This question concerns elliptical orbits.
(a) The equation of an ellipse of eccentricity ε = √(1 - b²/a²) and centre (x₀,y₀) in Cartesian co-ordinates is
(x-x₀)²/a² + (y-y₀)²/b² = 1
where a is the semi-major axis (i.e. along its longest diameter) and b is the semi-minor axis (i.e. along its shortest diameter).
Show that the polar equation
r = α/(1+ε cos θ)
describes an ellipse in the x-y plane with the following features:
centre (x₀,y₀) = (-αε/(1-ε²), 0)
and
semi-major axis a = α/(1-ε²)
and
semi-minor axis b = α/√(1-ε²)
Sketch the ellipse, noting the aforementioned features as well as the locations of the foci which are at a distance f = √(a²-b²) either side of the centre along the semi-major axis. What constraints are there on the eccentricity? You may find it useful to consider the relationship between cartesian coordinates and polar coordinates: x = r cos θ, y = r sin θ, noting that x² + y² = r².
(b) A Sun-planet system has total energy E and angular momentum L. Mₛ is the mass of the Sun and mₚ the mass of the planet.
(i) Angular momentum is like a 'moment' of momentum. So, for an object undergoing rotational motion, its angular momentum at a distance r from an axis of rotation with a linear speed v (the component perpendicular to r) at that point, will have angular momentum equal to mvr. Furthermore, Newton's second law applies to angular momentum just as it does to momentum if you replace 'resultant force' with 'resultant moment/torque' and 'momentum' with 'angular momentum'.
Why must L be a constant? Show that the rate at which an imaginary line drawn between the orbital axis and the planet sweeps out area is given by
dA/dt = L/(2mₚ)
What relevance does this have for Kepler's second law?
(ii) Show that
E = (1/2)mₚṙ² + L²/(2mₚr²) - k/r
where ṙ = dr/dt and k is a constant to be determined.
If time, t, is re-parametrised in terms of angle, θ, the solution to this differential equation is r = α/(1+ε cos θ) with α = L²/μk and ε² = 1 + 2EL²/μk².
(iii) Find a formula for the time period of the planet in terms of the semi-major axis of its orbit. The total area of an ellipse is πab.
(c) A satellite in low earth orbit at an altitude of 320 km is to be transferred to a geosynchronous orbit using an elliptical orbit which is tangential to both the initial and final orbits at its periapsis and apoapsis (the smallest and largest distances from a focus) respectively. How elliptical is the transfer orbit, what is the total velocity change, Δv, required to achieve the transfer and how long will it take? State any assumptions you make.
(d) A significant part of a manned mission to Mars involves transfer from a low earth orbit of altitude about 300 km to a low orbit around Mars of altitude about 250 km. What is the Δv budget needed to make the trip using a similar transfer to that in (c), and what is the shortest length of time for a return trip using this method? What are your assumptions and how good are they in this context? To three significant figures, the eccentricities of Earth and Mars are 0.0167 and 0.0934 respectively, while Mars takes 1.88 Earth years to orbit the Sun.