The Mass of a Black Hole calculator computes the estimated mass based on the radius of the event horizon.
INSTRUCTIONS: Choose units and enter the following:
- (R) This is the radius of the event horizon
Mass of the Black Hole (M): The calculator returns the mass in Solar Masses (multiple of the mass of our Sun). However, this can be automatically converted to compatible units via the pull-down menu.
Astronomy Calculators
The Math / Science
The Mass of a Black Hole equation builds on the very simple principle of the gravitational attraction of any two masses to define the radius of a black hole. The calculated radius, R, defines the event horizon of a black hole, in that any mass, M, that is compressed smaller than this radius, R, becomes a black hole.
Derivation
Given that the escape velocity of a body can be computed as: `v_"(escape)" = sqrt((2*M*G)/R)` we can recognize that this velocity must not exceed the speed of light. Therefore we can re-write the equation as `v_"(escape)" = sqrt((2*M*G)/R) < c`. Rewriting this again to solve for R, which represents the radius of the black hole, we get:
`R < (2 * M * G)/c^2` ; where G is the universal gravitational constant and M is the input mass of the black hole. Interpreting this in terms of a mass compressed by gravity, it tells us any mass, M, with a radius less than R is by definition a black hole.
Therefore, the formula for the mass of a black hole (solving for M) is:
`M = (R*c^2)/(2*G)`
where:
- M = Mass of the black hole
- R = Radius of the Event Horizon
- c = Speed of Light
- G = Universal Gravity Constant.
Example
If we were to calculate the radius of the sun, were the sun's present mass became a black hole, we would get the following radius:
`M_"(SUN)" = 1.989*10^30 kg`
And if we enter this mass into this equation, we find the sun's radius as a black hole would be approximately: 2953.9 meters.
Since the sun's present radius is approximated as 696,342 km, this equation would tell us the sun's radius as a black hole would be about `4.24 *10^-6` of its present size (0.00000624 of its present size, which is about 6 millionths of the sun's present size).
Reference
University Physics 12th Edition, Chapter 12, Equation #12.29
Astronomical Units
Because of the enormity of space and the size of the objects studied, the field of astronomy employs units not commonly used in everyday life. Nonetheless, these units do translate into common units at a grand scale, and vCalc provides automatic conversions between units for calculator inputs and answers via the pull-down menus. The following is a brief description on the distance, mass and time units employed in the field of astronomy
Astronomy Distance Units
Astronomical Unit (au): Within our solar system, a common measure of distance is au, which stands for astronomical units. A single astronomical unit is the mean distance from the Sun's center to the center of the Earth.
| Astronomical Unit (au) | Distance from Sun (au) |
|---|---|
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Light Travel in Time: Light is a primary observable when studying celestial bodies. For this reason, the distance to these objects are measured in the amount of time it would take light to travel from there to the Earth. We can say that an object is one light-year away, and that means that the object is at a distance where it took an entire year for light from the object to travel to Earth. Since the speed of light is 299,792,458.0 meters per second, one can compute the distance equal to a light year as follows:
1 light year = 299,792,458.0 (meters / second) x 31,536,000 (seconds / year) = 9,460,528,405,000,000 meters
The same exercise can be used for light traveling shorter periods of times, light seconds, light minutes, light hours and light days. Since even these units are not enough when computing distances across the universe, there is also a light relative distance of kilo-light years (1000 light years), or the distance light travels in a thousand years!
| Light Second | Light Minute | Light Hour | Light Day | Light Year | Kilo-Light Year |
|---|---|---|---|---|---|
|
299,786 km 186,278 miles 0.002 au |
17,987,163 km 11,176,705 miles 0.12023 au |
1,079,229,797 km 670,602,305 miles 7.214 au |
25,901,515,140 km 16,094,455,343 miles 173.14 au |
9,460,528,405,000 km 5,878,499,814,210 miles 63,240 au 0.306 parsecs |
9,460,528,405,000,000 km 5,878,499,814,210,000 miles 63,240,000 au 306 parsecs |
Angle Shift Seen from Earth: Because the Earth goes around the Sun, our observation of distant objects such as stars results in an angular shift when observed at opposite sides of the elliptical orbit. This shift is used as the basis of a unit knows as a parsec. A parsec was traditionally defined as the distance where one astronomical unit subtends an angle of one arcsecond. A parsec was redefined in 2015 to 648000/π astronomical units. Proxima Centauri, is the nearest star to the Sun and is approximately 1.3 parsecs (4.2 light-years) from the Sun. A mega-parsec is a million parsecs.
| Parsec | Mega-parsec |
|---|---|
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Astronomy Mass Units
Astronomical units also apply to the mass of enormous objects such as moons, planets and stars. For this reason, astronomy also employs mass units that compare other objects to ones familiar to us. For example, stars are often measured in mass units of solar masses. This is a comparison of their mass to the mass of our sun (one solar_mass). For planets, astronomers use Earth masses and Jupiter masses for understanding the relative size of rocky planets and gas giants.
| Earth Masses | Jupiter Masses | Solar Masses |
|---|---|---|
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Astronomy Time Units
Astronomers use the same time units as everyone else, from the very small nanoseconds, to seconds, minutes, hours, days and years. This is true with two exceptions known as sidereal days and sidereal years. These refer to time relative to the celestial objects (the fixed stars). The Earth rotates every 24 hours relative to the Sun. But we are moving in a circle around the Sun. In comparison, the Earth rotates every 23 hours, 56 minutes and 4.0905 seconds (23.9344696 hours) compared to the stars in the celestial sphere. This is known as a sidereal day.
In the same vein, a sidereal year is the time it takes the Earth to complete one orbit around the Sun relative to the celestial sphere. Where a year is 365 days, a sidereal year is 365.256363004 days, or 1,224.5 seconds more than a calendar year.
| Sidereal Day | Sidereal Year |
|---|---|
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Astronomy Calculators
The Astronomy Calculator Suite includes functions that are useful for studying astronomy and include the following:
- Kepler's 3rd Law formula T² = (4π • R³)/(G • M)
- Small Angle Formula
Small Angle Formula α = S / D
- Flux and Magnitude formulas
- Telescope formulas
- Mass formulas
- Mass from Luminosity
- Mass from Acceleration and Radius
- Mass from Speed and Separation
- Mass of extra-solar planet from mass of star, radius of star orbit and radius of exoplanet orbit
- Mass of extra-solar planet from mass and velocity of star and velocity of planet
- Black Hole Mass Calculator
- Black Hole Event Horizon Calculator
- Relative Size Formulas
- Wavelength Formulas
- Luminosity and other Formulas
- Distance from absolute and apparent magnitude
- Radius from Speed and Period
- Speed of Circular Motion
- Hubble Time
- Luminosity from Mass
- Hubble's Law
- Exoplanet Calculator
- Orbital Radial Acceleration
- Distance to Barycenter
- Sakuma-Hattori Equation
- Astronomical Distance Calculator provides the distance from the Earth to numerous astronomical bodies (e.g. Sun, Moon, planets, stars, Milky Way's Center and Edge, Andromeda Galaxy)
- Astronomical Distance Travel Time Calculator computes the time to travel to distant parts of space at different velocities.
- 3D Vector Calculator
- Drake Equation Calculator
- Seager Equation Calculator
- Friedman Equation Calculator