Gravitation & Astrophysics

a) [3] A planet has a radius $R$ and the gravitational field strength at its surface is $g$. A satellite is placed in a circular orbit of radius $r$ around the planet.

Obtain an expression for $T$, the period of orbit of the satellite, in terms of $R, g$ and $r$, explaining the relevant physics in each of your steps.

b) [1] Both the Earth and Jupiter can be assumed to have circular orbits about the Sun. Jupiter has an orbital period, $T_{\text {Jupiter }}=11.9$ years. What is the orbital radius of Jupiter in au?

Note: an au (astronomical unit) is the average distance between the Earth and the Sun.

c) [3] The density of the Earth changes sharply between the core and the mantle. By taking just two suitable values of the density of the core and mantle of the Earth from the graph in Figure 1, estimate a value of the mass of the Earth using this data. Your values and calculation must be shown.

Figure 1: credit: AllenMcC. [CC BY $3.0$ (https://creativecommons.org/licenses/by/3.0)], from Wikimedia Commons

d) [9] A planet of mass $M_{p}$ has an orbiting moon of mass $M_{m}$. The mass of the moon is about $1 / 8$ the mass of the planet.

(i) Sketch a set of equipotential lines for the planet-moon system. Ignore the orbital motion of the moon. Without detailed calculation, indicate on your sketch the relative positions of the neutral point (a point of zero field strength) and the centre of mass of the planet-moon system.

A rocket of mass $m$ is launched from the planet. The rocket engine is used very briefly at take-off when it is close to the planet's surface, and it almost immediately reaches speed $v$, after which the fuel is exhausted and there is no further propulsion. It is then in free-fall and will have just enough energy to reach the moon. Assume that you can ignore relative motion of the planet and moon, and ignore air resistance.

Obtain expressions and calculate values for:

(ii) The distance, $d$, from the planet's centre, where the speed of the rocket is lowest.

(iii) The launch speed $v$ of the rocket from the planet, in terms of the variables $M_{p}, M_{m}$, the radius of the planet, $R_{p}$, and the separation of the centres of the moon and planet, $D$. For this question part, you may take $M_{p}=6.00 \times 10^{24} \mathrm{~kg}, M_{m}=7.50 \times 10^{23} \mathrm{~kg}$, $D=3.84 \times 10^{8} \mathrm{~m}$.

e) [4] The Earth can be considered to consist of concentric spheres of material, the outer sphere being the crust with a density of $2500 \mathrm{~kg} \mathrm{~m}^{-3}$. If the mass of the Earth is $M_{\mathrm{E}}=6.0 \times 10^{24} \mathrm{~kg}$, what will be the fractional change in the gravitational field strength when descending to the bottom of a gold mine, a distance $d=3000 \mathrm{~m}$ below the surface of the Earth?

Hint: the field inside a hollow spherical shell is zero.

f) [5] A planetary sized object similar to the Earth was formed from a very large spherical cloud of interstellar dust collapsing under gravity to form a spherical body of uniform density $\rho$ and radius $R$

(i) Find an expression for the change in gravitational potential energy as the radius of the protoplanet increases from $r$ to $r+\delta r$ by the formation of a thin spherical shell of material of mass $\delta m$.

(ii) Now derive an expression for the total transfer of gravitational potential energy in the formation of the planet. Use the following terms: the gravitational constant $G$, and the final radius and mass, $R$ and $M$ respectively.

(iii) If all of this energy was used to heat the planet formed at a very low initial temperature, estimate the temperature that the planet would have reached.

For this planet, take $R=6400 \mathrm{~km}, M=4.0 \times 10^{26} \mathrm{~kg}$, and the average specific heat capacity of the material of the planet is $700 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$.

a) [4] In 2017 an object that became known as Oumuamua was observed in the Solar System.

Figure 13 shows an artist's impression.

Figure 13: (Picture: ESO)

The speed of the object was measured to be $87.7 \mathrm{~km} \mathrm{~s}^{-1}$ at a closest approach to the Sun of $38.1 \times 10^{6} \mathrm{~km}$ and astronomers deduced that Oumuamua did not belong to the Solar System. Make a suitable calculation to confirm this deduction.

b) [21] In January 2019 the New Horizons spacecraft made a fly by of a distant object in the Solar System called Ultima Thule, shown in Figure 14 below. The planetoid consists of two lobes of diameters $2 r_{1}=19.5 \mathrm{~km}$ and $2 r_{2}=14.2 \mathrm{~km}$. The period of rotation about its short axis is $T=15$ hours. (The long axis is a line passing through the centres of the two lobes, and a short axis is perpendicular to the long axis.)

