A free-falling capsule is at a distance from the centre of a brown dwarf of mass

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory

Relativity and Gravitation, Ch Tidal forces and the Newtonian tidal field

Problem

A free-falling capsule is at a distance

r = 6.00\times10^{7}\ \mathrm{m}

from the centre of a brown dwarf of mass

M = 5.00\times10^{28}\ \mathrm{kg}

, treated as a point mass; no forces other than gravity act. Inside the capsule the

x

axis points along the outward radius and the

y

axis is transverse. A free test particle of negligible mass sits at separation

\vec\xi

from the capsule's centre, of magnitude

\xi

, in the

x

–

y

plane and making an angle

\theta

with the

+x

axis (so

\xi_x = \xi\cos\theta

\xi_y = \xi\sin\theta

). Work to first order in

\xi/r

. ((a)) From the point-mass tidal field in these axes, derive the two components of the relative acceleration

\Delta\vec a

and show that its magnitude is

\dfrac{GM\xi}{r^{3}}\sqrt{1+3\cos^{2}\theta}

. ((b)) Show that the sum of the three diagonal entries of the tidal field vanishes here, and state in one sentence what that means for a small ball of dust released in the capsule. ((c)) For

\xi = 3.00\ \mathrm{m}

and

\theta = 30.0^{\circ}

, evaluate

\Delta a_x

\Delta a_y

, the magnitude of

\Delta\vec a

, and the angle between

\Delta\vec a

and the outward radius.

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