A1 Dynamical Astronomy
The gravitational force from a
spherically symmetric body
mass of the ring as
s
R
θ
dMring D .R/ dVring
φ
r
R0
D .R/2R sin R d dR
m
D 2R2 .R/ sin dR d:
From the diagram we can make the substitution
M
dMring
Figure 1: The geometry used in the proof.
cos D
r
R cos :
s
By the cosine rule, s is
s D .r 2
2rR cos C R2 /1=2 :
This proof is not examinable, and for the enSubstituting into the expression for dFring , we must
thusiast only! It explains why a spherically symmet-
now sum over all rings in a shell (i.e., integrate from
ric object, such as a star or planet, can be considered as
0 to ) and over all shells in the sphere (i.e., integrate
a point mass in gravitational calculations, even when
R from 0 to R0 ) to get the total gravitational force:
it is very close. Isaac Newton worried about this proof
Z R0Z .r R cos /.R/2R2 sin for years, so if it appears difficult at first sight you are
d dR
F D Gm
in good company. We will use the notation and aps3
0
0
Z R0Z proach of Carroll & Ostlie Modern Astrophysics.
rR2 .R/ sin D 2Gm
d dR
2
2
2rR cos /3=2
0
0 .r C R
Z R0Z The gravitational force exerted by a spherically
R3 .R/ sin cos symmetric object of mass M on a point mass, m, can
2Gm
d dR:
2
2
2rR cos /3=2
0
0 .r C R
be calculated by considering the sum of the contributions from rings of matter in M , centred along the line
of length r connecting the point mass with the centre
of the extended object (Figure 1). All points on a particular ring are the same distance from the point mass,
and therefore exert the same force on it. In addition,
the resultant force from the ring must be along r, as
the mass is symmetric about it. Hence we need only
consider the component of gravity from each part of
the ring along the line connecting the mass centres.
The integration over can be carried out by making
the substitution u D s 2 D r 2 C R2 2rR cos , so
that sin d D du=.2rR/. After integrating over u,
the force equation now becomes
Z
Gm R0
4R2 .R/ dR:
F D 2
r
0
The integrand of this expression is simply the mass,
dMshell , of a single shell of matter, thickness dR and
volume 4R2 dR, giving a force contribution of
Let the radius of the extended mass be R0 and the
Gm dMshell
dF
D
:
shell
mass of the ring under consideration be dMring . The
r2
gravitational force exerted by this ring on the point i.e., the gravitational attraction from the shell is as if
mass m is
all its mass was concentrated at its centre. The total
expression for F is just the sum of these:
m dMring
Mm
dFring D G
cos ;
F DG 2 ;
s2
r
so the attraction of the whole sphere is as if all its mass
where s is the distance between the rim of the ring were concentrated at its centre.
and m, and the angle it makes with the central line.
GW
Assuming that the mass density of the extended object,
2016
, is a function of radius only, we can calculate the