)
for the sources we have used is as:
The Source Luminosity Distance, D (in Mpc)
SDSS J081432.11+560956.6 2938
SDSS J085001.17+462600.5 3037
SDSS J095317.09+283601.5 4017
Now, 1Mpc = 3:086 × 1024 cm, and
Luminosity (L) is calculated from flux (fν) and luminosity distance (D) as follows:
L = 4πD2fν (23)
16We will draw the spectral curve for our sample sources between the log10 νLν on y-axis and log10ν on
x-axis. Here, ν is the frequency of the various filters of SDSS, GALEX and WISE. For this, we have
done the following procedure:
First converting the magnitudes given for our sample sources for different filters into flux by using
equation (21) for SDSS and GALEX, and for WISE, converting the Vega magnitudes into AB
magnitudes by using conversion table as given above and using equation (21) to convert these
magnitudes into fluxes.
Then using conversion between Mpc and cm, we have converted luminosity distance (D) into cm.
Then using equation (23) we can find the corresponding luminosity.
Then proceeding for spectral curve fitting for each sample sources, we first outline the log10 ν and
log10 νLν values need for plotting for each source in a table:
For SDSS J081432.11+560956.6
log10 ν log10 νLν
For W4 filter = 13.133539 44.689949
For W3 filter = 13.397940 44.705902
For W2 filter = 13.814247 44.800156
For W1 filter = 13.945469 44.752945
For z filter = 14.531479 44.888474
For i filter = 14.602060 44.975056
For r filter = 14.681241 45.006229
For g filter = 14.806180 45.151169
For u filter = 14.903090 45.140060
For NUV filter = 15.120574 45.188513
For FUV filter = 15.296665 45.217579
and the plotting is:
44.6
44.7
44.8
44.9
45
45.1
45.2
45.3
13 13.5 14 14.5 15 15.5
log vL (luminosity in erg/sec)
log v (frequency in Hz)
SDSS J081432.11+560956.6
Luminosity
17For SDSS J085001.17+462600.5
log10 ν log10 νLν
For W4 filter = 13.133539 44.795948
For W3 filter = 13.397940 44.297409
For W2 filter = 13.814247 44.561291
For W1 filter = 13.945469 44.621300
For z filter = 14.531479 44.757229
For i filter = 14.602060 44.811806
For r filter = 14.681241 44.750961
For g filter = 14.806180 45.755878
For u filter = 14.903090 45.572739
For NUV filter = 15.120574 44.250126
For FUV filter = 15.296665 44.194176
and the plotting is:
43.2
43.4
43.6
43.8
44
44.2
44.4
44.6
44.8
45
13 13.5 14 14.5 15 15.5
log vL (luminosity in erg/sec)
log v (frequency in Hz)
SDSS J085001.17+462600.5
Luminosity
For SDSS J095317.09+283601.5
log10 ν log10 νLν
For W4 filter = 13.133539 45.008858
For W3 filter = 13.397940 44.607937
For W2 filter = 13.814247 44.757797
For W1 filter = 13.945469 44.789398
For z filter = 14.531479 44.880119
For i filter = 14.602060 44.934700
For r filter = 14.681241 44.933865
For g filter = 14.806180 45.050804
For u filter = 14.903090 45.031693
For NUV filter = 15.120574 45.065144
For FUV filter = 15.296665 45.029198
18and the plotting is:
44.6
44.65
44.7
44.75
44.8
44.85
44.9
44.95
45
45.05
45.1
13 13.5 14 14.5 15 15.5
log vL (luminosity in erg/sec)
log v (frequency in Hz)
SDSS J095317.09+283601.5
Luminosity
Refrences:
1. C. Megan Urry and Paolo Padovani, Unified Schemes for radio-loud active galactic nuclei, Astronomical Society of the Pacific, September 1995
2. G. Calderone, G. Ghisellini, M. Colpi and M. Dotti, Black hole mass estimate for a sample of
radio-loud narrow-line Seyfert 1 galaxies, Royal Astronomical Society, March 2013
3. Osterbrock, QJRAS, 25, 1, 1984
4. Juhan Frank, Andrew King and Derek Raine, Accretion Power in Astrophysics, 3rd edn.
Cambridge University Press, 2002
5. Bradley W. Carroll and Dale A. Ostlie, An Introduction to Modern Astrophysics, 2nd edn.
Pearson, 2007
6. Bradley M. Peterson, Measuring the Masses of Supermassive Black Hole