Figure 14: The two approximately spherical lobes making up Ultima Thule.

Credits: NASA/Johns Hopkins Applied Physics Laboratory/Southwest Research Institute, National Optical Astronomy Observatory

If we assume that the two lobes are held together by a weak gravitational attraction, we can model the object as two spheres of the same density just touching. (i) Find the centre of mass of Ultima Thule, as measured from the centre of the largest sphere.

(ii) By considering the resultant force acting on one sphere, estimate the density of Ultima Thule.

(iii) Using the fact that the density can be taken to be similar for the two lobes, sketch a graph of the gravitational field strength along the long axis.

Note: the field strength varies linearly with radius inside a spherical body of uniform density.

You may find it helpful to sketch the potential along the long axis. [8]

In fact, the two lobes are not bound by gravity, so the density and the masses of the lobes are not known. From the appearance of the two lobes, it looks as though they were moving at a low relative speed when they collided.

As an example, consider two spherical iron masses with the same radii as the lobes of Ultima Thule. The masses are $m_{1}=3.06 \times 10^{16} \mathrm{~kg}$ and $m_{2}=1.18 \times 10^{16} \mathrm{~kg}$ for the larger and smaller lobes respectively. We can assume that they approached from a great distance apart along a line through their centres, with a negligible initial relative velocity.

(iv) Calculate the amount of energy liberated when these two iron spheres collided.

(v) Calculate the relative speed of approach. [7]

This energy of collision may have initially led to an increase in temperature of the two spheres (you may neglect any energy of rotation), which we could calculate from knowing the specific heat capacity of iron. However, at the very low temperatures of the outer Solar System the molar heat capacity of a material is not constant. For example, at low temperatures the molar heat capacity of iron, $c$, varies as

$$ c=\frac{12 \pi^{4} R}{5 \theta_{\mathrm{D}}^{3}} T^{3} $$

where:

. $c$ is the molar heat capacity in $\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$

. $R$ is the universal gas constant, $R=8.31 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$

• $\theta_{\mathrm{D}}$ is a constant, $\theta_{\mathrm{D}}=464 \mathrm{~K}$ for iron
• $T$ is the temperature in kelvin
• molar mass of iron $=56 \mathrm{~g} \mathrm{~mol}^{-1}$

(vi) Calculate the number of moles of iron involved in the collision.

(vii) Using your value of the energy liberated on collision, calculate the maximum temperature reached by the iron spheres, assuming the initial temperature was $4 \mathrm{~K}$ and radiative losses are in the first instance negligible. [6]

a) [3] A hump-back bridge has a circular cross section. It crosses a river of width $18 \mathrm{~m}$ and the height of the centre of the bridge above the surrounding ground is $4.0 \mathrm{~m}$. What is the maximum speed at which a car can drive over the bridge whilst still remaining in contact with it?

Hint: if two chords of a circle, $\mathrm{AB}$ and $\mathrm{CD}$ meet at a point $\mathrm{S}$ then $\mathrm{AS}\cdot\mathrm{SB} =\mathrm{CS} \cdot \mathrm{SD}$

b) [3] The Sun orbits the centre of the Milky Way galaxy at a radius of 30000 light years, and a period of orbit about the galaxy of 200 million years. Assuming that the mass distribution of matter in the galaxy is concentrated mainly in a central uniform sphere and the Earth lies in one of the arms, as illustrated schematically in Figure 10,

(i) obtain an expression for the period of orbit $T$ of the Sun in terms of its radius of orbit, $r, G$ and $M_{\mathrm{g}}$, the mass of the galaxy.

(ii) Determine a value of the mass of the galaxy.

(iii) If the Sun represents the mass of an average star, estimate the number of stars in the Milky Way.

Figure 10: Schematic of the Milky Way galaxy, viewed in the plane, with the Sun along one 'arm', marked by the arrow.

c) [14] A satellite orbits the spherical Earth of radius $R_{\mathrm{E}}$ in a circular orbit of radius $r$.

