Accretion Disk Temperature Calculator

Overview: What This Accretion Disk Temperature Calculator Does

This calculator estimates the effective temperature of a thin accretion disk at a chosen radius around a central object such as a star, white dwarf, neutron star, or black hole. It is designed for students, educators, and researchers who need a quick, order-of-magnitude temperature based on a standard thin-disk model.

You enter the central mass (in solar masses), the mass accretion rate (in solar masses per year), the radius in the disk (in kilometers), and a radiative efficiency factor. The tool converts these to SI units and applies the Shakura–Sunyaev thin-disk formula to return a temperature in kelvin.

What Is an Accretion Disk?

An accretion disk forms when gas or plasma spirals inward toward a massive central body. Instead of falling straight in, the material orbits due to angular momentum, spreading into a flattened, rotating disk. Viscous stresses in the disk cause material to slowly drift inward, converting gravitational potential energy into heat and radiation.

Accretion disks appear in many astrophysical environments:

• Protostars accrete gas from their surrounding molecular clouds.
• White dwarfs in binary systems accrete material from a companion star.
• Neutron stars and stellar-mass black holes accrete from companion stars, powering X-ray binaries.
• Supermassive black holes in galactic nuclei accrete gas and dust, forming active galactic nuclei and quasars.

In many of these systems, the accretion disk can outshine the central object at some wavelengths. Knowing how disk temperature varies with radius is crucial for predicting the spectrum and interpreting observations across bands from optical to X-ray.

Thin Disk Temperature Formula

The calculator is based on the standard, optically thick, geometrically thin disk model introduced by Shakura and Sunyaev. In this approximation, the vertical thickness of the disk is much smaller than its radius, and the energy generated by viscous dissipation is radiated locally from the disk surface.

The effective temperature at a radius r in a Newtonian thin disk is

where:

• T is the effective temperature of the disk surface at radius r (in K).
• G is the gravitational constant.
• M is the mass of the central object.
• Ṁ is the mass accretion rate.
• σ is the Stefan–Boltzmann constant.
• r is the radial distance from the center of the potential well (in m).

The temperature rises toward smaller radii and typically peaks near the inner edge of the disk. Beyond that point, relativistic effects and detailed disk physics are needed to refine the estimate.

Units and Conversions Used in the Calculator

To make the inputs more intuitive, the calculator accepts values in commonly used astrophysical units and then converts them internally into SI units before applying the formula.

• Central mass M is entered in solar masses. The conversion uses 1 solar mass = 1.989 × 10³⁰ kg.
• Accretion rate Ṁ is entered in solar masses per year. The conversion uses 1 year ≈ 3.154 × 10⁷ s, so the rate is turned into kg s⁻¹.
• Radius r is entered in kilometers, then converted to meters.
• Radiative efficiency η is dimensionless and can be used as a scaling factor if the implemented model includes efficiency explicitly.

After conversion, the code evaluates the thin-disk temperature expression to give an output in kelvin, which can be compared with typical stellar and disk temperatures.

How to Use the Accretion Disk Temperature Calculator

To compute a temperature estimate:

1. Enter the central mass M in solar masses. For example, a 10 M_⊙ black hole would use M = 10.
2. Enter the mass accretion rate Ṁ in solar masses per year. Low-mass X-ray binaries might have Ṁ ≈ 10⁻⁹–10⁻⁸ M_⊙ yr⁻¹, while luminous quasars can approach Ṁ ≈ 1 M_⊙ yr⁻¹ or more.
3. Enter the radius r in kilometers at which you want the temperature. You can try several radii to see how T(r) falls off.
4. Set the radiative efficiency η, typically between 0.05 and 0.3 for many black hole accretion scenarios. If you are unsure, leaving η ≈ 0.1 gives a reasonable order-of-magnitude value.
5. Run the calculation. The tool will display the estimated effective temperature in kelvin.

By varying the radius, you can map out how the disk cools with distance. By changing the mass and accretion rate, you can compare, for instance, a stellar-mass black hole to a supermassive black hole at the center of a galaxy.

Interpreting the Temperature Result

The computed temperature is an effective blackbody temperature for the disk surface at the chosen radius. It is not a full spectral model but a single temperature that characterizes the local radiative flux. You can use it to infer approximately where in the electromagnetic spectrum the disk emission at that radius will be strongest.

