Equation of the trajectory of a typical space object around any planet, in polar coordinates (r, θ) (i.e. a general conic section geometry), is given as follows. (h is angular momentum, μ is gravitational parameter, e is eccentricity, r is radial distance from the planet center, θ, is angle between vectors \(\vec{e}\) and \(\vec{r}\).
GATE AE - 2018,
Answer: C
The trajectory equation is \(r = \dfrac{(h^{2}/\mu)}{1 + e\cos\theta}\).
Solving the two-body equation of motion gives the conic \(r = p/(1 + e\cos\theta)\), with the semi-latus rectum \(p = h^{2}/\mu\) and \(\theta\) (the true anomaly) measured from the eccentricity vector, which points to periapsis. At \(\theta = 0\) the denominator is largest, giving the closest point, \(r = p/(1 + e)\), as it must; \(e = 0\) gives a circle, \(0 \lt e \lt 1\) an ellipse, \(e = 1\) a parabola and \(e \gt 1\) a hyperbola.
The form with \(1 - e\cos\theta\) would put periapsis at \(\theta = 180^{\circ}\), which is what you get by measuring the angle from the opposite direction, not from \(\vec{e}\). The forms with \(e \pm \cos\theta\) are not conics in this form: with \(e = 0\) they do not even give a circle.
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