(i) Write down an expression for the speed $v$ of the satellite in terms of $r, G$ and $M_{\mathrm{E}}$, the mass of the Earth, explaining the physics needed to obtain your expression.

(ii) Evaluate the speed of a satellite in a low orbit, if it was just grazing the Earth.

(iii) Calculate the period of the satellite orbit.

(iv) The satellite instead orbits at a height $h \ll R_{\mathrm{E}}$ above the surface of the Earth. Obtain an expression for the fractional change $\frac{\Delta T}{T}$ of the orbital period $T$ in terms of $h$ and $R_{\mathrm{E}}$. Hence what is the period of orbit of the satellite orbiting at a height of $200 \mathrm{~km}$ ?

(v) Sketch a graph of the period of orbit $T$ against $h$ for $h \ll R_{\mathrm{E}}$.

(vi) Show that when a large mass $m_{1}$ moving at speed $u$ collides elastically with a stationary small mass $m_{2}$, the kinetic energy lost by $m_{1}$ is $\approx 2 m_{2} u^{2}$

(vii) When a small satellite of mass $m$ and cross-sectional area $A$ passes through a gas of density $\rho$ it will make numerous elastic collisions with molecules of the gas. Obtain an expression for the rate of loss of energy of the satellite in terms of $\rho, A$ and $v$.

(viii) An Earth satellite of mass $20 \mathrm{~kg}$ is in a circular orbit at a height small compared to the radius of the Earth. If air resistance causes its total energy to decrease by $10 \mathrm{~kJ}$ per revolution, what is the fractional change in its speed per revolution?

(ix) By considering an expression for total energy of a satellite in orbit, show whether the speed increases or decreases.

(x) By simply using values for the period and speed of an Earth orbiting satellite in a grazing orbit, if the cross-sectional area of the satellite is $2.4 \mathrm{~m}^{2}$, estimate the density of the atmosphere for this rate of energy loss.

d) [5] In the previous parts of this question, we assumed that the mass of the Earth was very much larger than the satellite. In the remainder of this question, we relax this approximation.

Many stars are found as part of binary systems, consisting of two stars of unequal mass in circular orbits about one another.

(i) Draw a diagram of the orbits of such a binary system labelling the positions of the two stars at two successive times in their orbits.

(ii) Suppose the stars have mass $m$ and $k m$ with $k>1$, what are their relative orbital radii?

(iii) One day the more massive star explodes in a supernova. After the explosion, the masses of the stars are now equal, and the exploding material has moved to a great distance away. Describe using diagrams the motion of the star system if the binary system were to remain bound. Assume the supernova causes the larger star to lose mass in a spherical symmetrical manner.

(iv) Find the critical value of the initial mass ratio $k$, above which the binary system becomes unbound after the explosion described in (iii).

a) [6] DART was fitted with the most up-to-date ion drive available (NEXT-C) to ensure it could accelerate up to the required impact velocity with only a fraction of the mass of fuel that would be needed for conventional rockets. It does this by applying a very small thrust for a very long time. The fuel used was singly ionised xenon-131 ions.

The accelerating potential was 1800 V .

(i) What was the ejection speed of the xenon? Assume the xenon ions started from rest.

(ii) The current of ions was 3.52 A . What is the thrust generated?

(iii) The probe hit Dimorphos 306 days after launch. If the ion drive had been run continuously for that time (and there was enough xenon to do so), what would have been the speed at impact?

Assume linear acceleration of the probe in deep space starting from rest, and take the average mass of the probe to be 600 kg during the acceleration phase.

Dimorphos was discovered through dips in the brightness of Didymos as it passes in front and behind the asteroid (known as transits).

b) [3] Radar observations of Didymos measured its diameter to be 780 m (assuming a spherical shape) and the centre-to-centre distance to Dimorphos to be 1.20 km (assuming a circular orbit). The relative depth of the dips in brightness meant that the diameter of Dimorphos was derived to be 164 m (again assuming a spherical shape), and the period of the orbital motion was 11 h 55 min .

(i) By considering the forces acting on each of the objects in their circular orbits, calculate the total mass of the binary system.

