Download Chapter 4. Orbits

Chapter 4. Orbits
1.
Two-Body with m << M
This is the case for the Solar System and for most planetary orbits including exoplanetary systems (stars other than the Sun that have planets orbiting them). The nice thing
about this case is that you can neglect the motion of the heavier, central object (the star).
It merely sits at the focus of the ellipse traced out by the orbit of the lighter object (the
planet). Other examples where this comes into play is artificial satellites orbiting the Earth,
in which case the Earth plays the role of the heavy object, M.
Note that as long as m is very much less than M it doesn’t matter what the mass of
m is. A piece of chalk in orbit around the Sun would follow exactly the same orbit as the
Earth (neglecting non-gravitational forces). Newton solved this two-body problem using
vector calculus and the exact solution is usually reproduced for you in a mechanics course,
normally at about the sophomore level. It would take us way to far afield to do that exact
solution here, so we simply highlight the results and show how they are useful in astronomy.
The result is that the small object follows an elliptical orbit with the massive object at
one focus of the ellipse (just like Kepler’s first law of planetary motion). Check the drawing
of an ellipse given in one of the class lectures to make sure you understand the meanings
of the following terms: semi-major axis (a), which is one-half of the long axis of the ellipse,
semi-minor axis (b), which is one half of the short axis of the ellipse, eccentricity (e) which
is defined by the expression
r
b2
e= 1− 2
a
. Note from this definition that when b=a (i.e. when you have a circle), then e = 0. When
b ¡¡ a, you have the case of a very eccentric orbit and this is called high eccentricity. In this
case e approaches 1. Most planetary orbits are nearly circular. The Earth’s eccentricity is
only 0.017. Comets, on the other hand can have very eccentric orbits. Exoplanetary systems
often have planets with high eccentricities.
The distance from the center of the ellipse to a focus is given by the product of the semimajor axis and the eccentricity, as you can prove to yourself with a little geometry. When the
small object is closes to the large object it is said to be at perihelion (for something orbiting
the Sun, or equivalently perigee for an Earth-orbiting object or periastron for something
orbiting another star). When it is furthest from the central object it is said to be at aphelion,
apogee or apastron. Note that the points of perihelion and aphelion occur when the small
body crosses the major axis. Note also that the distance from the Sun to Planet at these
–2–
times is a(1-e) and a(1+e) for perihelion and aphelion respectively, as a little algebra and
geometry will easily prove.
Orbital speed is easiest to calculate in the m << M case by using the principle of
Conservation of Energy. There are two forms of energy in this system, the kinetic energy of
the small mass ( 21 mv2 ) where v is the speed of m (magnitude of its velocity vector), and the
potential energy (PE) of the system. The potential energy of two objects M and m separated
by a distance d is
GMm
PE = −
d
where the minus sign indicates that by convention, we refer to potential energy as “negative”
energy. The total energy (E) is the sum of these two and it remains constant according to the
principle of conservation of energy. The value of the constant can be derived for a circular
orbit of radius (r) as follows:
F = ma
so,
mv 2
GMm
=
r2
r
therefore,
v = vcircular =
r
GM
r
which defines the concept of circular velocity.
Now, if we calculate the total energy of a mass moving in a circular orbit, we have
1
GMm
E = KE + P E = mv 2 −
2
r
or, substituting for the circular velocity from above
GMm
1 GMm
1 GM
−
=−
E= m
2
r
r
2 r
. In the general case of an eliiptical orbit, the total energy is also still a constant and has
the value − 12 GMa m . In other words, the semi-major axis (a) replaces the radius of the circle
for the elliptical case. Hence, in general, we may write that:
v2 −
GM
2GM
=−
r
a
Note that this leads to Kepler’s second law, since as r increases (i.e. the distance from the
planet to the Sun, formerly called d), v decreases, in just such a way that the line between
the planet and the Sun sweeps out equal areas in equal times.
–3–
2.
Unbound Orbits and Escape Speed
We are still here considering the case of m << M. Note that the total energy of such
a system -GMm/a is negative. This is indicative of having a gravitationally bound system,
one that will have a closed (elliptical) orbit. We can see that as the total energy of such a
system increases (i.e. gets closer to zero) that means the semi-major axis is getting larger
and larger. The orbiting object is moving further from the central object and is becoming
“less bound” to it. In the extreme case, a can go to infinity which means that the orbiting
object never really comes back, it escapes the gravitational “clutches” of its more massive
companion. The speed which we would need to give the planet in order for it to escape from
an initially circular orbit of radius r is easy to calculate. Just set the total energy equal to
zero in the expression above, or equivalently let a go to infinity and you immediately get for
the velocity of the “orbiting” object v2 = 2GM/r. Therefore, the so-called “escape” speed
for an object is defined to be:
r
√
2GM
vescape =
= 2vcircular
r
.
