Accretion Disk Temperature Profile Calculator

Shakura-Sunyaev accretion disk peak temperature, luminosity, and emission band.

🌡 Accretion Disk Temperature Profile Calculator
Compact Object Mass
M☉
Accretion Rate
M☉/yr
Inner Disk Radius Type
Peak Disk Temperature
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Inner Disk Radius
—
Disk Luminosity
—
Eddington Ratio
—
Wien Peak Wavelength
—
Emission Band
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🌡 What Is an Accretion Disk Temperature Profile?

An accretion disk temperature profile describes how temperature varies with distance from the central compact object in an accretion disk. When matter falls toward a black hole or neutron star, it forms a rotating disk and releases gravitational energy as heat and radiation. The radial temperature structure determines what wavelengths the disk emits most strongly, which is key to interpreting X-ray binary light curves, AGN spectra, and tidal disruption events.

This calculator implements the Shakura-Sunyaev (1973) thin-disk model, the standard framework used in observational high-energy astrophysics. It applies to sub-Eddington accretion where the disk is geometrically thin and optically thick, so each annulus radiates as a blackbody. Given a mass M and accretion rate Mdot, the model predicts the peak disk temperature, luminosity, and the wavelength at which the disk radiates most intensely (the Wien peak).

The temperature profile has a characteristic shape: cool at large radii, rising steeply inward, reaching a maximum just outside the inner edge (at 1.36 times the ISCO radius), then dropping to zero at the inner boundary due to the no-torque condition. The inner boundary is the innermost stable circular orbit (ISCO) for black holes, or the stellar surface for neutron stars.

Real astrophysical applications include measuring BH spin (the peak temperature shifts with ISCO size), constraining BH masses from disk spectra, modeling multi-wavelength variability of AGN, and interpreting X-ray binary state transitions. While more sophisticated models (KERRBB, BHSPEC) include relativistic corrections and spectral hardening, the Shakura-Sunyaev result captures the essential physics and remains the starting point for all disk modeling.

📐 Formula

T(r) = [(3 G M &Mdot;) / (8 π σ r³)]¹⁄&sup4; × [1 − √(rin/r)]¹⁄&sup4;
T(r) = disk temperature at radius r [K]
G = gravitational constant = 6.674 × 10-8 cm3 g-1 s-2
M = compact object mass [g]
&Mdot; = accretion rate [g/s]
σ = Stefan-Boltzmann constant = 5.67 × 10-5 erg cm-2 s-1 K-4
r = disk radius [cm]
rin = inner disk radius; for Schwarzschild BH: 6 G M / c2 (ISCO)
Peak at rpeak = (49/36) rin ≈ 1.361 rin
Disk luminosity L = G M &Mdot; / (2 rin)
Eddington luminosity LEdd = 4 π G M mp c / σT
Wien peak λmax = 2.898 × 10-3 m⋅K / Tpeak

📖 How to Use This Calculator

Steps

1
Enter compact object mass in solar masses. Use the preset buttons to load common systems: stellar BH (10 M☉), AGN (108 M☉), neutron star (1.4 M☉), X-ray binary (7 M☉).
2
Set accretion rate in solar masses per year. Typical stellar BH X-ray binaries accrete at 1e-10 to 1e-8 M☉/yr; AGN accrete at 0.01 to several M☉/yr.
3
Choose inner radius type. For black holes select ISCO (automatic). For neutron stars select NS surface and enter the stellar radius in km (typically 10 to 13 km).
4
Read the results. The calculator shows peak temperature, inner radius, disk luminosity, Eddington ratio, Wien wavelength, and emission band. An Eddington ratio below 30% confirms the thin-disk model is valid.

💡 Example Calculations

Example 1 — Stellar Black Hole in an X-ray Binary

10 M☉ BH accreting at 10-8 M☉/yr (Cygnus X-1 like)

1
ISCO = 6 G M / c2 = 6 × 6.674e-8 × 10 × 1.989e33 / (2.998e10)2 = 8.86 × 107 cm = 88.6 km (6 rg).
2
Mdot = 1e-8 M☉/yr = 6.30 × 108 g/s. Peak radius r_p = (49/36) × 88.6 km = 120.5 km.
3
T_peak = 3.460 × 106 K. Disk luminosity L = 4.72 × 1037 erg/s = 1.23 × 104 L☉. Eddington ratio = 3.75% L_Edd. Wien peak at 838 pm (soft X-ray).
Peak Temperature = 3.460 × 106 K (X-ray, λmax = 838 pm)
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Example 2 — Active Galactic Nucleus (AGN)

108 M☉ SMBH accreting at 0.1 M☉/yr (Seyfert-1 galaxy)

1
ISCO = 6 rg = 8.86 × 1013 cm = 5.92 AU. Mdot = 0.1 M☉/yr = 6.30 × 1015 g/s.
2
Peak radius r_p = (49/36) × 5.92 AU = 8.06 AU. T_peak = 6.154 × 104 K, peaking in far-UV at 47 nm.
3
Disk luminosity L = 4.72 × 1044 erg/s. Eddington ratio = 3.75% L_Edd. This UV emission is the "Big Blue Bump" seen in AGN spectra.
Peak Temperature = 6.154 × 104 K (Ultraviolet, λmax = 47.1 nm)
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Example 3 — Accreting Neutron Star (Low-Mass X-ray Binary)