(ii) Assuming both asteroid and moon have the same density, calculate the mass of Didymos and Dimorphos.

c) [10] The DART probe had a mass of 570 kg at impact and was travelling at a speed of $6.14 \mathrm{~km} \mathrm{~s}^{-1}$ (relative to the rest frame of Dimorphos).

(i) By conserving momentum and assuming a head-on inelastic collision, determine the change in Dimorphos' orbital speed, $\Delta v$. Give your answer in $\mathrm{mm} \mathrm{s}^{-1}$.

(ii) The impact will have made the orbit elliptical, although by considering the total energy of the orbit you can calculate the radius of an equivalent circular one.

i. obtain an expression for the total energy of a satellite, $m_{\mathrm{s}}$, in orbit about a central mass, $m_{\mathrm{c}}$, in terms of $G, m_{\mathrm{c}}, m_{\mathrm{s}}$ and the radius of orbit $r$.

Using this approach, calculate

ii. the expected decrease in orbital radius, $\Delta r$, and hence

iii. the expected decrease in orbital period, $\Delta T$, giving your answer in minutes.

d) [6] Less than a month after the impact, it was announced that the observed orbital period shortening was 32 minutes. This is larger than the one in part (c) due to the momentum transfer enhancement factor, $\beta$, which quantifies how much the ejecta from the surface and deformation of Dimorphos has amplified the effect. To calculate $\beta$ accurately is quite complicated and will need months of observations, but a very rough estimate can be made using the approximation $\beta \propto \Delta v$.

Treating the value in part (c) as the $\beta=1$ case, find the observed value of $\Delta v$ and hence a ballpark estimate for the value of $\beta$.

a) [5] Infrared measurements by JWST suggest that in this region some O and B class stars will be formed. An example of such a star has a mass of $23 M_{\text {Sun }}$, radius of $10.0 R_{\text {Sun }}$, luminosity of $1.47 \times 10^{5} L_{\mathrm{Sun}}$, and effective surface temperature of 35800 K .
(i) Given that the peak wavelength in a blackbody spectrum (i.e. the wavelength $\lambda_{\max }$ at which the power emitted is a maximum) is inversely proportional to the effective surface temperature, $T$, so $T \lambda_{\max }=$ constant, verify by calculation that photons emitted at the star's peak wavelength will be ionising.

(ii) In the Pillars of Creation, $n_{\mathrm{H}} \approx 10^{9} \mathrm{~m}^{-3}$ and ionisation will lead to an equilibrium temperature inside the sphere of 8000 K for which $\alpha=3.1 \times 10^{-19} \mathrm{~m}^{3} \mathrm{~s}^{-1}$. Assuming that the star emits monochromatically at just the peak wavelength, estimate $N$ and hence estimate $r_{\mathrm{s}}$. Give your answer in light years (ly).

(iii) The height of the pillar on the left is 5 ly. How many such large stars will be needed to photoevaporate the pillar (given long enough), such that it is no longer visible? Assume they are evenly spread out throughout the pillar.

As well as the ionising effects on its surroundings, a new, bright star will exert radiation pressure, and this can trigger further collapses of regions of the nebula into other stars. Assuming the region outside the Strömgren Sphere is purely neutral hydrogen, the mass that will be collapsed by an external pressure $p_{0}$ is given as:

$$ M_{\text {collapse }}=\frac{1.18 k_{\mathrm{B}}^{2} T_{\mathrm{neb}}^{2}}{p_{0}^{\frac{1}{2}} G^{\frac{3}{2}} m_{\mathrm{H}}^{2}} $$

This is known at the Bonner-Ebert mass, where $k_{\mathrm{B}}$ is the Boltzmann constant, $T_{\text {neb }}$ is the temperature of the neutral hydrogen in the nebula, and $m_{\mathrm{H}}$ is the mass of a hydrogen atom. The radiation pressure is given as:

$$ p_{\mathrm{rad}}=\frac{L}{4 \pi r^{2} c} $$

where $L$ is the luminosity of the star (the power output), $r$ is the distance from the star, and $c$ is the speed of light.

b) [2] Taking the temperature of the nebula to be $T_{\text {neb }}=50 \mathrm{~K}$, and taking $p_{0}=p_{\mathrm{rad}}$, calculate the cloud mass that will begin to collapse just outside the Strömgren Sphere and form a new star (cluster). Give your answer in solar masses.

