7 Einstein's Modification of the Orbit Equation
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4.8 Universality of Newton’s Law
93
increases u by a very small constant quantity. The third term,
˛ 2
e cos 2.Â
6p2
Â0 /
(4.75)
is very small and periodic. However, the term
˛
eÂ sin.Â
p2
Â0 /
(4.76)
is periodic, and steadily increasing in amplitude as Â increases. So this is bound to
have some effect with increased time. Considering only the observable effects,
uD
1
Œ1 C e cos.Â
p
Â0 / C
˛e
Â sin.Â
p2
Â0 /
(4.77)
Let kÂ D ˛Â=p and neglect ˛ 2 , then Eq. (4.77) can be written as
uD
1
f1 C eŒcos.Â
p
Â0 / C kÂ sin.Â
Â0 /g
1
Œ1Ce cos.Â Â0 kÂ/
p
(4.78)
Â0 C kÂ is the angular coordinate of the perihelion. So the planet is moving in an
ellipse with a moving line of apsides, which is a slowly rotating ellipse. The angular
change of the line of apsides is ! D 2 ˛=p per period. Substituting values for
the planets we have
Mercury: ! D 43:03 arcsec per century.
Venus: ! D 8:63 arcsec per century.
Earth: ! D 3:84 arcsec per century.
For Mercury this small effect was observable and remained unexplained until
Einstein’s relativity. It was fudged in Newcomb’s planetary theories and called an
empirical term (Danby 1962, pp. 66–67).
4.8 Universality of Newton’s Law
Newton’s law follows from Kepler’s first two laws of planetary motion; therefore,
any two bodies traveling around each other according to Kepler’s first two laws
are subject to Newton’s law. When two stars are observed moving around each
other, these are called visual binaries. In other cases of binaries, the two stars have
different spectra, continuously shifting spectra, or eclipse each other. For visual
binaries, when the fainter star is plotted with respect to the brighter, the orbits are
ellipses, and the law of areas is followed. However, a projection of the true orbit is
being plotted, so the brighter star is not at the focus. It is unlikely in any case that
the orbital plane is perpendicular to the line of sight.
94
4 Central Force Motion
An ellipse always projects into another ellipse. Since the apparent orbit as plotted
is an ellipse, the true orbit must be an ellipse. The law of areas of the apparent
orbit will hold for the true orbit, because the law of areas depends on ratios, which
are not affected by projections. For the true orbit, the theory of central forces and
conservative fields applies, and an ellipse with the brighter star at the focus can be
found. It can be shown that for visual binaries. elliptical motion requires the center
of attraction to be at the focus of the ellipse.
So Newton’s law is the only plausible law governing Keplerian motion within and
exterior to the solar system. Also, this law has explained deviations from Keplerian
motion, when the relativity effect is included, and has led to the correction of
Kepler’s third law (Danby 1962, pp. 73–76).
References
McCuskey, S.W.: Introduction to Celestial Mechanics. Addison-Wesley, Reading (1963)
Danby, J.: Fundamentals of Celestial Mechanics. The Macmillan Company, New York (1962)
Chapter 5
The Two-Body Problem
5.1 Introduction
Assume that the masses are spherically symmetrical and homogeneous in concentric
layers. So they attract one another as if the mass were concentrated at spherical
centers, i.e. gravitationally they act like two mass particles separated by the distance
between the centers. The two masses are assumed to be isolated from other masses,
so the only force acting is the inverse square force of their mutual attractions along
the line joining the centers. In astronomical applications, the distance between
centers is large, compared to the diameters of the spheres. This is not true for
artificial satellites.
The dynamics of the motion of two masses presents two problems for celestial
mechanics and astrodynamics:
1. Given the position and velocity, or three positions, of a mass as a function of time,
find the elements of the orbit. This is the computation of orbits to be considered
in Chap. 6.
2. Given the orbital elements, or parameters defining the orbital motion, find
the position in space of the mass at a given time. This we will take up now (the
first problem is by far the more difficult), with the first step being defining the
classical orbital elements.
5.2 Classical Orbital Elements
Assume that a mass m is rotating counterclockwise in orbit when viewed from the
planet’s north pole. If the orbital plane and the fundamental plane of some reference
frame intersect, then we define two points of interest on the line of intersection,
© Springer-Verlag Berlin Heidelberg 2016
P. Gurfil, P.K. Seidelmann, Celestial Mechanics and Astrodynamics: Theory
and Practice, Astrophysics and Space Science Library 436,
DOI 10.1007/978-3-662-50370-6_5
95
96
5 The Two-Body Problem
e(eccentricity vector)
zˆ
uˆ A
Periapsis
f
i
Orbital plane
r
Descending
Node
܁
ω
Ω
Reference plane
yˆ
i
܀
Ascending
Node
Reference frame
xˆ (vernal equinox)
ˆl (line of nodes)
Fig. 5.1 Definitions of the right ascension of the ascending node,
and the inclination, i. Also shown is the true anomaly, f
, the argument of periapsis, !,
as shown in Fig. 5.1: The first is the ascending node, denoted by . This point
marks the location on the line of intersection when moving eastward; the second is
the descending node, denoted by . This point marks the location on the line of
intersection when moving westward. The line connecting to is called the line
of nodes (LON); we will use the notation Ol to denote a unit vector that lies along
the LON.
