How to use: Introduction: How this accretion disk temperature calculator works
This calculator estimates the effective temperature of a thin accretion disk at a chosen radius around a compact object or star. In astrophysics, accretion disks appear when gas carries angular momentum and therefore settles into orbit instead of falling straight inward. Frictional and turbulent stresses inside the disk gradually move matter inward and convert gravitational energy into heat. That heat is then radiated away, and in the standard thin-disk picture the local surface emission can be described by an effective blackbody temperature. The purpose of this page is to turn that idea into a quick, usable estimate from four inputs: central mass, accretion rate, radius, and radiative efficiency.
The result is most useful when you want an order-of-magnitude answer. For example, you may want to know whether a disk annulus is likely to radiate mainly in the optical, ultraviolet, or X-ray band; whether a chosen radius is plausibly hot enough to explain an observed signal; or how strongly the temperature changes when you vary the mass or accretion rate. The calculator is not a full spectral synthesis tool, but it is a practical way to build intuition and compare scenarios consistently.
The model used here follows the familiar Newtonian thin-disk temperature scaling. That means the disk is treated as geometrically thin, optically thick, and radiatively efficient enough that locally generated heat escapes rather than being carried inward. Those assumptions are often reasonable for teaching, back-of-the-envelope estimates, and broad comparisons across systems, but they become less reliable near the innermost stable orbits of compact objects or in flows that are thick, magnetically dominated, or strongly time variable.
What the inputs mean
Central Mass M is the mass of the object being orbited, entered in solar masses. A value of 1 corresponds to one solar mass, while 10 would represent a typical stellar-mass black hole and much larger values can represent supermassive black holes in galactic nuclei. Increasing the central mass tends to increase the local disk temperature at a fixed radius and accretion rate, although the full astrophysical interpretation depends on how that radius compares with characteristic scales such as the Schwarzschild radius or the innermost stable circular orbit.
Accretion Rate Ṁ is the rate at which mass flows through the disk, entered in solar masses per year. This quantity controls how much gravitational energy is being released. A higher accretion rate generally means a hotter disk because more power is dissipated per unit time. Since accretion rates in astronomy can span many orders of magnitude, it is common to enter very small decimal values such as 10−8 solar masses per year for X-ray binaries or much larger values for active galactic nuclei.
Radius r is the distance from the center at which you want the temperature, entered in kilometers. This is the most sensitive input in the thin-disk temperature law because the temperature scales as roughly r−3/4. In plain language, moving inward heats the disk quickly, while moving outward cools it quickly. If you are exploring a system, it is often helpful to run the calculator several times at different radii to see how the temperature profile changes across the disk.
Radiative efficiency η is a dimensionless number between 0 and 1. In the current calculation code, efficiency is used to estimate the luminosity term shown in the result panel, while the temperature itself is computed from the thin-disk expression based on mass, accretion rate, and radius. A value near 0.1 is a common illustrative choice for black hole accretion, but the appropriate number depends on the physical system and on how you are interpreting the luminosity estimate.
The formula behind the calculator
The page preserves the general mathematical structure of the original calculator and also includes the specific thin-disk temperature relation used for the astrophysical estimate. In a generic calculator, one often writes the output as a function of several inputs:
For this specific problem, the effective temperature at radius r in a standard thin disk is estimated with:
Here, G is the gravitational constant, M is the central mass, Ṁ is the mass accretion rate, σ is the Stefan–Boltzmann constant, and r is the radius measured in meters after unit conversion. The code converts the user inputs into SI units, evaluates the quantity inside the fourth root, and then reports the temperature in kelvin. It also computes a peak wavelength from Wien’s law and a luminosity estimate from L = η Ṁ c², which is why efficiency appears in the form even though it does not directly modify the temperature expression in the current script.
The original page also included a generic weighted-sum example, which is preserved below as a mathematical reference block:
That expression is not the physical disk-temperature law, but it illustrates the broader idea that outputs depend on scaled inputs. For the astrophysical calculation on this page, the physically important scaling is the fourth-root dependence on mass and accretion rate and the strong inverse dependence on radius.