Radiation pressure can also be important in some stars in keeping them in equilibrium against the gravitational forces trying to compress them. The gravitational pressure at the centre of a star is given by:

$$ p_{\text {Grav }}=\int_{0}^{R_{\text {star }}} \frac{G M(r) \rho(r)}{r^{2}} d r $$

where $M(r)$ is the mass enclosed within radius $r$, and $\rho(r)$ is the density. Since in real stars both $M$ and $\rho$ are complicated functions of $r$, this integral is not straightforward to do and generally can only be done numerically. In equilibrium, this will be balanced by contributions from the gas pressure and blackbody radiation pressure:

$$ p_{\mathrm{tot}}=p_{\mathrm{gas}}+p_{\mathrm{rad}}=\frac{\rho_{c} k_{\mathrm{B}} T_{c}}{\bar{m}}+\frac{4 \sigma}{3 c} T_{c}^{4} $$

where $\rho_{c}$ and $T_{c}$ are the central density and central temperature, respectively, $\sigma$ is the StefanBoltzmann constant, and $\bar{m}$ is the average mass of the particles. Generally, in the core of the star is a mix of fully ionised hydrogen and helium, so

$$ \frac{\bar{m}}{m_{\mathrm{H}}}=\frac{2}{1+3 X+0.5 Y} $$

where $X$ and $Y$ are the mass fractions of hydrogen and helium, respectively.

c) [10] At the centre of the Sun, lots of its initial hydrogen has already been turned into helium, so $X=0.35, Y=0.65, \rho_{c}=1.53 \times 10^{5} \mathrm{~kg} \mathrm{~m}^{-3}$ and $T_{c}=1.57 \times 10^{7} \mathrm{~K}$.

(i) Estimate the pressure at the centre of the Sun, $p_{\text {tot }}$.

(ii) Calculate $p_{\text {rad }}$ as a percentage of $p_{\text {tot }}$. Comment on your answer.

(iii) Consider a solar mass, Earth-sized white dwarf and assume it has a constant density equal to its average density. Carry out the integration and estimate $p_{\text {Grav }}$. How many times larger is it than the solar $p_{\text {tot }}$ ?

$R_{\text {Earth }}=6370 \mathrm{~km}$

The photons generated by the nuclear fusion in the core of the Sun are gamma ray photons, however by the time they reach the surface of the Sun they are mostly visible light. This is because the photons spend a very long time being absorbed and then re-emitted by particles in the plasma, and initially this is via Compton scattering, which lowers the energy of the photon with each emission, and then later via Thomson scattering, which does not affect the photon energy but does mean it continues to bounce around in the star.

Throughout all the scattering, the photons follow a "random walk", meaning that after $N$ steps of average step size $\ell$ (known as the mean free path), the final displacement, $d$, is

$$ d=\ell \sqrt{N} $$

d) [4] The energy density of blackbody photons is $\frac{E}{V}=\frac{4 \sigma}{c} T^{4}$ and the mean temperature of the Sun is $4.73 \times 10^{6} \mathrm{~K}$.

(i) By considering the total energy stored as EM radiation in the Sun and the current luminosity, estimate the time it takes for a photon to diffuse from the core to the surface. Give your answer in years.

(ii) Hence find the mean free path, $\ell$, and the number of interactions, $N$.

Unlike photons, neutrinos produced in the fusion reactions hardly interact at all with the dense, opaque plasma in the core. Consequently, they can tell us about the energy production happening now, which can be compared to the observed luminosity and alert us to any changes over the photon diffusion timescale. In 2018, the Borexino experiment detected, for the first time, neutrinos from the proton-proton I branch from the Sun. This branch is responsible for $92 \%$ of the Sun's energy output, and each neutrino corresponds to 13.1 MeV of energy provided to the star by fusion reactions. The measured neutrino flux at the position of the Earth was $(6.1 \pm 0.5) \times 10^{10}$ neutrinos $\mathrm{cm}^{-2} \mathrm{~s}^{-1}$.

e) [4] Use the photon output luminosity to predict the neutrino flux at Earth and compare it to the measured value. Comment on the implications of your answer.

a) [16] The Hope probe and its launch from the Earth are considered first, followed by the transfer to Mars.

