Orbits of planets and satellites

The asteroid Bennu is in an approximately circular orbit around the Sun. Scientists consider that Bennu may collide with Earth within the next century.

In 2060 Bennu will be 7.49×10⁸m from the Earth.

Show that the mass of Bennu is about 7.810¹⁰kg.

mean radius of Bennu = 245 m average density of Bennu = 1260 kg m^–3

Calculate the magnitude of the force F between Bennu and Earth in 2060.

mass of Earth = 5.9810²⁴kg

The orbital time of Bennu around the Sun is 1.17 years.

Calculate the mass of the Sun.

average distance of Bennu from the Sun = 1.7010¹¹m

A satellite of mass m orbits the Earth with speed v.

What is the speed of a satellite of mass 2m orbiting at the same distance from the Earth?

Galileo is credited with inventing the first telescope in 1610. The picture shows an early demonstration of the telescope.

(Source: © CPA Media Pte Ltd/Alamy Stock Photo)

A converging lens was positioned at one end of the telescope. A diverging lens was placed at the other end and a person looked through this lens.

The converging lens produced an image at a distance equal to the focal length of the lens.

Explain what can be concluded about the object being viewed.

The final image produced by the telescope is described as virtual and upright.

State what is meant by virtual and upright.

Virtual

Upright

The image, I, produced by the converging lens is at a distance from the diverging lens equal to the focal length of the diverging lens, as shown. This image acts as an object for the diverging lens.

The distance equal to the focal length on the other side of the lens is marked with F.

Draw the ray diagram for the diverging lens.

Galileo’s first telescope had a magnification of 10, and a distance between the centres of the two lenses of 90 cm.

Calculate the focal length of each lens.

Galileo was the first person to observe Jupiter’s larger moons.

Ganymede is Jupiter’s largest moon. The distance between the centre of Ganymede and the centre of Jupiter is 1.07×10⁶km. Ganymede takes 171 hours to complete an orbit around Jupiter.

Calculate the mass of Jupiter.

Astronomers observing stars at the centre of our galaxy have suggested that many of them are orbiting a supermassive black hole. The mass of this black hole is 9.2 10³⁶kg.

Calculate the orbital period for a star in a circular orbit at a distance of 1.9 10¹⁴m from a black hole of this mass.

The star S0-2 is in a highly elliptical orbit around the position of the black hole.

At its point of closest approach, S0-2 is at a distance of 1.8 10¹³m from the centre of the black hole.

At the most distant point of its orbit, S0-2 is 2.7 10¹⁴m from the black hole.

Show that the change in gravitational potential between the closest and most distant points in this orbit is about 3 10¹³J kg⁻¹.

At its point of closest approach, the star is travelling at a speed of 8.1 10⁶m s⁻¹.

Calculate the speed of S0-2 at the furthest point in its orbit using the change in gravitational potential.

mass of S0-2 = 2.4 10³¹kg

Trigonometric parallax and Hubble’s law are two methods used to determine astronomical distances. Explain whether either of these methods is suitable to determine the distance to S0-2.

Astronauts on the 1971 Apollo 14 mission to the Moon brought back many rock samples. It is now believed that one of these contains a piece of rock that originated on Earth about 4 billion years (4 × 10⁹ years) ago.

The piece of rock is believed to have been launched into space when an asteroid struck the Earth.

The rock sample contains uranium. The radioactive decay of uranium allows it to be used to determine the time since the rock was formed on the Earth.

The uranium isotope²³⁸₉₂U becomes the lead isotope²⁰⁶₈₂Pb through a series of radioactive decays.

Calculate the number of α particles and the number of β particles emitted for one nucleus of²³⁸₉₂U to decay to become a nucleus of²⁰⁶₈₂Pb.

The half-life of²³⁸₉₂U is 4.47 10⁹ years. The half-lives of the other stages in the decay to²⁰⁶₈₂Pb are relatively so short that they can be ignored.

