# Formula First Cosmic Velocity Velocity Orbital radius Mass Gravitational constant

## Velocity

`$$ v $$`Unit

`$$ \frac{\mathrm m}{\mathrm s} $$`

Tangential velocity of the body (e.g. satellite) on the orbit around a planet, a star or another central body. Only with this velocity the body remains without propulsion on its circular orbit around the central body without falling on the central body or moving further away from it.

In order for the body to orbit the earth at the earth's surface without propulsion, the following velocity of the body is required:`\[ v ~=~ \sqrt{ \frac{6.67 \cdot 10^{-11} \frac{\mathrm N \, \mathrm{m}^2}{\mathrm{kg}^2} ~\cdot~5.97 \cdot 10^{24}\,\mathrm{kg} }{6.38 \cdot 10^6 \,\mathrm{m}} } ~=~ 7.9 \, \frac{\mathrm{km}}{\mathrm s} \]`

## Orbital radius

`$$ r $$`Unit

`$$ \mathrm{m} $$`

The desired radius of the circular path, that is, the distance of the body from the center of the central body. The condition is: \( r \geq r' \), where \(r'\) is the radius of the central body.

The further away the body (e.g. satellite) should orbit the central body (larger radius \(r\)), the smaller its orbital velocity \(v\) must be.

## Mass

`$$ M $$`Unit

`$$ \mathrm{kg} $$`

Mass of the central body around which a satellite should orbit. In the case of the earth the mass is: \( M ~=~ 5.972 \cdot 10^{24} \, \mathrm{kg} \).

The larger the mass of the central body, the larger the orbital velocity \(v\) of the satellite must be to orbit the central body without propulsion.

## Gravitational constant

`$$ G $$`Unit

`$$ \frac{\mathrm{N} \, \mathrm{m}^2}{\mathrm{kg}^2} = \frac{\mathrm{m}^3}{\mathrm{kg} \, \mathrm{s}^2} $$`

Gravitational constant is a physical constant and has the value: \( G = 6.674 \cdot 10^{-11} \frac{\mathrm N \, \mathrm{m}^2}{\text{kg}^2} \).