Figure 10: The Hope probe and the gravitational field of the Earth.

1. (i) At what distance $r$ from the centre of the Earth, with $r > R_\mathrm{E}$, does the gravitational field strength reduce to $1\%$ of its value at the surface? Express this as a fraction of the radius of the Earth.
2. (ii) The Hope probe was launched from Tanegashima Space Centre in Japan, which lies on $30^\circ$ latitude above the equatorial plane. Calculate the tangential speed of the space centre.
3. (iii) Estimate the escape speed of a satellite that is launched from the surface of the Earth, with no kinetic energy contribution from Earth's rotation, so that it completely escapes the Earth's gravitational field.
4. (iv) What is the percentage reduction in energy for a satellite to escape the Earth's gravitational field when
   1. i. launched from Tanegashima Space Centre in Japan,
   2. ii. it is already in a low Earth orbit (close to the surface of the Earth)?

The trajectory of the space probe between Earth and Mars can be represented as part of an elliptical orbit about the Sun, with the Sun at one focus of the ellipse, shown schematically in Fig. 11. Thus the space probe cruises without the use of propulsion in an elliptical orbit about the Sun to reach its destination. For an elliptical orbit, $r v \sin\theta = \text{constant}$, where $r$ is the displacement of the body from one focus, $v$ is the velocity vector, and $\theta$ is the angle between them.

Figure 11: The transfer ellipse about the Sun, with point P near Earth and point A near Mars.

1. (v) By considering the influence of the Sun only, use the idea of total energy to produce expressions for the velocity of the probe at point P (near Earth), $v_\mathrm{P}$, and at point A (near Mars), $v_\mathrm{A}$, in terms of the gravitational constant $G$, the Solar mass $M_\mathrm{S}$, the radius of orbit of the Earth $R_{\mathrm{E}_o}$, and the radius of orbit of Mars $R_{\mathrm{M}_o}$.
2. (vi) Determine the difference between $v_\mathrm{P}$ and the Earth's orbital velocity $v_\mathrm{E}$, and the difference between $v_\mathrm{A}$ and Mars' orbital velocity $v_\mathrm{M}$. Thus calculate the total change in tangential velocity needed at point A to allow the probe to cruise to its meeting point with Mars; assume the probe already has sufficient kinetic energy to escape the gravitational potential well of the Earth.
3. (vii) Calculate the speed of the probe on close approach to Mars.
4. (viii) On the approach to Mars, the speed of the probe must be slowed to match the orbital speed of Mars to allow it to be captured in an orbit around the planet. This deceleration is achieved by using six thrusters expending half of the fuel over a $27\,\mathrm{minute}$ period. Assume the decelerating force and the rate of fuel consumption, $k$, are constants. For the period of time in which the thrusters are in operation:
   1. i. Sketch a graph of the variation of acceleration against time; append appropriate values to the axes.
   2. ii. Produce an expression for the change in velocity with time $v(t)$ in terms of the total decelerating force $F$, the initial probe mass $m_0$, the rate of fuel consumption $k$, initial velocity $v_0$ and time $t$.
   3. iii. Calculate the force provided by each thruster.

b) [9] The iridium satellite telephone system uses a constellation of 66 satellites orbiting at an altitude of $780\,\mathrm{km}$ above the surface of the Earth.

1. (i) Calculate the horizon radius of the area covered by any one satellite (this is the radius of a flat circle within the horizon, i.e. the radius of one of the small circles illustrated in Fig. 14).

   

   Figure 14: The iridium constellation; each satellite covers a flat circle within its horizon.

2. (ii) In practice, the effective radius is approximately $2350\,\mathrm{km}$. The power distributed by the satellite transmitter to this area is about $200\,\mathrm{W}$. Calculate the intensity of the signal at the Earth's surface.
3. (iii) The movement of the satellites relative to the Earth's surface introduces the Doppler Effect. If the wavelength of the signal from the satellite is $\lambda$, the speed of the satellite $v$, and the speed of light $c$, produce an expression for the shifted wavelength $\lambda'$ observed by the receiver as the satellite recedes.
4. (iv) Using a suitable approximation, provide an expression for the apparent shift in frequency, $\Delta f$.
5. (v) Now consider the satellite moving above the receiver on the Earth's surface. For the effective radius of coverage in (ii) calculate $f_\mathrm{min}$ and $f_\mathrm{max}$ as observed, given the transmission frequency of $1620\,\mathrm{MHz}$, and thus state the bandwidth that must be set aside to manage Doppler shift.

