In December 1991, the astronomers Alex Wolszczan and Dale Frail, using observations at the Arecibo dish in Puerto Rico, established an amazing fact about the pulsar PSR B1257+12.
A pulsar is a rapidly rotating neutron star---a super dense object about 10 km in radius, but with a mass equal to that of the Sun. PSR B1257+12 rotates about 161 times per second! Imagine a ball bearing 20 km in diameter spinning this fast! Actually, a regular ball bearing would be unable to withstand the stresses and would fly apart. But the crust of neutron stars is enormously stronger than any terrestrial material and so can withstand the immense stresses.
A neutron star arises from the collapse of the core of a massive star that detonates, resulting in one of the most violent events in the universe: a supernova. The curious and amazing thing about the pulsar PSR B1257+12, which is about 1300 light years from Earth, is that it has 3 planets in orbit around it! This was totally unexpected, because one expects that any planets orbiting a star would have been destroyed by the violence of the supernova explosion.
The discovery of planets around, of all things, a pulsar encouraged those searching for evidence of planets around stars similar to the Sun. On July 4, 1995, early in the morning, and after months of careful observations, the French astronomers Mayor and Queloz became convinced that they had, finally, discovered the first extrasolar planet orbiting a Sun-like star.
Since 1995, many extrasolar planets have been discovered. (See,
for example, Upsilon
Andromedae.) But, as is often true in science, these discoveries have forced us to revise
much of what we thought we understood. The extrasolar systems are quite
different from our solar system and puts into question the standard theory
of the origin of solar systems.
The basic idea is this: According to Newton's laws, the star and its planets orbit their common center of gravity. Because stars are so much more massive than their planets the center of gravity usually lies within the star. The star, and planets, move about that point. As the star moves, it sometimes approaches us and sometimes moves away from us. When it moves towards us its light is blue-shifted; when it moves away, the light is red-shifted. From these periodic shifts one can infer the speed with which the star orbits the center of gravity, as well as the period of its orbit.
The line joining the star, the center of gravity, and the planet always remains straight; it's as if the star and planet were joined by a rigid rod. Therefore, the planet takes the same amount of time to orbit the center of gravity as does the star. See animation.
Let's look at the dynamics of a planet-star system in a bit
more detail.
The star of mass M* and planet of mass m revolve about their common center of gravity according to the "balance equation"
am = a*M*
We see that the center of gravity is closer to the larger mass. Also, by the law of momentum conservation we have that
mV = M*V*
where V is the speed of the planet and V* that of the star.
Question: Given that Jupiter is about 1/1000th the mass of the Sun, where is the center of gravity of the Sun-Jupiter system, relative to the Sun? How fast does the Sun travel around the center of gravity given that Jupiter takes about 12 years to complete one orbit? (See Problem Set 2)
Consider the motion of the star about the center of gravity. If P is the period of the star's orbit (and therefore that of the planet) and V* is the star's orbital speed then
V* = 2pa*/P.
According to Newton's law of gravity the gravitational force between the star and planet is
F = GmM*/(a+a*)2
The gravitational force on the star is equal to the mass of the star times its acceleration. For simplicity we shall assume that the star moves on a circular orbit. Therefore, for circular motion, we can write
GmM*/(a+a*)2 = M*(V*2/ a*)
The expression on the right-hand side, in parentheses, is the acceleration for motion in a circle at a constant speed V*. After a bit of algebra we arrive at
V* = (2pG/P)1/3 m/(m+M*)2/3
which relates the orbital speed V*
of the star, its period P,
and the mass m of the planet. This formula assumes that we are viewing the
extrasolar system edge-on, in which case we can measure the full speed V*,
using the Doppler effect. In practice, we must account for the
possibility that the orbital plane of the star is tilted relative to our
viewpoint, as illustrated in the figure below.
What we measure is not V* but the velocity component K in our direction, which is less than or equal to V*. We can get an expression for K by multiplying the formula for V* by sin(i):
V*sin(i) = K = (2pG/P)1/3 m sin(i)/(m+M*)2/3
Since astronomers are currently unable to measure the angle i, the mass that is measured is actually m times sin(i), which necessarily is less than or equal to the mass m; that is, at present only lower bounds on the mass of extrasolar planets can be measured using this method of detection.
Here is a figure indicating how the amplitude K is expected to change as a function of time. For a circular orbit we obtain a sinusoidal curve.
All the extra-solar planets that have been discovered so far seem to be Jupiter-like objects. It is of course of considerably interest to determine if Earth-size planets exist beyond our solar system. Many groups around the world have initiated projects to search for such planets. NASA is currently engaged in such a project, called the Kepler mission, to search for Earth-size extra-solar planets.
