Orbits of planets & satellites

Mimas and Enceladus are moons of Saturn.

Mimas orbits at a height of 1.28 × 10⁸ m above the surface of Saturn. The orbital period of Mimas is 0.94 Earth days.

Enceladus orbits at a height of 1.80 × 10⁸ m above the surface of Saturn.

Calculate, in Earth days, the orbital period of Enceladus.

Titan is the largest moon of Saturn.

Figure 2 shows the variation of gravitational potential with distance from the centre of Saturn in the direction of Titan.

The dashed line shows the gravitational potential due to the mass of Saturn alone.

The solid line shows the total gravitational potential due to Saturn and Titan. The solid line ends at the surface of Titan.

A student suggests that, over the range of distances shown in Figure 2, the gravitational field strength due to the mass of Saturn alone is approximately constant.

Explain why the suggestion is correct.

At a point P, the resultant gravitational field strength due to Saturn and Titan is zero.

Determine, using the solid line in Figure 2, the distance from Saturn to P.

The centre of Titan is 12.22 × 10⁸ m from the centre of Saturn.

Show that the gravitational potential due to the mass of Titan at a distance of 12.08 × 10⁸ m from Saturn is about −7 × 10⁵ J kg⁻¹. Go on to determine the mass of Titan.

Ions orbit Jupiter in a ring called a plasma torus as shown in Figure 6. This question is about applying gravitational, magnetic and electric fields to a particle in the plasma.

An ion travels in a circular path at a speed of 74.2 km s⁻¹ and at a distance of 4.22 × 10⁸ m from the centre of Jupiter.

Student A suggests that the gravitational force acting on the ion due to Jupiter provides the centripetal acceleration of the ion.

Deduce whether this suggestion is correct.

Student B suggests that the magnetic field of Jupiter alone can cause the ion to move in the circular path.

Figure 7 shows the direction of the velocity v of the ion in the plasma torus. v is tangential to the plasma torus.

Figure 7

The ion has a positive charge.

Student B ’s suggestion requires a magnetic field to act at the position of the ion.

Draw an arrow on Figure 7 to show the direction of this magnetic field.

The ion has a charge of +3.20 × 10⁻¹⁹ C and a mass of 5.31 × 10⁻²⁶ kg.

The ion moves with a speed of 74.2 km s⁻¹ in a circle of radius 4.22 × 10⁸ m.

Calculate the magnitude of the magnetic flux density required for this motion. State, in fundamental (base) units, the unit for your answer. Ignore any contribution from the gravitational field.

The plasma torus contains two ions X and Y as shown in Figure 8. X has a charge of +2 e. Y has a charge of +1 e. Assume that each ion acts as a point charge.

Calculate the magnitude of the resultant electric field strength at point P in Figure 8 due to the two ions.

An object on the surface of a moon has an escape velocity v. The moon has a density ρ and radius R. Another moon X has a density 2ρ and radius .

What is the escape velocity for an object on the surface of X ?

A satellite S₁ is placed in a circular orbit around the Earth so that observations of the far side of the Moon can be made continuously.

S₁ has the same angular speed as the Moon so that the centres of the Earth, the Moon and S₁ are always in a straight line.

Figure 7 shows two positions of the Moon and S₁ as they orbit the Earth.

The resultant force on S₁ is due to the gravitational forces from the Earth and the Moon. The magnitude of the Earth’s gravitational field strength at the orbital radius

The magnitude of the Moon’s gravitational field strength at the orbital radius

Show that g_M is approximately 1.2 × 10⁻³ N kg⁻¹.

period of the Moon’s orbit = 27.3 days

Calculate the distance from S₁ to the centre of the Moon.

Another satellite S₂ is placed in a circular orbit between the Earth and the Moon. S₂ always views the near side of the Moon. S₂ also has the same angular speed as the Moon so that the centres of the Earth, the Moon and S₂ are always in a straight line. Figure 8 shows two positions of the Moon and S₂ as they orbit the Earth.

Explain how the resultant force on S₂ due to the gravitational fields of the Earth and the Moon causes S₂ to orbit with the same angular speed as the Moon. No calculations are required.

A synchronous orbit of the Earth has a radius R. A planet has a mass twice the mass of the Earth. A day on the planet is one quarter of an Earth day.

What is the radius of a synchronous orbit for this planet?

An asteroid has a mass of 2 × 10¹⁷ kg and an escape velocity of 40 m s⁻¹.

What is the order of magnitude of the radius of the asteroid?

Data are collected for the mass M, radius R and escape velocity u for each planet in the Solar System.

The data show that u is directly proportional to

A satellite is in a circular orbit at a height h above the surface of a planet of mass M and radius R.

