Gravitational potential

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.

Gravitational waves are disturbances in space-time that are produced by violent astrophysical events.

Define strain.

The maximum strain shown on the graph is approximately 10^–21. (ii)

Calculate the force that would produce the same strain in a copper wire of diameter 2 mm.

Young modulus of copper = 130 GPa

By considering the answer to (ii), suggest why it is very difficult to detect the strain caused by gravitational waves.

In this part of the question you will use Newtonian physics to estimate the power output of the black hole collision.

The diagram shows two identical black holes, each of mass M, orbiting each other with period T.

Prior to colliding, the black holes move at speed V in identical circular orbits of diameter D about the central point C.

Write down an expression for the gravitational force, F, acting between the black holes in terms of M, D and the universal gravitational constant, G.

The total orbital energy, E, of the black holes is the sum of their kinetic energy and gravitational potential energy. Show that E is given by the expression:

E = k (²G²M⁵)^1/3 (1/T^2/3)

where k is a numerical constant.

By considering times B and D, use your answer to (iii) to estimate the power radiated as gravitational waves in this time interval.

The Hipparcos space telescope used stellar parallax with a precision of 9.7 × 10^–4 arcseconds to determine the distance to stars.

One of the stars studied was Polaris A. Data about this star is in the table below.

Estimate the maximum stellar distance in parsecs that could be measured using Hipparcos.

Calculate the percentage uncertainty in the calculated value of the distance to Polaris A.

A continuous stream of particles called a solar wind flows from the surface of the star into the surrounding space.

These particles include helium nuclei of mass 6.6 × 10^–27 kg.

Assume that the atmosphere is modelled as an ideal gas.

Show that the typical kinetic energy of a helium nucleus in the atmosphere is about 10^–16 J.

The gravitational potential energy of a helium nucleus in the outer layer of the star is –2.3 × 10^–16 J.

Calculate the gravitational potential energy U at the maximum distance from the star that a helium nucleus could reach.

Calculate the distance from the centre of the star reached by this helium nucleus.

Explain why the star has a solar wind that reaches a much greater distance from the star than found in (iii).

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.

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.