A comet of mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector

Problem

A comet of mass

m

, with

m \ll M_\odot

so that

\mu \approx m

and

k = G_{\rm N}M_\odot m

, moves under the Sun's gravity

V(r) = -k/r

. At perihelion its distance is

r_{\rm p} = 8.0\times10^{10}\,\mathrm{m}

and its speed is

v_{\rm p} = 54.0\,\mathrm{km\,s^{-1}}

. Take

G_{\rm N}M_\odot = 1.327\times10^{20}\,\mathrm{m^3\,s^{-2}}

. ((a)) Starting from

\mathbf{A}_{\rm LRL} = \mathbf{p}\times\mathbf{L} - \mu k\hat{\mathbf{r}}

and using

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\times\mathbf{b})\cdot\mathbf{c}

, derive the polar orbit equation

r = p_{\rm orb}/(1+e\cos\varphi)

with

e = A/\mu k

and

p_{\rm orb} = \ell^2/\mu k

, where

A = |\mathbf{A}_{\rm LRL}|

, without evaluating any integral. ((b)) At perihelion,

\mathbf{p}\times\mathbf{L}

is parallel to

\hat{\mathbf{r}}

and the velocity is purely tangential. Use this and the given data to find the comet's eccentricity

e

. ((c)) For this one-body reduced problem define

D = \mathbf{p}\cdot\mathbf{r}

. Derive

\dot D = 2T + \mathbf{F}\cdot\mathbf{r}

and hence, from a long-time average,

2\langle T\rangle = -\langle V\rangle

, stating precisely the condition on the motion needed for

\langle \dot D\rangle

to vanish. Verify that the comet's orbit satisfies that condition.

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