Orbital Mechanics

Gravity

Any two objects that have mass attract each other by gravity. The strength of attraction depends on the mass of both objects. Isaac Newton defined the force of gravity to be:

F_g = G⋅m₁⋅m₂ r²,

where G is the gravitational constant, G = 6.67×10^-11 m³⋅kg^-1⋅s^-2, m is the mass of an object in kg, and r is the distance between the centres of mass of the objects in metres.

Centripetal Force

As an object orbits another, there is a centripetal force that acts on it to keep it moving along a circular or elliptical path. In this course, we will assume a perfectly circular orbit. For more information about elliptical orbits, see course "Orbital Mechanics (Advanced)" .

The centripetal force can be calculated by:

F_c = mv² r,

where m is the mass of the object in kg, v is the velocity of the object in m⋅s^-1, and r is the radius of orbit in metres.

Orbital Speed

In a perfectly circular orbit, one object orbits another at a constant speed. Below is the derivation of the formula of orbital velocity. The mass of the bigger object is written as M, and the lighter is written as m. In a stable orbit, the centripetal force opposes the force of gravity, thus:

F_c = F_g

mv² r = GMm r²

v² = GM r

\( v = \sqrt{\frac{GM}{r}} \)

From this formula it can be calculated that the Earth orbits the Sun at around 29.8 km⋅s^-1, and the minimum orbital speed of an object around Earth is roughly 7.9 km⋅s^-1.

Escape Velocity

For an object to escape the gravitational field of another object, its kinetic energy has to overcome the potential energy. The potential energy for an object in a gravitational field is:

E_p = - GMm r,

where G is the gravitational constant, G = 6.67×10^-11 m³⋅kg^-1⋅s^-2, M and m are the masses of the objects in kg, and r is the distance between the centres of mass of the objects in metres.

Below is the derivation of the formula for escape velocity in a gravitational field. Since an object 'escapes' when its kinetic energy overcomes the potential energy:

E_k + E_p = 0

mv² 2 - GMm r = 0

v² 2 = GM r

v² = 2GM r

\( v = \sqrt{\frac{2GM}{r}} \)

You can apply this knowledge in the gravity simulation found below. Press "Add Object" to add a body.

Interactive Demonstration