It can be shown that the shape of an ellipse as the semi-major axis becomes huge
becomes indistinguishable from a parabola. The orbit of an object with zero total energy is
that of a parabola. It is interesting that most comets have orbits that are indistinguishable
from parabolas. We think that they are, in fact, gravitationally bound to the Sun and do
have a negative total energy, but it is so small that their orbits look exactly like parabolas.
This means that they are almost not moving at all when at their aphelion points and it turns
out that those points are generally way beyond the orbit of Pluto and as much as 1/3 of the
way to the nearest star!
Nonetheless, the comets do not have hyperbolic orbits, which is the case one gets if the
total energy is greater than zero. This corresponds to the object having some speed even
at huge distances from the Sun. If comets, for example, were interstellar “tramps”, i.e. not
members of our own Solar System but coming at us from elsewhere in space, then they would
have some speed already before they fell into the gravitational well of the Sun. In that case,
they would shoot by the Sun on more open curves than parabolas or ellipses ... namely
hyperbolic orbits. In the extreme case where their incoming speed was very large compared
to the speed they gained by coming close to the Sun, their orbits would hardly deviate from
a straight line. We use the hyperbolic orbit as a kind of gravitational “slingshot” to send
probes to various planets by paths that will save them energy (in the form of fuel). That is,
by falling in close to (say) Venus a satellite can gain a lot of energy that could ultimately
propel it to, say, Mars!
–4–
In general the shape of every 2-body orbit is a “conic section” – circle, ellipse, parabola
or hyperbola. All of these shapes are obtained by slicing a cone at various angles with respect
to its axis and its opening angle. See the figure in the links section of the Web site.
3.
Two-Body Orbits when m and M are comparable
If the masses are not greatly different then one cannot neglect the motion of one object.
Technically, this is always true, of course. Even the Earth has some effect on the Sun,
causing it to wobble slightly about its axis on a 1 year period in response to the Earth’s
changing gravitational pull. In fact, one can show that both objects in a two-body orbit move
along elliptical paths of the same eccentricity and the focus of both ellipses is the center of
mass of the system. When m << M applies, the center of mass for the whole system is
indistinguishable from the center of mass of the larger object, so we can totally ignore the
motion of M. Symmetry requires, of course, that the periods of the orbits of m and M also
be identical. A version of Kepler’s Third Law still applies, but it involves the sum of the
masses (m1 + m2 ) and the sum of the semi-major axes (a1 + a2 ). The relationship is:
P2 =
4π 2
(a1 + a2 )3
G(m1 + m2 )
4.
Three-Body Orbits
It is an interesting fact of Newtonian mechanics and his gravitational law that while the
two-body problem can be solved explicitly in all cases, the three body problem has no general
solution. It does, however, have some special cases that are stable. For example, if one has
two bodies really close together, in a tight binary and the third orbiting both of them at a
considerable distance, this can be a stable arrangement. The orbits of all the objects will be
nearly ellipses. But, in general, one will not find any stable orbit and a three-body system
may interact in such a way as to eject one member leaving the other two as a two-body
system. Of course, if the mass of one object is much larger than the others then one can get
a solar-system like arrangement in which the two low mass objects orbit the heavier mass
and do not interact with each other to any great extent. This, also, can be stable for a long
time.
–5–
5.
N-body Orbits and the Stability of the Solar System
As one adds more objects to the orbiting system the problems become more and more
complex and they are generally impossible to solve. One usually goes to numerical simulations on large computers and basically integrates the equations of motion, adding up the
forces from all the various objects and then calculating how each object will move and then
doing it again for the next “time step”. Because of round-off error and the necessity to divide
continuous things like time into discrete units, there is unavoidable error in doing this and
it becomes impossible, after awhile to model any system precisely. Even in the case of the
Solar System, we do not know how long into the future it will remain stable, beyond a few
million years. We can integrate forward to that extent and see that the Earth, for example,
will not get ejected by some crazy, chaotic interaction with Mars and Venus or something
like that. However, beyond that, we are not really sure. There are real variations in things
like the eccentricity of various planets’ orbits and the like that give cause for concern. It will
be interesting to learn more about other solar systems and see what sort of arrangements
they have gotten themselves into and the degree to which ours is unusual.
Star clusters and galaxy clusters can contain hundreds or thousands of stars and galaxies are collections of millions or billions of stars. The dynamics of these systems are quite
complex and fascinating. Saturn’s rings contains perhaps millions of individual rocks and
while they are too small for self-gravity to be important, they are subject to the gravitational influences of its many moons. These systems can produce unbelievably complex (and
beautiful!) patterns such as spiral arms and waves and tidal tails, etc. It is really amazing
what a simple law such as the gravitational force law can do when large numbers of particles
and interactions are involved. As mentioned above, it becomes impossible for us even to
predict exactly what will happen in the future, despite the fact that we may know exactly
where everything is now and we may know exactly the force law that is applying. Simple
underlying laws can generate amazing complexity, Look at a close-up of Saturn’s rings!