As a rough guide:

• T ≲ 5,000 K: emission peaks in the optical and near-infrared.
• 5,000 K ≲ T ≲ 50,000 K: ultraviolet becomes increasingly important.
• T ≳ 10⁶ K: soft and hard X-rays dominate.

In a real disk, different radii contribute different parts of the spectrum. The inner, hotter regions dominate the X-ray and far-ultraviolet output, while the cooler outer disk contributes optical and infrared light. This calculator focuses only on a single radius at a time, so it is most useful for exploring trends and approximate scales.

Worked Example

Consider a stellar-mass black hole with mass M = 10 M_⊙, accreting at Ṁ = 10⁻⁸ M_⊙ yr⁻¹. Suppose we want the temperature at a radius r = 300 km in the disk, with η = 0.1.

1. Convert mass: M = 10 × 1.989 × 10³⁰ kg ≈ 1.99 × 10³¹ kg.
2. Convert accretion rate: Ṁ = 10⁻⁸ M_⊙ yr⁻¹ ≈ 10⁻⁸ × 1.989 × 10³⁰ kg / 3.154 × 10⁷ s ≈ 6.3 × 10¹⁴ kg s⁻¹.
3. Convert radius: r = 300 km = 3.0 × 10⁵ m.
4. Insert into the thin-disk formula with the known values of G and σ to solve for T.

Evaluating the expression yields an effective temperature of the order of 10⁷ K at that radius, indicating that emission from this part of the disk lies firmly in the X-ray range. Changing M, Ṁ, or r in the calculator will show how the temperature responds to each parameter.

Comparison of Typical Disk Temperatures

The table below summarizes qualitative temperature ranges one might expect from thin accretion disks in different systems, for illustrative purposes only. Exact numbers depend strongly on mass, accretion rate, and radius, so always use the calculator for specific scenarios.

Use these ranges only as broad guides. The calculator lets you explore specific combinations of M, Ṁ, and r for your particular problem.

Assumptions and Limitations of This Model

The thin-disk temperature estimate used here rests on several simplifying assumptions. Understanding them is important to avoid overinterpreting the results.

Key Assumptions

• Geometrically thin disk: The vertical scale height is much smaller than the radius (H ≪ r), so vertical structure is treated in a simplified way.
• Optically thick and radiatively efficient: Locally generated heat is radiated away rather than being advected inward, and the surface radiates roughly like a blackbody.
• Steady-state accretion: The mass accretion rate Ṁ does not change with time or radius.
• Newtonian gravity: General relativistic corrections are neglected, which is reasonable only at radii well outside the innermost stable circular orbit (ISCO).
• Axisymmetric, time-averaged disk: Spiral arms, clumps, and short-term variability are ignored.

Important Limitations

• Near the ISCO or event horizon: For black holes, relativistic effects become dominant at small radii, changing the temperature profile and spectrum. The simple Newtonian expression may significantly misestimate T there.
• Non-blackbody emission: Real disks can deviate from blackbody emission due to scattering, line emission, Comptonization, and vertical temperature gradients.
• Magnetic fields and turbulence: Magnetorotational instability and magnetic coronae can alter both heating and radiation patterns.
• Time variability: Outbursts, flares, and state transitions in X-ray binaries and AGN are not captured by a steady-state model.
• Thick or radiatively inefficient flows: In regimes such as advection-dominated accretion flows (ADAFs), slim disks, or super-Eddington accretion, the thin-disk temperature law is not appropriate.

Because of these limitations, the output should be treated as an approximate guide rather than a definitive prediction, especially for extreme accretion regimes or very small radii.

Questions and Usage Notes

What does this accretion disk temperature represent?

The temperature from this calculator is the effective blackbody temperature of the disk surface at a specified radius in a standard thin-disk model. It is directly related to the local radiative flux via the Stefan–Boltzmann law, F = σT⁴, but does not describe the full detailed spectrum or vertical structure of the disk.

When is this model unreliable?

The model becomes unreliable very close to compact objects (where relativity is essential), in low-luminosity systems where radiatively inefficient flows dominate, and in highly time-variable or magnetically dominated disks. In such cases, more sophisticated numerical models or observational fitting tools are required.

This calculator is intended for educational and approximate scientific use. It is useful for building intuition, back-of-the-envelope estimates, and teaching exercises, but it should not replace detailed modeling when high precision is required.

Model note: The underlying physics follows the classical Shakura–Sunyaev thin accretion disk approximation, adapted to user-friendly input units.