We now define three angles that determine the orientation of the orbital plane
with respect to the reference frame: , the right ascension of the ascending node
(RAAN), also referred to as the longitude of the ascending node, an angle measured
from the vernal equinox (see Sect. 3.1) to the LON; !, the argument of periapsis,
which is an angle measured from the LON to the eccentricity vector (see Sect. 4.6.1);
and i, the inclination, an angle measured from the z axis of the reference frame, zO ,
to the vector normal to the orbital plane, uO A . These angles are shown in Fig. 5.1. It
is convenient to express the position vector in a perifocal coordinate system. This
coordinate system is centered at the attraction center. The fundamental plane is the
orbital plane. The unit vector PO is directed from the center to the periapsis (recall
O is normal to the fundamental plane, positive in the direction of the
Sect. 4.6.1), R
O is pointed toward the point where the true
orbital angular momentum vector, and Q
ı
anomaly is 90 , thus completing the right-hand Cartesian triad. Using the definition
of the true anomaly (Sect. 4.6.1), we can write the position vector in the perifocal
5.2 Classical Orbital Elements
97
frame as
rp D r Œcos f ; sin f ; 0T
(5.1)
where, as before, r D p=.1 C e cos f /.
We can transform from the orbital frame to the reference frame using three
consecutive clockwise rotations: a rotation about uO A by 0 Ä ! Ä 2 , mapping
the eccentricity vector, eO , onto the LON, Ol; a rotation about Ol by 0 Ä i Ä , mapping
uO A onto zO ; and a rotation about zO by 0 Ä Ä 2 , mapping Ol onto xO .
The composite rotation, transforming any vector in the orbital frame into the
inertial frame is given by
2
c c! s s! ci
T D 4 s c! C c s ! ci
s! si
3
c s! s c! ci s si
s s! C c c! ci c si 5
c! si
ci
(5.2)
where we used the compact notation cx D cos x, sx D sin x.
Transforming into inertial reference coordinates using Eqs. (5.1) and (5.2), we
obtain the position vector
3
2
cf C! c
ci sf C! s
p
4 ci c sf C! C cf C! s 5
rD
1 C e cos f
si sf C!
(5.3)
The true anomaly f depends on time and on the epoch of observation, T. Thus, the
inertial position and velocity depend on time t and the classical orbital elements,
given by
fa; e; i;
; !; Tg
(5.4)
5.2.1 Osculating Orbital Elements
In the two-body problem, the orbital elements are constant; f is time-varying.
However, in the presence of perturbations and/or thrust forces, the orbital elements
may become time-varying, and are referred to as osculating orbital elements.
In general, elliptical motion constitutes a correct approximation to the real
motion observed in the solar system. Thus, for example if, starting from an instant
t0 , all the perturbing forces were neglected, the movement of a body would become
exactly elliptical. It would represent the real movement quite well for a certain time,
even though strictly speaking, it would not be identical with the real movement as
regards position and velocity, except at the instant t0 . The elements of an ellipse that
would be followed by a body after a specific time t are thus said to be osculating, or
98
5 The Two-Body Problem
instantaneous, if starting from this instant, all the forces with the exception of the
central force were to disappear. The elements of such an unperturbed orbit can be
defined at any instant; they correspond to the elliptical orbit followed by a moving
body, which would have at the given instant the same position and velocity as the
real body. As in fact the real orbit is simply tangential to the osculating orbit, at an
instant tCıt the osculating orbit will be different, with different osculating elements.
It follows that the osculating elements in perturbed motion are no longer constant,
but are functions of time.
Osculating elements can be used to describe the perturbed motion of a body.
They possess the advantage of having a precise and simple geometrical significance
while having small variations.
The coordinates and velocity components of perturbed motion at an instant t are
those which would be obtained at this instant t, assuming that the orbit is elliptical,
from elements equal to the osculating elements at the same time t.
