Orbits of planets & satellites

An asteroid of mass 10 kg is in a circular orbit around the Sun at a distance of 3.5 AU from the centre of the Sun.

1 AU = 1.5 × 10¹¹ m

Use Kepler’s third law to calculate the period of the orbit of the asteroid in years.

An object orbiting the Sun at a distance of 1 AU has a period of 1 year.

Calculate the kinetic energy of the asteroid.

Use the graph to calculate the change in gravitational potential energy of the satellite from the moment of collision until it crosses the Earth’s orbit.

Eventually, the asteroid falls into a new orbit around the Sun. This orbit is elliptical.

State the other feature of this orbit predicted by Kepler’s first law.

A satellite is in geostationary orbit 36 000 km above the Earth’s surface. The Earth has a radius of 6400 km.

At what speed is the satellite moving relative to the centre of the Earth?

0 m s^–1 A

490 m s^–1 B

2.6 km s^–1 C

3.1 km s^–1 D

Your answer

The MAVEN spacecraft orbits Mars and studies its upper atmosphere.

The diagram below shows the orbit of MAVEN around Mars.

Mark an X on the diagram to show the point in the orbit where MAVEN has maximum

acceleration.

Explain how Kepler’s 1st law applies to MAVEN’s orbit around Mars.

The table shows data for four orbits around Mars.

Phobos and Deimos are moons of Mars.

An areostationary orbit for Mars is the equivalent of a geostationary orbit for Earth.

Show that Kepler’s 3rd law applies to this data.

Suggest two reasons why MAVEN was not placed in an areostationary orbit.

In June 2018, the spacecraft Hayabusa2 arrived at an asteroid called Ryugu.

The asteroid orbits the Sun in an elliptical orbit as shown below.

The diagram is not drawn to scale.

Indicate with a letter X on the orbit where the asteroid would be moving at maximum

speed.

Use Kepler’s second law to explain your answer to (a)(i).

Define gravitational potential.

Show that the magnitude of the gradient of the graph is equal to GM, where M is the mass of the asteroid and G is the gravitational constant.

Use the gradient of the graph to show that the mass M of the asteroid is about 4.6 × 10¹¹ kg.

A student has collected some data on the Solar System. The student plots a graph, but only two data points are shown below.

The distance from the centre of the Sun is r.

Which quantity y is represented on the vertical axis?

A Speed of a planet.

B Period of a planet.

C Gravitational potential of the Sun.

D Gravitational field strength of the Sun.

Your answer

Complete the equation for Kepler’s third law below. You do not need to define any of the terms.

The mass of Earth is 6.0 × 10²⁴ kg. (ii) Calculate the radius of the circular path of a satellite in a geostationary orbit around the Earth.

A binary star is a pair of stars which move in circular orbits around their common centre of mass.

In this question consider the stars to be point masses situated at their centres.

Fig. 3.1 shows a binary star where the mass of each star is m. The stars move in the same circular orbit.

Explain why the stars of equal mass must always be diametrically opposite as they travel in the circular orbit.

The stars are viewed from Earth in the plane of rotation. The stars are observed using light that has wavelength of 656 nm in the laboratory. The observed light from the stars is Doppler shifted.

Calculate the maximum change in the observed wavelength of this light from the orbiting stars. Give your answer in nm.

The centre of mass of the binary star lies at the surface of the star of mass 4 m. Draw on

Fig. 3.2 two circles to represent the orbits of both stars.

Explain why the smaller mass star travels faster in its orbit than the larger mass star.

Write an expression for the gravitational potential V_g at the surface of a planet of mass M and radius r.

The table below shows some data for Mercury and Pluto.

Show that the escape velocity v of a gas molecule on the surface of Pluto is given by the equation

where M is the mass of Pluto and r is its radius.

Calculate the escape velocity v of gas molecules on the surface of Pluto.

=........ m s⁻¹ v

Explain why Mercury has no atmosphere whilst Pluto still has a thin atmosphere. Use data from the table to support your explanation.