1.2 Orbits and Origins
Edmund Halley’s successful prediction showed that his comet is gravitationally
bound to the Sun. In the simple assumption of the restricted two-body problem,
the orbit of an inactive test particle around the Sun can be described as in Table 1.1.
For closed orbits (where the orbital eccentricity, e x , is less than 1), the Sun is at one
of the foci of an ellipse with the orbital period, P, being given by Kepler’s third law,
as seen in Eq. (1).
The eccentricity can be calculated from the periapsis and apoapsis distances, r p
and r a respectively, through
e x ¼
r a À r p
r a þ r p
ð1:4Þ
that are in turn related to the semi-major axis through
r p ¼ a s 1 À e x
ð
Þ
and r a ¼ a s 1 þ e x
ð
Þ
ð1:5Þ
The radial distance from the Sun, r h , at a given time can be computed from
r h ¼
a s 1 À e x
2
ð
Þ
1 þ e x cos f
ð1:6Þ
where f is the true anomaly. Even in the simple case of the two-body problem, the
introduction of the time dependence leads to an equation that can only be solved
numerically. The simplest way is to use Newton’s false-root method to iterate on the
value of the eccentric anomaly, E a , which arises from the definition
r h ¼ a s 1 À e x cos E a
ð
Þ
ð 1:7Þ
and is related to the mean anomaly, M a , through the equation
M a ¼ ñ t À T PERI
ð
Þ¼E a À e x sin E a
ð1:8Þ
where ñ is the mean motion of the object about the fixed mass in units of, for
example, [rad s
À1 ] and time is relative to the last time of periapsis. The true anomaly
is related to E a through the equation
Table 1.1 Orbit types
described through their
eccentricity
Orbit type
Eccentricity, e x
Circular
0.0
Elliptical
0.0 < e x < 1.0
Parabolic
1.0
Hyperbolic
e x > 1.0
6
1 Light Curves, Orbits, and Reservoirs
Edmund Halley’s successful prediction showed that his comet is gravitationally
bound to the Sun. In the simple assumption of the restricted two-body problem,
the orbit of an inactive test particle around the Sun can be described as in Table 1.1.
For closed orbits (where the orbital eccentricity, e x , is less than 1), the Sun is at one
of the foci of an ellipse with the orbital period, P, being given by Kepler’s third law,
as seen in Eq. (1).
The eccentricity can be calculated from the periapsis and apoapsis distances, r p
and r a respectively, through
e x ¼
r a À r p
r a þ r p
ð1:4Þ
that are in turn related to the semi-major axis through
r p ¼ a s 1 À e x
ð
Þ
and r a ¼ a s 1 þ e x
ð
Þ
ð1:5Þ
The radial distance from the Sun, r h , at a given time can be computed from
r h ¼
a s 1 À e x
2
ð
Þ
1 þ e x cos f
ð1:6Þ
where f is the true anomaly. Even in the simple case of the two-body problem, the
introduction of the time dependence leads to an equation that can only be solved
numerically. The simplest way is to use Newton’s false-root method to iterate on the
value of the eccentric anomaly, E a , which arises from the definition
r h ¼ a s 1 À e x cos E a
ð
Þ
ð 1:7Þ
and is related to the mean anomaly, M a , through the equation
M a ¼ ñ t À T PERI
ð
Þ¼E a À e x sin E a
ð1:8Þ
where ñ is the mean motion of the object about the fixed mass in units of, for
example, [rad s
À1 ] and time is relative to the last time of periapsis. The true anomaly
is related to E a through the equation
Table 1.1 Orbit types
described through their
eccentricity
Orbit type
Eccentricity, e x
Circular
0.0
Elliptical
0.0 < e x < 1.0
Parabolic
1.0
Hyperbolic
e x > 1.0
6
1 Light Curves, Orbits, and Reservoirs