5.2.2 Nonsingular Orbital Elements
While the angles ; i; ! may become degenerate is some cases (for instance, !
is undefined for circular orbits; both ! and are undefined for equatorial orbits),
the position and velocity vectors are always well-defined. However, occasionally
alternative orbital elements are used to alleviate these deficiencies. These alternative elements are collectively referred to as nonsingular orbital elements, see
Sect. 10.12. A thorough survey of these elements was performed by Hintz (2008).
5.3 Motion of the Center of Mass
Let an origin, O, define an inertial system with Newton’s laws of motion. The
positions of two masses are given by vectors r1 and r2 , and R is the vector to the
center of mass of the pair, C. r is the position vector of m2 relative to m1 , as shown
in Fig. 5.2.
From Newton’s law of gravitation, the force on m1 due to m2 is r12 k2 m1 m2 uO r and
that on m2 due to m1 is r12 k2 m1 m2 uO r . uO r is a unit vector in the direction of r, and
k2 is the constant of gravitation. The reason and significance of the notation k2 will
become apparent later. The equations of motion are
m1 rR 1 D
m2 rR 2 D
k 2 m1 m2
r
r3
k 2 m1 m2
r
r3
(5.5)
(5.6)
5.4 Relative Motion
99
1
Fig. 5.2 Motion of the center
of mass
1
2
2
Adding Eqs. (5.5) and (5.6) and integrating twice, we have
m1 r 1 C m2 r 2 D c 1 t C c 2
(5.7)
where c1 and c2 are vector constants. The left side of Eq. (5.7) is MR, by the
definition of the center of mass, with M D m1 C m2 . Thus,
RD
c2
c1 t
C
M
M
(5.8)
so the center of mass moves uniformly in a straight line in space.
This is in agreement with previous results and what one would expect, since
there is no external force acting on this system. This result is applicable to double
star observations, but of little interest otherwise (McCuskey 1963, pp. 32–33).
5.4 Relative Motion
The motions of m1 and m2 relative to the center of mass can be derived as follows.
Let r1 D R C r01 and r2 D R C r02 , where r01 and r02 denote position vectors to m1
and m2 from the center of mass, C, respectively, as shown in Fig. 5.3.
R D 0, from definitions of r0 and r0 , m1 rR 1 D m1 rR 0 and
Then r D r02 r01 . Since R
1
2
1
0
m2 rR2 D m2 rR 2 . Thus Eqs. (5.5) and (5.6) become
m1 rR 01 D
m2 rR 02 D
k2 m1 m2 .r02
r3
k2 m1 m2 .r02
r3
r01 /
r01 /
(5.9)
(5.10)
100
5 The Two-Body Problem
1
Fig. 5.3 Motion of two
masses relative to the center
of mass
′1
1
′2
2
2
and m1 r01 C m2 r02 D 0 due to the center of mass definition. So r02 can be eliminated
from Eq. (5.9) and r01 can be eliminated from Eq. (5.10). Thus,
m1 rR 01 D
m2 rR 02 D
Ä
m1
k 2 m1 m2 1 C
m2
Ä
m2
k 2 m1 m2 1 C
m1
r01
r3
(5.11)
r02
r3
(5.12)
Since
rD
M 0
M 0
r1 D
r
m2
m1 2
(5.13)
we may write by dividing though by m1 or m2 and replacing the sum of the masses
by M, and then substituting from above,
rR 01 D
k2 M 0
r D
r3 1
k2
rR 02
k2 M 0
r D
r3 2
2
D
k
Â
Â
m32
M2
m31
M2
Ã
Ã
r01
r103
(5.14)
r02
r203
(5.15)
The accelerations of m1 and m2 relative to the center of mass are given. They are the
same as Eqs. (5.5) and (5.6), with m1 and m2 , respectively, replaced by the adjusted
effective mass.
From Eqs. (5.14) and (5.15) and constants c1 and c2 of Eq. (5.8), the positions
of m1 and m2 can be determined for any time. However, the constants cannot be
determined, because they are with respect to an origin fixed in space. So the solution
must be for one mass with respect to the other.
5.5 The Integral of Areas
101
Consider m1 as the origin of the two-body system. Then, from the first parts of
Eqs. (5.14) and (5.15)
rR D
k2 M
r
r3
(5.16)
where r is the relative radius vector. This is the acceleration of m2 around m1 . In a
planetary system, m1 is the Sun and m2 is the planet. In the case of a satellite and a
planet, m1 is the planet and m2 is the satellite.
For computations, the equations of relative motion can be expressed in Cartesian
O so that r D xOi C yOj C zk.
O Hence, Eq. (5.16) is
form, with unit vectors Oi; Oj; k,
xR D
k2 Mx.x2 C y2 C z2 /
3=2
yR D
k2 My.x2 C y2 C z2 /
3=2
zR D k2 Mz.x2 C y2 C z2 /
3=2
(5.17)
The differential equations, Eqs. (5.5) and (5.6), in vector form, are three secondorder equations. Each solution introduces two constants of integration, which would
be the initial conditions. So there are twelve constants in the original system.
