Kepler's Third Law:
This law states that the square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (a) of its orbit. Mathematically:
```
T^2 ∝ a^3
```
This law implies that:
* Greater distance (larger 'a') leads to a longer orbital period (T).
* Shorter distance (smaller 'a') leads to a shorter orbital period (T).
Vis-viva Equation:
This equation relates the orbital speed (v) of a body to its distance (r) from the attracting body and the mass (M) of the attracting body.
```
v^2 = GM(2/r - 1/a)
```
Where:
* G is the gravitational constant.
* M is the mass of the attracting body.
* r is the distance between the orbiting body and the attracting body.
* a is the semi-major axis of the orbit.
From this equation, we can infer:
* Higher mass (M) leads to higher orbital speed (v).
* Greater distance (larger 'r') leads to lower orbital speed (v).
* The orbital speed is higher at periapsis (closest point to the attracting body) and lower at apoapsis (farthest point).
In Summary:
* Mass of the attracting body (M): Higher mass results in higher orbital speed.
* Distance between the bodies (r): Greater distance results in lower orbital speed.
It's important to note that Kepler's Third Law and the vis-viva equation describe the orbital motion of a body assuming a perfect circular orbit. In reality, most orbits are elliptical, and the orbital speed varies throughout the orbit.