There was no lead in the rock when it formed, so all the²⁰⁶₈₂Pb in the sample is a product of²³⁸₉₂U decay. In the sample, for every 103 uranium nuclei present at the start, 50 are now lead nuclei.

Show that the age of the sample is about 4 × 10⁹ years.

The gravitational potential between the Earth and the Moon due to the combined effect of their gravitational fields increases to a maximum value of −1.28 MJ kg⁻¹ at a point between them.

Calculate the minimum speed at which a rock would have to leave the Earth in order to reach the Moon.

In your calculation, you may assume the rock has zero kinetic energy when it has maximum potential energy.

mass of Earth = 5.97 10²⁴kg radius of Earth = 6370 km

Four billion years ago, the Moon had a different orbital period, because it was closer to the Earth than it is today.

Calculate the period of the Moon’s orbit four billion years ago, when the radius of its orbit was 1.34 10⁸m.

mass of Earth = 5.97 × 10²⁴kg

Astronauts on the 1971 Apollo 14 mission to the Moon brought back many rock samples. It is now believed that one of these contains a piece of rock that originated on Earth about 4 billion years (4 × 10⁹ years) ago.

The piece of rock is believed to have been launched into space when an asteroid struck the Earth.

The rock sample contains uranium. The radioactive decay of uranium allows it to be used to determine the time since the rock was formed on the Earth.

The uranium isotope²³⁸₉₂U becomes the lead isotope²⁰⁶₈₂Pb through a series of radioactive decays.

Calculate the number of α particles and the number of β particles emitted for one nucleus of²³⁸₉₂U to decay to become a nucleus of²⁰⁶₈₂Pb.

The half-life of²³⁸₉₂U is 4.47 10⁹ years. The half-lives of the other stages in the decay to²⁰⁶₈₂Pb are relatively so short that they can be ignored.

There was no lead in the rock when it formed, so all the²⁰⁶₈₂Pb in the sample is a product of²³⁸₉₂U decay. In the sample, for every 103 uranium nuclei present at the start, 50 are now lead nuclei.

Show that the age of the sample is about 4 × 10⁹ years.

The gravitational potential between the Earth and the Moon due to the combined effect of their gravitational fields increases to a maximum value of −1.28 MJ kg⁻¹ at a point between them.

Calculate the minimum speed at which a rock would have to leave the Earth in order to reach the Moon.

In your calculation, you may assume the rock has zero kinetic energy when it has maximum potential energy.

mass of Earth = 5.97 10²⁴kg radius of Earth = 6370 km

Four billion years ago, the Moon had a different orbital period, because it was closer to the Earth than it is today.

Calculate the period of the Moon’s orbit four billion years ago, when the radius of its orbit was 1.34 10⁸m.

mass of Earth = 5.97 × 10²⁴kg

Solar panels consisting of combinations of photovoltaic cells use energy in the radiation received from the Sun to generate electricity.

An advertisement for solar panels claims that the intensity of radiation from the Sun incident at the top of the Earth’s atmosphere is more than 2 kW m⁻².

Assess the validity of this claim.

radius of Sun = 6.96 × 10⁸m

surface temperature of Sun = 5790 K

distance from Sun to Earth = 1.50 × 10¹¹m

The average intensity of radiation from the Sun incident at the Earth’s surface over a 24-hour period has been determined to be 164 W m⁻².

The average intensity of radiation from the Sun at the Earth’s surface is much less than the intensity incident at the top of the Earth’s atmosphere.

Explain why.

It is claimed that the area of solar panels needed to generate 100 GW of power is about 0.5% of the surface area of the Earth.

Assess the validity of this claim.

radius of Earth = 6.4 × 10⁶m

typical efficiency of solar panels = 25%

Scientists are developing a space station equipped with large solar panels. The space station would be located in a geostationary orbit. The space station would transfer energy to Earth as microwaves.