If we ignore the motion of the center of gravity, the number of constants reduces
to six. So, the solution of Eqs. (5.17) will result in six constants. These six constants
can be determined, if we know the 3 position coordinates and 3 velocity components
at any instant. So to determine an orbit, six pieces of information are required; they
can be three observations of two angles each, or two observations of two angles and
a distance, each. An orbit is thus defined by six values: position and velocity or six
parameters (McCuskey 1963, pp. 33–35).
5.5 The Integral of Areas
The motion of m2 around m1 is a central force motion, so the areal velocity is
constant. That is
P D 1 .r
A
2
v/ D
1
huO A
2
(5.18)
where uO A is a unit vector with constant direction perpendicular to the orbital plane
defined by r and v. The components of areal velocity in Cartesian coordinates are
1
.yPz
2
1
.zPx
2
1
.xPy
2
1
c1
2
1
xPz/ D c2
2
1
yPx/ D c3
2
zPy/ D
(5.19)
102
5 The Two-Body Problem
where c1 ; c2 ; c3 are constants related to h by
q
c1 2 C c2 2 C c23 D h
(5.20)
From the initial coordinate and velocity components of m2 , the constants
c1 ; c2 ; c3 can be determined. When known, they must be related to the elements
of the orbit, which were defined in Sect. 5.2 (see also Fig. 5.1). In terms of ; i the
unit vector is
uO A D sin i sin
Oi
Oj C cos i kO
(5.21)
Oj C 1 h cos i kO
2
(5.22)
sin i cos
The areal velocity is
P D 1 h sin i sin
A
2
Oi
1
h sin i cos
2
Comparing Eqs. (5.19) and (5.20) we have
c1 D h sin i sin
c2 D h sin i cos
c3 D h cos i
q
h D c21 C c22 C c23
(5.23)
From the initial conditions, c1 ; c2 ; c3 are determined, and Eqs. (5.23) determine
and i. These elements orient the orbital plane with respect to a Cartesian coordinate
system (McCuskey 1963, pp. 35–36).
5.6 Elements of the Orbit from Position and Velocity
The orientation of the orbital plane is established by the constants c1 ; c2 ; c3 . The
size, shape, and orientation in the orbital plane must be determined. It will be a conic
section with a central force mass, M, at the conic focus. k2 D G is the gravitational
constant, m1 is at the focus, and m2 is the moving mass.
We seek the elements: a, the semimajor of an ellipse or semitransverse axis of a
hyperbola, respectively; q, the distance from the focus to the vertex of the parabola;
e, the eccentricity; !, the argument of periapsis, which is the angle in the orbital
plane between the line of nodes and the eccentricity vector, as explained in Sect. 5.2;
and T, the time of periapsis passage. The longitude of periapsis is !Q D C !. As
defined in Sect. 4.6.1, the true anomaly, f , is the angle in the orbital plane between
the eccentricity vector and the m2 position vector, as shown in Fig. 5.1.
5.6 Elements of the Orbit from Position and Velocity
103
The equation of the conic for the m2 motion is
rD
p
1 C e cos f
(5.24)
The coordinates, x0 ; y0 ; z0 , and velocity components, xP 0 ; yP 0 ; zP0 , are for m2 at time
t D 0. Then Eqs. (5.19) and (5.23) are
which determine
q
are r0 D
x20
C
y20
c1 D y0 zP0
z0 yP0 D h sin i sin
c2 D z0 xP0
x0 zP0 D
c3 D x0 yP0
y0 xP0 D h cos i
h sin i cos
(5.25)
; i, and h, as previously
indicated. The initial distance and speed
q
C z20 and v0 D
xP 20 C yP 20 C zP20 . Then, for an ellipse,
2
1
D
a
r0
v02
k2 M
(5.26)
2
r0
(5.27)
For a hyperbola,
v2
1
D 20
a
k M
For a parabola,
qD
h2
2k2 M
(5.28)
We can determine the eccentricity from the values of h by
e2 D 1
h2
k2 Ma
(5.29)
where there is a minus sign for an ellipse and a plus sign for a hyperbola. The angle
! can be calculated as follows. Set the argument of latitude as u D f C !, then
r cos u D x cos
r sin u D . x sin
C y sin
C y cos / cos i C z sin i
(5.30)
Once and i are known, and from initial conditions r0 is known, Eq. (5.30) yields
u0 at time t0 . From Eq. (5.24),
e cos f D
p
r
1
(5.31)