1.4 M☉ neutron star with 10 km radius, Mdot = 10-9 M☉/yr

1
Inner radius = NS surface = 10 km = 106 cm. The NS ISCO is at 12.6 km, just inside the star, so the stellar surface sets r_in.
2
Peak radius r_p = (49/36) × 10 km = 13.6 km. Mdot = 1e-9 M☉/yr = 6.30 × 107 g/s. T_peak = 6.114 × 106 K.
3
Disk luminosity L = 5.856 × 1036 erg/s = 1530 L☉. Eddington ratio = 3.33% L_Edd. Wien peak at 474 pm (soft X-ray).
Peak Temperature = 6.114 × 106 K (X-ray, λmax = 474 pm)
Try this example →

❓ Frequently Asked Questions

What is the Shakura-Sunyaev accretion disk model?+
The Shakura-Sunyaev (1973) model describes a geometrically thin, optically thick accretion disk where each annulus radiates as a blackbody. Viscous heating drives angular momentum outward and mass inward. It predicts T(r) proportional to r^(-3/4) far from the inner boundary, modified by the no-torque factor [1 - sqrt(r_in/r)]^(1/4). It is the standard thin-disk solution widely used in X-ray binary and AGN physics.
How hot do accretion disks get around stellar mass black holes?+
Stellar-mass BH accretion disks (5 to 20 solar masses) reach peak temperatures of 1 to 30 million K, placing the peak emission in soft X-rays (0.1 to 2 keV). Hotter disks (higher Mdot or lower mass) emit harder X-rays. This is why X-ray binaries in the soft (high) state are easily detected by X-ray telescopes like ROSAT, Chandra, and NICER.
Why are AGN disks cooler than stellar BH disks despite higher mass?+
The ISCO radius scales linearly with mass (r_in proportional to M), while T_peak scales as M^(-1/4) times Mdot^(1/4) divided by r_in^(3/4). Since r_in grows faster than Mdot for typical AGN vs. BH X-ray binary conditions, AGN disks are much cooler (30,000 to 300,000 K) and peak in UV or optical, not X-rays. The X-ray emission from AGN is instead produced in a hot corona above the disk.
What is the innermost stable circular orbit (ISCO)?+
The ISCO is the smallest radius where stable circular orbits exist around a compact object. For a non-spinning Schwarzschild BH it is 6 gravitational radii = 6 G M / c^2 = 8.85 km per solar mass. For a maximally spinning prograde Kerr BH it shrinks to 1 rg. The ISCO sets the inner edge of the accretion disk and largely determines the peak disk temperature and total radiative efficiency.
How is black hole spin measured using disk temperature?+
The X-ray continuum fitting (CF) method fits the disk spectrum with a relativistic disk model (KERRBB or BHSPEC) and infers the spin from the color-corrected temperature and normalization. A rapidly spinning BH has a smaller ISCO, so its disk is hotter and more luminous at fixed Mdot. Combined with an independent mass and distance measurement, the spectrum constrains the spin parameter a to about 10% precision for bright X-ray binaries.
What is the Eddington luminosity and why does it limit accretion?+
The Eddington luminosity L_Edd = 4 pi G M m_p c / sigma_T is the luminosity at which radiation pressure on ionized hydrogen exactly balances gravity. For a 10 solar mass BH it is about 1.26 x 10^39 erg/s. Above L_Edd, radiation drives an outflow and standard thin-disk theory breaks down. Super-Eddington systems (ultra-luminous X-ray sources, some tidal disruption events) require thick-disk or slim-disk models.
Can this calculator be used for white dwarf accretion disks?+
Yes. A white dwarf (0.5 to 1.4 solar masses, radius about 5000 km) acts like the NS case: set rType to NS and enter the WD radius in km (about 5000 to 7000 km). Accretion rates are typically 1e-10 to 1e-8 M☉/yr for cataclysmic variables. WD disk peak temperatures are typically 10,000 to 100,000 K, emitting in UV and optical, consistent with CV spectra observed with HST and IUE.
What is the Big Blue Bump in quasar spectra?+
The Big Blue Bump is a broad spectral feature peaking in far-UV (around 100 nm, redshifted into optical for high-z quasars) seen in most unobscured AGN. It is interpreted as thermal emission from the accretion disk around a supermassive BH. Supermassive BHs accreting at a few tenths of a solar mass per year produce disk temperatures of 30,000 to 100,000 K, matching the observed UV peak.
Why does the disk temperature drop to zero at the ISCO?+
The Shakura-Sunyaev model applies a no-torque boundary condition at r_in: the viscous stress goes to zero there, meaning no energy is extracted from matter inside the ISCO. This forces T(r_in) = 0 via the [1 - sqrt(r_in/r)] factor. In practice magnetic stresses can still dissipate energy inside the ISCO, but the no-torque condition is a good approximation for most thin-disk calculations.
What are typical accretion rates for different astrophysical systems?+
Low-mass X-ray binaries in quiescence: 1e-12 to 1e-10 M☉/yr. Outburst state: 1e-9 to 1e-7 M☉/yr. Cataclysmic variables: 1e-11 to 1e-8 M☉/yr. Seyfert-1 AGN: 0.001 to 1 M☉/yr. Quasars: 0.1 to 50 M☉/yr. Tidal disruption events at peak: up to 1000 M☉/yr. Gamma-ray burst central engines: up to 10^-2 solar masses per second, far above Eddington.
How does the disk luminosity relate to accretion rate and BH mass?+
This calculator uses L_disk = G M Mdot / (2 r_in), equal to the gravitational energy released as matter falls to the ISCO. An equivalent form is L = eta times Mdot times c^2 where eta = G M / (2 r_in c^2) = 1/12 approximately 0.083 for a Schwarzschild BH. This 8.3% efficiency compares to 0.7% for hydrogen fusion, making BH accretion the most efficient steady energy source known.
Tags: Accretion Disk Temperature Shakura Sunyaev Disk Black Hole Accretion Astrophysics Calculator Calculator Formula Free Online

