Kepler’s laws and the shape of the orbit of the celestial bodies

Kepler’s laws and the shape of the orbit of
the celestial bodies




Figure 1: The change of coordinate system

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, Consider a cartesian system with it’s unit vectors e⃗x and e⃗y . In
O, the origin of our system, we consider a star which has the mass M
and in B, we consider a celestial body which has the mass m << M
and orbits the star anticlockwise. Because m << M we can consider
that the star is fixed in O. We will study the movement of the celestial
body around the star in one system of reference solitary bounded by
the celestial body (Figure 1). Let’s consider that the position vector
of the celestial body makes the angle θ with Ox axis. We will note
the unit vectors of the new system with e⃗r , a unit vector which is
oriented in the direction of the increase of vector ⃗r and e⃗θ , a unit
vector which is oriented in the direction of the increase of angle θ and
it is perpendicular to e⃗r .




Figure 2: The new unit vectors

By using Figure 2, we can write the unit vectors e⃗r and e⃗θ depending
on the unit vectors e⃗x and e⃗y :

e⃗r = e⃗x · cos θ + e⃗y · sin θ (1)

e⃗θ = −e⃗x · sin θ + e⃗y · cos θ (2)
We can write the position of celestial body relative to he star as
r · e⃗r . Now, let’s calculate the derivatives with respect to time of the 2


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