[25 marks]

a) [5] Consider a planet of mass $m_{\mathrm{p}}$, orbiting a star of mass $M_{\mathrm{s}}$ in a circular orbit of radius $r$ which takes a time $T$ to complete one orbit.

1. (i) By balancing centripetal and gravitational forces, show that the orbital speed $v \propto r^{-1/2}$.
2. (ii) Hence, show that the total energy of the planet (KE + GPE), $E_{\mathrm{total}} \propto r^{-1}$.
3. (iii) By considering the speed of a planet in its orbit around the star in terms of $r$ and $T$, show that the relationship between orbital period and orbital radius is $T^{2} \propto r^{3}$. (This is known as Kepler's Third Law.)
4. (iv) Given the Earth has an orbital radius $r_{\mathrm{E}} = 1.50 \times 10^{11}\ \mathrm{m}$ and orbital period $T_{\mathrm{E}} = 365$ days, calculate the mass of the Sun in kg.

Earth's orbital properties can be given in different units, namely $T_{\mathrm{E}} = 1$ year, and $r_{\mathrm{E}} = 1$ au. (An “au” stands for astronomical unit.)

1. (v) Using your result from part (iii), if you use the Earth values of $r_{\mathrm{E}} = 1\ \mathrm{au}$ and $T_{\mathrm{E}} = 1\ \mathrm{year}$ orbiting a star of mass $M_{\mathrm{s}} = 1\ M_{\mathrm{Sun}}$, what is the value (and units) of $G$ in just these given quantities and units?

For the following calculations, you may choose to work in standard SI units like metres, kilograms and seconds (and so use your value of $M_{\mathrm{Sun}}$ from part (iv)), or in au, solar masses and years, as that can often greatly simplify the formulae (and so use your value of $G$ from part (v)).

b) [8] Disaster strikes and the Sun disappears! This causes all the planets to be perturbed from their orbits. In this question we will just look at Venus and Earth. Assume that the loss of the Sun's gravity was felt instantaneously by both planets, and that both are travelling in coplanar circular orbits.

Venus has a mass of $0.815\,M_{\mathrm{Earth}}$, where $M_{\mathrm{Earth}} = 5.97 \times 10^{24}\ \mathrm{kg}$, and an orbital radius of $0.723\ \mathrm{au}$.

Venus and Earth happened to be at such a point in their orbit when the Sun disappeared that they are on a collision course.

1. (i) Calculate the orbital speeds of Earth and Venus, $v_{\mathrm{E}}$ and $v_{\mathrm{V}}$.
2. (ii) Determine the time, $t$, until they crash. Give your answer in days.
3. (iii) Show that the distance $d$ from the Sun's original location when they crash is $\sim 1.5\ \mathrm{au}$.
4. (iv) Show that the angle between their velocity vectors as they approach is $\sim 13^{\circ}$.
5. (v) Find the Earth--Sun--Venus angle at the moment the Sun disappeared in order for this crash to occur.

c) [7] The collision between them is inelastic, causing them to stick together and make a new, heavier planet: “Venearth”.

1. (i) Show that the speed of Venearth, $v$, just after the collision is $\sim 6.7\ \mathrm{au\ year^{-1}}$.
2. (ii) Determine the angle between Venearth's velocity vector and Earth's initial velocity vector before the collision.
3. (iii) Calculate the percentage of kinetic energy lost in the collision.

d) [5] At the moment of the collision, a new star magically appears in the location that the Sun used to be. Again, assume its gravitational effects are felt instantaneously throughout the Solar System.

1. (i) What is the minimum mass of this new star necessary for Venearth to still be bound within the Solar System? Give your answer in units of $M_{\mathrm{Sun}}$.

The new planet Venearth will follow an elliptical orbit whose orbital period will be equal to a circular orbit with the same total energy as Venearth.

1. (ii) If in fact, the new star has a mass equal to $M_{\mathrm{Sun}}$, what will be the orbital period of Venearth? Give your answer in years.