# Practical Exercise 3: MEASURING MERCURY AND VENUS

Determine the radii of circular orbits of Mercury and Venus when you know that:

a) the greatest elongation of Mercury is 23° and of Venus 47°,

b) synodic orbital period of Mercury is 116 days, Venus 584 days.

a) From Figure 1 and the greatest elongation we will determine the main half-axes of the planets: Mercury a = a sin 23° ≐ 0,39 au, Venus: a☿ = a sin 23° ≐ 0,39 au, Venus: a = a sin 47° ≐ 0,73 au.

b) For mutual angular motion of the inferior planet and the Earth it applies: . We will start from the definition of angular velocity .We start from the definition of angular velocity , after substitution we get:  We start from the definition of angular velocity , after substitution we get:

i.e. . We start from the definition of angular velocity  after substitution we get:  i.e.

For Mercury we get: . We start from the definition of angular velocity after substitution we get:  i.e.

For Mercury we get: T ≐ 88 days ≐ 0,241 years, for Venus: .

We start from the definition of angular velocity , substitution we get:

i.e. .  . For Mercury we get: T ≐ 88 days ≐ 0,241 years , for Venus: T ≐ 245 days ≐ 0,615 years.

From the 3rd Kepler’s law a' =  we get: a ≐ 0,391 au, . We start from the definition of angular velocity ,substitution we get:

i.e. .  .For Mercury we get: T ≐ 88 dní ≐ 0,241 years , for Venus: T≐ 245 days ≐ 0,615 years From the 3rd Kepler’s law a' = we get: a ≐ 0,391 au, a ≐ 0,723 au.