Limitations and assumptions: Units, assumptions, and interpretation
The calculator accepts user-friendly astrophysical units and converts them internally. One solar mass is taken as 1.989 × 1030 kg, one year as 3.154 × 107 s, and one kilometer as 1000 m. This matters because the thin-disk formula is evaluated in SI units. If your source data are already in cgs or another astronomical convention, convert carefully before entering values. A unit slip of even a factor of 1000 in radius can change the temperature by a large amount because of the cubic radius term inside the formula.
When you read the result, remember that the reported temperature is an effective surface temperature. It is not the full gas temperature throughout the disk, and it is not a complete radiation-transfer model. Real disks can show color corrections, scattering atmospheres, line emission, coronae, magnetic heating, and relativistic effects. Even so, the effective temperature remains a useful summary quantity because it tells you the local radiative scale and gives a first clue about the wavelength range where emission from that annulus may be strongest.
As a rough guide, temperatures of a few thousand kelvin correspond to optical or near-infrared emission, tens of thousands of kelvin push into the ultraviolet, and temperatures above about 106 K are associated with soft or hard X-ray emitting regions. These are broad signposts rather than strict boundaries, but they help connect the numerical output to physical intuition.
Worked example
Suppose you want to model a stellar-mass black hole with M = 10 solar masses, accreting at Ṁ = 10−8 solar masses per year, and you want the temperature at r = 300 km. If you set the efficiency to η = 0.1, the calculator will use that value for the luminosity estimate while still computing temperature from the thin-disk relation above. Converting the inputs gives a central mass of about 1.99 × 1031 kg, an accretion rate of about 6.3 × 1014 kg s−1, and a radius of 3.0 × 105 m. Plugging those values into the formula yields a temperature on the order of 107 K.
That result tells you the chosen annulus is extremely hot and would be associated with X-ray emission rather than visible light. If you keep the same mass and accretion rate but move to a larger radius, the temperature falls rapidly. If instead you keep the radius fixed and increase the accretion rate, the temperature rises, but more gently because of the fourth-root dependence. This is why radius changes often dominate quick scenario testing.
Typical astrophysical contexts
Thin-disk estimates are used across a wide range of systems. Around protostars, disk temperatures are usually much lower and often matter for dust survival, chemistry, and infrared emission. Around white dwarfs in cataclysmic variables, the disk can reach ultraviolet-emitting temperatures. Around neutron stars and stellar-mass black holes, the inner disk can become hot enough to radiate strongly in X-rays. Around supermassive black holes, the much larger physical scale shifts characteristic temperatures downward compared with stellar-mass systems, so optical and ultraviolet emission often dominate broad portions of the disk.
| System type | Typical central mass | Accretion rate range | Inner disk temperature scale |
|---|---|---|---|
| Protostar | 0.1–10 M⊙ | 10−8–10−5 M⊙ yr−1 | 103–104 K (optical/IR) |
| White dwarf accretion disk | 0.6–1.4 M⊙ | 10−10–10−8 M⊙ yr−1 | 104–105 K (UV) |
| Neutron star / stellar-mass BH | 1.4–20 M⊙ | 10−10–10−8 M⊙ yr−1 | 106–107 K (X-ray) |
| Supermassive BH (AGN/quasar) | 106–109 M⊙ | 10−3–10 M⊙ yr−1 | 104–106 K (optical/UV) |
These ranges are intentionally broad. They are included to help you interpret the output, not to replace a system-specific calculation. The calculator is the better tool when you have a particular mass, accretion rate, and radius in mind.
Limits of the model
This estimate should be treated as a physically motivated approximation. It works best for steady, optically thick, geometrically thin disks that radiate locally. It becomes less trustworthy very close to compact objects, where general relativity changes the energy release profile and the inner boundary condition matters. It is also less suitable for advection-dominated flows, slim disks, super-Eddington accretion, strongly magnetized environments, or rapidly changing outbursts. In those cases, the output is still useful as a rough scale, but not as a precision prediction.
If you are using the result for research planning, classroom work, or observational intuition, the calculator is a convenient first pass. If you need detailed spectra, relativistic corrections, or vertical structure, you should follow up with a more specialized model. Used in that spirit, the tool is valuable because it makes the assumptions visible and lets you test how sensitive the answer is to each input.
Arcade Mini-Game: Accretion Disk Temperature Calculator Calibration Run
Use this quick arcade run to practice separating useful scenario inputs from common planning mistakes before you rely on the calculator output.
Start the game, then use your pointer or arrow keys to catch useful inputs and avoid bad assumptions.