The Two Body Problem
Two point masses with total energy less than their gravitational binding energy will have elliptical orbits with the center of mass as one of the foci. Curiously enough, one can transform the coordinates so that one of the bodies (usually the more massive or the brighter, called the primary) is at one of the foci of a relative ellipse which describes the relative orbit of the other body or secondary. Still stranger is the fact that an ellipse not looked at along a normal appears to be an ellipse but with foci "out of position"... When we look at visual binary stars this is usually the case and it is a simple prcedure to determine the orbital inclination.
Some properties of the ellipse, one of the conic sections:
The long or major axis is 2a where a is the semimajor axis.
The short or minor axis is 2b where b is the semiminor axis.
The area of an ellipse is pab.
The circumference C of an ellipse is given by elliptic integrals E, C=4aE, but for small e it is about p times the square root of 2(a2+b2).
The eccentricity e is defined by
e2=(a2-b2)/a2.
If e=0 the figure is a circle, e=1 is a parabola. The ellipse below has e=0.75 like the orbit of Nereid.
The red dots are the foci.
The foci can be used to define an ellipse: The locus of points whose total distance from two points (foci, separated by 2ae) is a constant (=2a). Since the path from one focus to the next via the ellipse is constant or "stationary" in Fermat's parlance, if a ray of light from one focus strikes the ellipse it must pass through the other focus. Hence, elliptical mirrors are ideal surfaces for imaging objects, and are used in microscope objectives and flashlamp pumped lasers. Hence, the name "foci".
Note that in orbits, the point of closest approach (perigee for earth orbit, perihelion for the sun, periastron for stars) is a(1-e). The farthest distance (apogee, aphelion, I've never seen apastron used) is a(1+e).
Check out the Earth's orbit. The semimajor axis is a=1 (astronomical unit, AU) and the orbital eccentricity was 0.0167 so the distance to the Sun varies between 0.9833 and 1.0167 AU. The insolation varies between the ratio squared, about a factor of (1.0167/0.9833)2=1.069, so Southern summers (perihelion is early in January) are hotter than their Northern counterparts and winters colder. (But the hot summer is shorter and southern extreme seasonal variations are buffered by the fact that the southern hemisphere has more oceans than the north. What will happen 13,000 years from now?)
If the total energy eqals the gravitational potential the masses are asymptotically bound (e=1) and the orbit is a parabola.
If the total energy exceeds the gravitational binding energy the masses are not bound and the trajectory(s) is(/are) a hyperbola(e) with e>1.