Calculation of the Local Gravity

The local gravity, which we also know as gravitational acceleration, acceleration due to gravity or location factor can easily be calculated from the mass, the distance as well as the gravitational constant.

How this works, I would like to show you in this info using example calculations for different locations on or above the surface of different celestial bodies, after introducing the necessary formula.

The Formula for Calculating the Local Gravity

To calculate the local gravity, we only need information about the mass M and zje radius r of the respective celestial body as well as the gravitational constant G, which always has the value 6.6743*10^−11 m³/kg*s².

The formula for calculating the local gravity from these values ​​is:

\[g=\frac{G*M}{r^2}\]

As a first example, we would like to look at the calculation of the mean surface gravity of the Earth in the next section.

Calculation of the Earth's Surface Gravity

The earth has a mass of 5.9722*10^24 kg and an average diameter of around 12,750 kilometers. For the formula we need the mass in kilograms and the radius in meters (that is 12,750 km / 2 * 1000 = 6,375,000 m). If we insert these values ​​into the formula, we get the following result:

\[g=\frac{6.6743\cdot 10^{-11}\frac{m^3}{kg\cdot s^2}*5.9722\cdot 10^{24}kg}{6,375,000m^2}\thickapprox 9.81\frac{m}{s^2}\]

So, the mean local gravity respectively the average gravitational acceleration on the earth's surface is accordingly (rounded) 9.81 m/s².

At the equator, the diameter of the earth is with 12,756.27 km slightly larger than the average. If we use this diameter as a basis instead, we get a local gravity of about 9.79 m/s² for the equator. At the poles, however, the diameter of the earth is 12,713.50 km, what is slightly smaller than the average. If we use this value instead of the equator diameter in the formula, we get a slightly higher surface gravity of around 9.83 m/s², because we are closer to the center of our planet while maintaining the same mass. You can find out more about this difference between the equator and the poles in the article about why the local gravity is larger at the poles than at the equator.

Calculation of the Local Gravity above the Earth's Surface

To calculate the gravitational acceleration at locations above the Earth's surface, we can simply increase the value of the radius by our desired distance. In this case, our r from the formula has to be understood as the distance d from the center of the Earth.

For our example, we want to calculate the acceleration due to gravity 1,000 km above the earth and use a value for r increased by this height in our formula:

\[g=\frac{6.6743\cdot 10^{-11}\frac{m^3}{kg\cdot s^2}*5.9722\cdot 10^{24}kg}{7,378,135m^2}\thickapprox 7.33\frac{m}{s^2}\]

So, instead of 6,378,135 meters, we use 7,378,135 m (6,378.135 km + 1,000 km) and the result is a gravitational acceleration of rounded 7.33 m/s² for 1,000 km above the Earth's equator.

Calculation of the Moon's Local Gravity

Of course, our formula not only works for the Earth but also for other planets and celestial bodies such as our moon. Our moon has a mass of 7.3477*10^22 kg and a radius of 1,737 km (1,737,000 m). If we insert these values ​​into the formula, we get the following result:

\[g=\frac{6.6743\cdot 10^{-11}\frac{m^3}{kg\cdot s^2}*7.3477\cdot 10^{22}kg}{1,737,000m^2}\thickapprox 1.625\frac{m}{s^2}\]

So, the local gravity on our moon's surface is 1.625 m/s² and is therefore about six times smaller than on our home planet.

Mass and Radius of other Planets and Moons

In our examples, we looked at our home planet Earth as well as at our own moon. The masses and radii of other planets and moons in our solar system can be found in addition to the local gravitoes already calculated for these planets and moons in the article Local Gravity for different Places and Planets. In the first section of this article you will also find a list of gravitational accelerations for different heights above the Earth's surface, which you can reproduce using the approach from the second section of this info.