A space station in a geostationary orbit is above the equator and has a period of 24 hours.

Explain one advantage of locating the space station in a geostationary orbit.

Calculate the height h of the space station above the equator when it is in a geostationary orbit.

mass of Earth = 6.00 × 10²⁴kg

24 hours = 8.64 10⁴s

Solar panels consisting of combinations of photovoltaic cells use energy in the radiation received from the Sun to generate electricity.

An advertisement for solar panels claims that the intensity of radiation from the Sun incident at the top of the Earth’s atmosphere is more than 2 kW m⁻².

Assess the validity of this claim.

radius of Sun = 6.96 × 10⁸m

surface temperature of Sun = 5790 K

distance from Sun to Earth = 1.50 × 10¹¹m

The average intensity of radiation from the Sun incident at the Earth’s surface over a 24-hour period has been determined to be 164 W m⁻².

The average intensity of radiation from the Sun at the Earth’s surface is much less than the intensity incident at the top of the Earth’s atmosphere.

Explain why.

It is claimed that the area of solar panels needed to generate 100 GW of power is about 0.5% of the surface area of the Earth.

Assess the validity of this claim.

radius of Earth = 6.4 × 10⁶m

typical efficiency of solar panels = 25%

Scientists are developing a space station equipped with large solar panels. The space station would be located in a geostationary orbit. The space station would transfer energy to Earth as microwaves.

A space station in a geostationary orbit is above the equator and has a period of 24 hours.

Explain one advantage of locating the space station in a geostationary orbit.

Calculate the height h of the space station above the equator when it is in a geostationary orbit.

mass of Earth = 6.00 × 10²⁴kg

24 hours = 8.64 10⁴s

In 1990, the Hubble Space Telescope (HST) was launched into a low Earth orbit above the Earth’s atmosphere.

HST orbits the Earth in a circular orbit with a speed of 7.59 km s^–1.

mass of Earth = 5.97 × 10²⁴ kg radius of Earth = 6.37 10⁶ m

Show that the height of HST above the surface of the Earth is about 6 × 10⁵ m.

Calculate the increase in the gravitational potential energy as HST is raised, from its initial position at the Earth’s surface, to its orbital height.

mass of HST = 11 600 kg

A student suggests that giving HST more energy than that required in (ii) would result in the satellite orbiting at a greater height and with a greater speed.

Assess the validity of the student’s suggestion.

The transmission of electromagnetic radiation through the atmosphere is shown on the graph.

State one advantage shown by this graph of positioning a telescope above the atmosphere.

High resolution images from HST allow astronomers to make detailed measurements of very distant galaxies. The graph shows how the recessional velocities of distant galaxies depend on their distance from Earth.

Determine an age for the universe.

The International Space Station (ISS) is in a low Earth orbit. Astronauts in ISS have an apparent weight of zero. In order to determine their mass, the astronauts must secure themselves to a platform which is set into oscillation and moves with simple harmonic

Explain why the astronauts in the ISS have an apparent weight of zero.

State what is meant by simple harmonic motion.

Describe how, using a stopwatch and a ruler, the following quantities could be determined for the oscillating platform:

the frequency of oscillation

the maximum speed of the platform.

The platform continues to move with simple harmonic motion at the same frequency, but its amplitude is doubled.

Explain how the maximum kinetic energy of the platform will change.

The diagram shows two black holes, A and B, orbiting each other.

Assume that the centre of mass C of the system is the centre of a circular orbit for each black hole as shown in the diagram.

Black hole A is in an orbit of radius 2.910¹⁰m and black hole B is in an orbit of radius 3.610¹⁰m. Both orbit with the same period, so the total distance between them is 6.510¹⁰m.

Calculate the force between the black holes.

mass of Sun, M = 1.9910³⁰kg mass of black hole A = 36M mass of black hole B = 29M

By considering the orbit of one black hole about C, determine the period of the orbit.