Thermal models

Michal Bursa

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Thermal radiation

thermal radiation [ˈθɜr məl ˌreɪ diˈeɪ ʃən]

noun

electromagetic radiation generated by the thermal motion of particles in matter

Examples:

infrared radiation emitted by animals (detectable with an infrared camera)
cosmic microwave background radiation (detectable with a radio telescope)

Thermal radiation

thermal radiation [ˈθɜr məl ˌreɪ diˈeɪ ʃən]

noun

electromagetic radiation generated by the thermal motion of particles in matter

Examples:

infrared radiation emitted by animals (detectable with an infrared camera)
cosmic microwave background radiation (detectable with a radio telescope)

Accretion disks

accretion disk [ə-ˈkrē-shən ˈdisk]

noun

a disk of usually gaseous matter surrounding a massive celestial object (such as a black hole) in which the matter gradually spirals in and accretes onto the object as a result of gravitational attraction and viscous interaction

We can find them at

black-hole and neutron-star X-ray binaries
active galactic nuclei
quasars
gamma-ray bursts
young stellar objects (protoplanetary disks), planets, etc

Accretion disk spectra

Accretion disk equations - I.

Total torque:	$ G(R) = 2\,\pi\,R \; \nu\,\Sigma\,R^2\,\Omega^\prime $
Net torque on a ring:	$ G(R+dR) - G(R) = \cfrac{\partial G}{\partial R}\,dR $
Torque caries out work:	$ \Omega\,\cfrac{\partial G}{\partial R}\,dR = \left[ \color{blue}{\cfrac{\partial}{\partial R}(G\Omega)} - \color{red}{G\Omega^\prime} \right]\,dR $ $\color{blue}{\partial(G\Omega)/{\partial R}}$: rate of flow of rotational energy (ang. mom.), $\color{red}{G\Omega^\prime}$: local rate of loss of mechanical energy
Local rate of energy release: (per unit time and area)	$ D(R) = \cfrac{G\,\Omega^\prime dR}{2\cdot 2\pi\,R\,dR} = \cfrac{G\,\Omega^\prime}{4\pi R} = \cfrac{9}{8}\nu\,\Sigma\,\cfrac{\mathcal{G}\,M}{R^3}$

Accretion disk equations - II.

Conservation laws

$$ \dot{M} = - 2\,\pi\,R\,\Sigma\,v^{\scriptscriptstyle R} $$ $$ R\cfrac{\partial}{\partial t}\left( \Sigma\,R^2\,\Omega \right) + \cfrac{\partial}{\partial R} \big(\underbrace{R\,\Sigma\,v^{\scriptscriptstyle R}}_{-\dot{M}/2\pi} \;\; \underbrace{R^2\Omega}_{L}\big) = \cfrac{1}{2\pi}\cfrac{\partial G}{\partial R}$$

Stationary case $(\partial/\partial t = 0)$:

$$ -\cfrac{\dot{M}}{2\,\pi} \, L = \cfrac{G}{2\pi} + {\rm const} $$ $$ \cfrac{\dot{M}}{2\,\pi} \, \big(L - L_{\rm in} \big) = -\cfrac{G}{2\pi}$$ (torque vanishes at the inner edge)

Accretion disk equations - III.

$$ L(R) = R^2\,\Omega(R)$$ $$ \Omega(R) = \Omega_{\rm K}(R) = \sqrt{\mathcal{G}\,M\,R^{-3}}$$ $$ G(R) = 2\,\pi\,R \; \nu\,\Sigma\,R^2\,\Omega^\prime $$ $$ \cfrac{\dot{M}}{2\,\pi} \, \big(L - L_{\rm in} \big) = -\cfrac{G}{2\pi}$$ $$ D(R) = \cfrac{G\,\Omega^\prime}{4\pi R} $$ $$ \color{blue}{D(R) = \cfrac{3\,\mathcal{G}\,M\,\dot{M}}{8\,\pi\,R^3} \left[ 1 - \left(\cfrac{R_{\rm in}}{R}\right)^{1/2} \right]} $$ local rate of enery release = cooling radiative flux [erg/s/cm²]

Accretion disk spectra

Local radiative flux and effective temperature

$$ F(R) = \cfrac{3\,\mathcal{G}\,M\,\dot{M}}{8\,\pi\,R^3} \left[ 1 - \left(\cfrac{R_{\rm in}}{R}\right)^{1/2} \right] $$ $$ F = \sigma_{\rm\scriptscriptstyle SB} T^4 $$ $$ T_{\rm eff}(R) = \Big( F(R) / \sigma_{\rm\scriptscriptstyle SB} \Big)^{1/4} $$

Local black-body spectrum

$$ I_\nu(E, T) = \cfrac{2}{h^2 \, c^2} \cfrac{E^3}{\exp{E/kT}-1}$$

Integrated observed energy spectrum

$$ F_\nu(E) = \int I_\nu(E, T) \, d\Omega = \int I_\nu(E, T) \, \cos\theta \, dS/D^2 = \int I_\nu(E, T) \, \cos\theta /D^2 \, r\, dr $$

Accretion disk spectra

Spin measurements from continuum fitting

Idea: By fitting the shape of the thermal component, we may estimate the spin of the BH.

measure flux $F$ received from the disk
measure maximum temperature $T$ from the spectrum
assuming black-body radiation:
$L=4\pi D^2 F=\cfrac{GM\dot{M}}{2 R_{\rm in}}$
$R_{\rm in} = \sqrt{\cfrac{3}{4\pi\sigma}} \cfrac{L^{1/2}}{T^2}$
$R_{\rm in} \rightarrow a$
independent measurements of $D$, $M$ and $i$ are needed

