Gravitational Binding Energy

## Description

Gravitational binding energy is the energy needed to completely disperse a celestial body. If GBE is broken, the particles of the body will not reform or be bound to each other's gravity, but instead drift off infinitely in the direction they were moved towards. There are precise calculations for this via integration, but a good approximation can be achieved with the following formula:

Where U = GBE in joules, M = the mass of the body in question in kilograms, r = its radius in meters, and G = the gravitational constant of 6.67408x10^-11.

For stars, however, the formula is slightly different:

U is GBE in joules, G is the gravitational constant of 6.67408x10^-11, M is mass in kilograms, r is radius in meters, and n is the polytropic value attributed to the type of star. While this formula is not perfectly accurate, it is widely applicable, and is still within the acceptable margin for error.

Ignoring this formula often leads to vast underestimates of the energy required to destroy astronomical objects (for example, some people assume it scales linearly with mass or volume).

Please note that this formula only works on objects that are mostly held together by their own gravity (meaning: large objects in space such as asteroids, moons, planets, stars, etc.) It also doesn't work on black holes, for obvious reasons.

### List of approximate GBE values for various objects

* Earth's moon (Luna): 1.24e29J
* Earth: 2.487e32J (Calculated with a more accurate method than the above formula)
* OTS 44: 6.906e37J
* VB 10: 3.139e40J
* The sun (Sol): 5.693e41J
* Rigel A: 3.817e42J

SD.net's Planetary Parameter Calculator gives a simplified approach to finding a celestial body's GBE.

Here is a calculator for the GBE of stars.

## Limitations

Gravitational Binding Energy only works as quantification for the destruction of things as long as they are not smaller than their own Schwarzschild radius. In reality objects of such a nature would always be black holes, but in fiction one frequently finds giant celestial objects which have not collapsed into such despite being inside their own Schwarzschild radius.
The trouble with such objects is that, according to proper physics, dispersing them should be impossible. I.e. their gravitational binding energy would be considered either infinite or just simply undefined. Hence quantifying the energy to disperse them becomes more difficult.
In case of actual black holes one would do it as described on the black hole feats in fiction page. In other cases there are several options to go through.

First, it is a good idea to check if there are other ends for the quantification of mass or size that are also valid and return physics-wise more reasonable results. The mass of rocky planets can, for example, be calculated by assuming they have a similar density to other rocky planets, like Earth or Mars. However, if the planet has a regular surface gravity another option is to quantify the planet's total mass via that. The planetary parameter calculator can, for example, be used for that purpose or the formula mass = (9.81 m/s^2 / gravitational constant) \* (planet radius)^2. In some cases using density might give a very large result, while using surface gravity gives a more feasable one.

If all reasonable results still end of with the celestial body being inside its own Schwarzschild Radius, then two options remain. Either one quantifies the object like a black hole creation feat or one uses regular destruction values on the material the celestial object is made from.

## References

1. ↑ https://en.wikipedia.org/wiki/Gravitational\_binding\_energy
2. ↑ "An Introduction to the Study of Stellar Structure" by S. Chandrasekhar, Chapter IV formula 90
3. ↑ https://www.astro.princeton.edu/~gk/A403/polytrop.pdf