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What is an accretion disk temperature profile?

The temperature profile T(r) describes how disk temperature varies with radius. In the thin-disk (Shakura-Sunyaev) model it rises steeply inward from the outer disk, reaches a maximum near the inner edge, then drops to zero at the inner boundary where the no-torque condition is imposed.

What formula does this calculator use?

The Shakura-Sunyaev formula: T(r)^4 = (3 G M Mdot) / (8 pi sigma r^3) times [1 - sqrt(r_in / r)]. The peak occurs at r = (49/36) r_in and is computed analytically. This is the standard thin-disk solution valid for sub-Eddington accretion rates.

What is the ISCO and why does it matter?

The innermost stable circular orbit (ISCO) is the smallest orbit from which matter can stably orbit a black hole. For a non-spinning Schwarzschild BH it lies at 6 gravitational radii. Matter inside the ISCO falls quickly into the BH without significant radiation, so the ISCO sets the inner boundary of the accretion disk and controls the peak temperature.

How hot do black hole accretion disks get?

Stellar-mass BH disks (about 10 solar masses) accreting at typical X-ray binary rates reach peak temperatures of 1 to 30 MK, peaking in soft X-rays. Supermassive BH disks in AGN (a hundred million solar masses) are cooler (10,000 to 100,000 K) and peak in ultraviolet, because r_in is much larger even though Mdot is vastly higher.

What is the Eddington luminosity?

The Eddington luminosity L_Edd = 4 pi G M m_p c / sigma_T is the luminosity at which radiation pressure balances gravity. Standard thin-disk models apply for L below roughly 30% L_Edd. At higher accretion rates the disk puffs up into a slim or thick disk and the temperature profile changes.

Why does the temperature peak slightly outside the inner edge?

The factor [1 - sqrt(r_in/r)] in the temperature formula goes to zero at r = r_in (no-torque inner boundary condition) and rises outward. The competing r^(-3/4) decline produces a maximum at r = (49/36) r_in, about 1.36 times the ISCO radius.

Can I use this calculator for neutron stars?

Yes. Select the Neutron Star preset or choose inner radius type NS and enter the stellar radius (typically 10 to 13 km). The NS surface acts as the inner boundary of the accretion disk, and the formula gives the disk temperature just outside the star. The calculation does not include the boundary layer where disk material impacts the stellar surface.

What is the Wien peak wavelength shown?

Wien displacement law gives the wavelength where a blackbody radiates most intensely: lambda_max = 2.898 mm K / T. The calculator applies this to the peak disk temperature to identify the emission band (X-ray, UV, visible, infrared), indicating which telescope would detect the disk emission most strongly.

📌 Quick Tips

💡For a Schwarzschild (non-spinning) BH the ISCO is 6 gravitational radii. Kerr BHs have smaller ISCOs (down to 1 rg for maximal prograde spin), so real AGN disks are hotter than this calculator shows.
💡Accretion rate is often expressed as a fraction of the Eddington rate. Sub-Eddington disks (below 10%) are thin and well described by the Shakura-Sunyaev model used here.
💡Peak disk temperature scales as T_peak proportional to M^(-1/4) times Mdot^(1/4) divided by r_in^(3/4). Doubling the mass at fixed Mdot lowers the peak temperature.
💡For neutron stars the inner radius is the stellar surface (about 10 km), not the ISCO. Use the NS preset and adjust the radius for different equations of state.
💡The disk luminosity shown (L = G M Mdot / 2 r_in) is the gravitational power released at the ISCO. A radiative efficiency of eta = 0.1 (10%) is typical; super-Eddington sources deviate significantly.