Spin measurements from continuum fitting

Credit: R. Narayan, J. McClintock

XSPEC Thermal Models

DISK	non-relativistic $\alpha$-disk with opacity given by free-free absorption (Kramers law)
DISKM	$\alpha$-disk with $\tau_{r\phi} \sim P_{\rm gas}$
DISKO	$\alpha$-disk with $\tau_{r\phi} \sim P_{\rm rad}$
BBODY	single-temperature black-body model (normalized to luminosity)
BBODYRAD	single-temperature black-body model (normalized to emitting area)
DISKBB	MCBB disk with $T_{\rm eff} \sim r^{-3/4}$ (non-zero torque boundary)
DISKPBB	MCBB disk with $T_{\rm eff} \sim r^{-p}$ (non-zero torque boundary; for radial advection)
DISKPN	$\alpha$-disk with $T_{\rm eff}(r)$ given by Paczynski-Wita potential
EZDISKBB	Shakura-Sunyaev disk model (zero-torque boundary)
GRAD	relativistic $\alpha$-disk for Schwazschild black hole ($r_{\rm in}=6\,r_{\rm g}$)
KERRD	relativistic $\alpha$-disk for extreme-Kerr black hole ($r_{\rm in}=6\,r_{\rm g}$)
KERRBB	relativistic $\alpha$-disk (NT) with color-corrected blackbody, self-irradiation and limb-darkening
BHSPEC	relativistic $\alpha$-disk (NT) with surface emission computed by stellar atmospheres-like calculations (vertical structure + radiative transfer)
AGNSPEC	like BHSPEC, but for AGNs ($M \sim 10^7 M_\odot$)
KERRBB2	KERRBB with $f_{\rm col}$ taken from BHSPEC
KYNBB	relativistic $\alpha$-disk (NT) with color-corrected blackbody, obscuration and polarization
SLIMBH	relativistic slim disk model

search for: XSPEC models

XSPEC Thermal Models

DISK	non-relativistic $\alpha$-disk with opacity given by free-free absorption (Kramers law)
DISKM	$\alpha$-disk with $\tau_{r\phi} \sim P_{\rm gas}$
DISKO	$\alpha$-disk with $\tau_{r\phi} \sim P_{\rm rad}$
BBODY	single-temperature black-body model (normalized to luminosity)
BBODYRAD	single-temperature black-body model (normalized to emitting area)
DISKBB	MCBB disk with $T_{\rm eff} \sim r^{-3/4}$ (non-zero torque boundary)
DISKPBB	MCBB disk with $T_{\rm eff} \sim r^{-p}$ (non-zero torque boundary; for radial advection)
DISKPN	$\alpha$-disk with $T_{\rm eff}(r)$ given by Paczynski-Wita potential
EZDISKBB	Shakura-Sunyaev disk model (zero-torque boundary)
GRAD	relativistic $\alpha$-disk for Schwazschild black hole ($r_{\rm in}=6\,r_{\rm g}$)
KERRD	relativistic $\alpha$-disk for extreme-Kerr black hole ($r_{\rm in}=6\,r_{\rm g}$)
KERRBB	relativistic $\alpha$-disk (NT) with color-corrected blackbody, self-irradiation and limb-darkening
BHSPEC	relativistic $\alpha$-disk (NT) with surface emission computed by stellar atmospheres-like calculations (vertical structure + radiative transfer)
AGNSPEC	like BHSPEC, but for AGNs ($M \sim 10^7 M_\odot$)
KERRBB2	KERRBB with $f_{\rm col}$ taken from BHSPEC
KYNBB	relativistic $\alpha$-disk (NT) with color-corrected blackbody, obscuration and polarization
SLIMBH	relativistic slim disk model

search for: XSPEC models

XSPEC Thermal Models: diskbb

diskbb

The spectrum from an accretion disk consisting of multiple blackbody components with $T_{\rm eff} \sim r^{-3/4}$.
(Mitsuda et al. 1984; Makishima et al. 1986).

par1	temperature at inner disk radius (keV)
norm	$(R_{in}/D_{10})^2 \cos\theta$ $R_{in}$ is "an apparent" inner disk radius in km, $D_{10}$ the distance to the source in units of 10 kpc, and $\theta$ the angle of the disk ($\theta = 0$ is face-on). On the correction factor between the apparent inner disk radius and the realistic radius see Kubota et al. (1998).

XSPEC Thermal Models: kerrbb

kerrbb

A multi-temperature black-body model for a thin, steady, general relativistic accretion disk around a Kerr black hole. The effect of self-irradiation of the disk is considered, and the torque at the inner boundary of the disk is allowed to be non-zero.
(Li et al. 2005)

par1	torque at the inner boudary
par2	BH spin
par3	disk inclination angle
par4	BH mass
par5	mass accretion rate
par6	source distance
par7	spectral hardening factor (explain)
par8	self-irradiation flag
par9	limb-darkening flag
norm	fixed to 1, if inclination, mass and distance are frozen

Color correction factor

$$ I_\nu(\nu, R) = \color{red}{f_{\rm c}^{-4}} B_\nu(\color{red}{f_{\rm c}}T_{\rm eff}) = \frac{2 h \nu^3 c^{-2} \color{red}{f_{\rm c}^{-4}}} {\exp\left[h \nu/ k \color{red}{f_{\rm c}} T_{\rm eff}(R)\right]-1} $$