What is the linear speed of the satellite?

Which statement is not true for a satellite in a geostationary orbit?

The lines in Figure 4 show the shape of the gravitational field around two stars G and H.

Compare, with reference to Figure 4, the masses of G and H.

X and Y are two points in the field.

Annotate Figure 4 to show the field direction at X and the field direction at Y.

A spherical asteroid P has a mass of 2.0 × 10²⁰ kg.

The gravitational field strength at its surface is 0.40 N kg⁻¹.

Calculate the radius R of P.

Sketch, on Figure 5, the variation of the gravitational field strength g with distance r. The distance r is measured from the centre of P.

Explain what is represented by the area under the graph between r = R and r = 2 R on Figure 5.

Asteroid P approaches the two stars G and H. Figure 6 shows one position of P close to H.

The gravitational force on P from G is 6.38 × 10¹² N.

Calculate the magnitude of the acceleration of P.

Explain why P cannot have a circular orbit around H.

The Moon orbits the Earth in 27 days.

What is the angular speed of the Moon’s orbit?

The radius of the Earth is R and the acceleration due to gravity at the surface of the Earth is g.

What is the escape velocity for a mass m at the surface of the Earth?

A planet has a mass M and a radius R. Loose material at the equator only just remains in contact with the surface of the planet. This is because the speed at which the planet rotates is very large.

What is the period of rotation of the planet?

Satellites N and F have the same mass and move in circular orbits about the same planet. The orbital radius of N is less than that of F.

Which is smaller for N than for F ?

The Global Positioning System (GPS) uses satellites to support navigation on Earth.

One GPS satellite is in a circular orbit at a height h above the surface of the Earth. The Earth has mass M and radius R.

Calculate the orbital period of the satellite when h equals 2.02 × 10⁷ m.

Figure 3 shows the orbital plane of the satellite inclined at an angle to the equator. X, Y and Z are locations on the Earth.

X is at the North Pole, Y is on a high mountain and Z is on the equator.

The satellite is to be launched from one of the locations.

State and explain which launch site X, Y or Z minimises the amount of fuel required to send the satellite into its orbit.

The satellite has a mass of 1630 kg.

Calculate the gravitational potential energy of the satellite when in the orbit in Question 02.2.

A different satellite is in a higher circular orbit.

Explain how the linear speed of this satellite compares with the linear speed of the satellite in Question 02.1.

Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M.

The moon has an orbital period T. T is related to r by

where k is a constant for this planet.

Table 2 gives data for two of the moons of the planet Uranus.

Calculate the orbital radius X of Umbriel.

Calculate the mass of Uranus.

Table 3 gives data for three more moons of Uranus.

Deduce which moon in Table 3 has the greatest escape velocity for an object on its surface. Assume the effect of Uranus is negligible.

A spring mechanism can project an object vertically to a maximum height of 1.0 m from the surface of the Earth.

Determine whether the same mechanism could project the same object vertically to a maximum height greater than 100 m when placed on the surface of Ariel.

What is the angular speed of a satellite in a geostationary orbit around the Earth?

Define gravitational potential at a point.

Figure 2 shows the positions of equipotential surfaces at different distances from the centre of the Moon.

Explain how the equipotential surfaces in Figure 2 show that the gravitational field is not uniform.

Calculate, using Figure 2, the escape velocity at the surface of the Moon.

radius of Moon = 1.74 × 10⁶ m

What is the angular speed of a satellite in a geostationary orbit around the Earth?

A planet of mass M and radius R rotates so quickly that material at its equator only just remains on its surface.

What is the period of rotation of the planet?

Satellites N and F have the same mass and are in circular orbits about the same planet. The orbital radius of F is greater than that of N.

Which is greater for F than for N ?

Which graph shows the relationship between the time period T and the orbital radius r of a planet in orbit around the Sun?

Derive an expression to show that for satellites in a circular orbit

where T is the period of orbit and r is the radius of the orbit.

Pluto is a dwarf planet. The mean orbital radius of Pluto around the Sun is 5.91 10⁹ km compared to a mean orbital radius of 1.50 10⁸ km for the Earth.

Calculate in years the orbital period of Pluto.

A small mass released from rest just above the surface of Pluto has an

Assume Pluto has no atmosphere that could provide any resistance to motion.

Calculate the mass of Pluto.

Give your answer to an appropriate number of significant figures.

Figure 8 shows the variation in gravitational potential with distance from the centre of Pluto for points at and above its surface.

A meteorite hits Pluto and ejects a lump of ice from the surface that travels vertically at an initial speed of 1400 m s¹.

Determine whether this lump of ice can escape from Pluto.

What is the kinetic energy of X ?