1.3 Non-gravitational Forces
Comets can be considered massless for Solar System dynamics calculations and their
motion, incorporating perturbation by the planets, can be calculated directly through
numerical integration. However, additional complexity arises from two other
sources.
First, comets on eccentric orbits with small semi-major axes can experience
non-negligible relativistic effects resulting in radial acceleration towards the Sun
(Shahid-Saless and Yeomans 1994). The perihelion precession of the orbit is
probably the most measurable effect and can be determined through the equation
Δω ¼
0:0384
a s 1 À e x
2
ð
Þ
ð1:20Þ
where Δω is the precession of the argument of perihelion is arcseconds per comet
orbit about the Sun and a s is again the semi-major axis in units of [AU].
The second, and more physically interesting, complexity arises from the
outgassing and mass loss from the cometary surface which provide additional forces
(non-gravitational forces or NGFs) on the nucleus that are sufficient to modify both
its orbit and its rotational characteristics. The equation of motion, ignoring the
relativistic effects (see Beutler (2005) for how these can be included) and assuming
the Sun is at the barycentre of Solar System, can be written as
d
2
r h
dt 2 ¼ ÀGM ⨀
r h
r
3
h
þ
dR p
dr h
þ A 1 g r h
ð Þ r h
z}|{ þ A 2 g r h
ð Þ T
z}|{ þ A 3 g r h
ð Þ n
z}|{ ð1:21Þ
where r h is the heliocentric distance with the bold face indicating that it is being used
vectorially in this equation rather than the usual scalar magnitude. Here,
g r h
ð Þ ¼ c NGF
r h
r 0
Àm NGF
1 þ
r h
r 0
n NGF
Àk NGF
:
ð1:22Þ
In the second term on the right hand side, R p is a planetary disturbing function
describing the influence of the other Solar System bodies on the comet’s motion. The
NGFs are represented by the last three terms and the coefficients, A 1,2,3 . This semiempirical approach was introduced in the 1970s by Marsden and is conceptually
straightforward. The outgassing from the nucleus produces accelerations. A radial
acceleration arises because outgassing is predominately from the illuminated surface
which leads to a reaction force in the anti-Sun direction. Terms arise in the other two
directions (transverse, T
z}|{
, and normal, n
z}|{ , to the orbital plane) as a result of the
rotation of the nucleus and the obliquity (the angle between the rotation axis and the
normal to the orbital plane) combined with the thermal inertia of the nucleus and
latitudinal variations. The form of g(r h ) is intended to account for the variation in
outgassing strength with heliocentric distance. r 0 is the heliocentric distance inside
1.3 Non-gravitational Forces
23
Comets can be considered massless for Solar System dynamics calculations and their
motion, incorporating perturbation by the planets, can be calculated directly through
numerical integration. However, additional complexity arises from two other
sources.
First, comets on eccentric orbits with small semi-major axes can experience
non-negligible relativistic effects resulting in radial acceleration towards the Sun
(Shahid-Saless and Yeomans 1994). The perihelion precession of the orbit is
probably the most measurable effect and can be determined through the equation
Δω ¼
0:0384
a s 1 À e x
2
ð
Þ
ð1:20Þ
where Δω is the precession of the argument of perihelion is arcseconds per comet
orbit about the Sun and a s is again the semi-major axis in units of [AU].
The second, and more physically interesting, complexity arises from the
outgassing and mass loss from the cometary surface which provide additional forces
(non-gravitational forces or NGFs) on the nucleus that are sufficient to modify both
its orbit and its rotational characteristics. The equation of motion, ignoring the
relativistic effects (see Beutler (2005) for how these can be included) and assuming
the Sun is at the barycentre of Solar System, can be written as
d
2
r h
dt 2 ¼ ÀGM ⨀
r h
r
3
h
þ
dR p
dr h
þ A 1 g r h
ð Þ r h
z}|{ þ A 2 g r h
ð Þ T
z}|{ þ A 3 g r h
ð Þ n
z}|{ ð1:21Þ
where r h is the heliocentric distance with the bold face indicating that it is being used
vectorially in this equation rather than the usual scalar magnitude. Here,
g r h
ð Þ ¼ c NGF
r h
r 0
Àm NGF
1 þ
r h
r 0
n NGF
Àk NGF
:
ð1:22Þ
In the second term on the right hand side, R p is a planetary disturbing function
describing the influence of the other Solar System bodies on the comet’s motion. The
NGFs are represented by the last three terms and the coefficients, A 1,2,3 . This semiempirical approach was introduced in the 1970s by Marsden and is conceptually
straightforward. The outgassing from the nucleus produces accelerations. A radial
acceleration arises because outgassing is predominately from the illuminated surface
which leads to a reaction force in the anti-Sun direction. Terms arise in the other two
directions (transverse, T
z}|{
, and normal, n
z}|{ , to the orbital plane) as a result of the
rotation of the nucleus and the obliquity (the angle between the rotation axis and the
normal to the orbital plane) combined with the thermal inertia of the nucleus and
latitudinal variations. The form of g(r h ) is intended to account for the variation in
outgassing strength with heliocentric distance. r 0 is the heliocentric distance inside
1.3 Non-gravitational Forces
23