Question

As Uranus occults a star the relative motion between Uranus and Earth is 23 km/s. An observer on Earth sees the star disappear for 37 minutes and 2 seconds, and notes that the center of Uranus passed directly in front of the star. 1. Based on these observations, what value would you calculate for the diameter of Uranus? 2. How would the result be different if the center of Uranus had not passed directly in front of the star? 3. Could you tell from the speed given whether this observation was done when Uranus was near the same side as the Sun or near the opposite side of the Sun from the Earth?

Answer 1

a) The Uranus has to move a distance equal to its diameter to cross a star.

The relative speed of the Uranus is given to be

v = 23 km/s

The time taken to cross the star, which can be considered as a point object is

t = 37 min 2 seconds = 37*60+2 = 2222 s

So, the diameter if Uranus = distance it traveled = speed*time = 23*2222 = 51106 km

b) If the center of Uranus has not passed directly in front of the star, the distance Uranus has to travel to cross the star will be lesser.

This means that the distance we get will be a smaller value than the diameter of Uranus.

distance for which the star is covered if it is not at the center distance for which the star is covered if it is at the center

c) The speed of the earth in its orbit is approximately 30 km/s

Also, we know that all the planets move around the sun in the same direction.

So, if the Earth and the Uranus was on the same side as viewed from sun, they will be travelling in the same direction, and the relative speed will be smaller than individual speed.

If the Earth and the Uranus was on the opposite side as viewed from sun, they will be travelling in the opposite directions, and the relative speed will be higher than individual speed.

Here, the relative speed is less than the speed of earth.

So, the planets are on the same side as viewed from the sun.

So, as viewed from the earth, the sun and Uranus will be on opposite sides.