Double Star Astronomy
Part 4: Orbital & Dynamic Elements
Historically, the orbital motions of double stars provided the first evidence that Newton's description of gravitational attraction and the laws of motion (in the Principia mathematica, 1687) applied not just to the planets and periodic comets of the solar system but equally to the celestial motions of the stars around each other and around the Galaxy. The 19th century analysis of the orbits of a small number of nearby, short period binary stars, many of them eclipsing variable and spectroscopic stars, revealed the variety of stellar masses and dimensions, which laid the foundation for theories of stellar structure and evolution.
I first present the classical view of Kepler and Newton, then how to "build" a binary star from mass, orbital radius and orbital energy. The orbital elements are defined and illustrated in the graphical construction of the relative orbit and the method used to recreate the inclination.
Kepler's Laws. The effects of gravity within the solar system were first presented in the Epitome of Copernican Astronomy, Books IV & V (1621) by Johannes Kepler. By analyzing measurements of the motion of Mars made by Tycho Brahe, Kepler deduced his three principles of planetary motion (diagram, below):
First Law. The orbit of every planet is an ellipse with the Sun at one of the two focal points of the ellipse.
The Sun or more massive star is located at the focus 1, and the orbit describes the motion of a planet or the less massive star in a binary.
Second Law. A line from the star at 1 to another star or planet sweeps over equal areas in equal intervals of time.
Therefore the ratio between two areas swept out by a planet is equal to the ratio between the two time intervals: a1/a2 = (t1-t2)/(t3-t4). This describes orbital velocity as greatest at periastron or smallest orbital separation between the two bodies, and slowest at apastron or point of largest orbital separation.
Third Law. The square of the orbital period of a planet is proportional to the cube of the semimajor axis of its orbit.
The semimajor axis is the distance r measured from the center of the ellipse to the point of periastron or apastron. If the ellipse is a circle, r is the radius of the circle.
These are often imprecisely called Kepler's "Laws," although they are not physical laws in the scientific sense but empirical principles or generalizations. However they are the phenomena that scientific laws must explain.
Newton's Mechanics. The geometric formulation of the laws of motion described by Galileo was accomplished by Isaac Newton's Philosophiae Naturalis Principia Mathematica (1687) the mathematical principles of natural philosophy, as science was then called.
Newton's "thought experiment" was to imagine a powerful cannon at the peak of a very high mountain (at V, diagram left). According to Newton's first law of motion, a cannonball fired from the perfectly level cannon would tend to travel forever in a straight line at a fixed velocity and kinetic energy. But the continuous downward pull of Earth's gravity would bend the path into a parabolic trajectory until the cannonball hit the Earth at D.
If the powder charge in the cannon were increased, the initial velocity of the cannonball would be greater, its kinetic energy would be greater, and it would travel farther, to E or even to F. Eventually, if enough powder were used to impart a sufficiently high initial velocity, the cannonball would circle the Earth and return to V in a closed orbit.
This illustrates that planetary orbits are possible because the orbital velocity balances the gravitational acceleration, and also suggests that circular orbits contain the minimum orbital velocity or lowest energy for a given orbital radius. Higher energy orbits would be increasingly elliptical, up to the point where the orbital energy was sufficient to produce an escape velocity and the observed section of the trajectory or "orbit" would be in the form of a parabola or hyperbola.
Newton showed by a geometrical proof (not by the calculus that he invented for numerical analysis) that an elliptical orbit must be produced by an inverse square mutual attraction between two orbiting bodies:
Fd2 = Fd1·(d1/d2)2
As the distance between two bodies is changed, the gravitational attraction between them is changed by the square of the ratio of the distances. The corresponding kinetic energy necessary to sustain the orbit is changed in the same proportion.
The Dynamical Equations. Newton's key insight was that gravity was a force continuously exerted on masses, and was therefore a form of acceleration. This linked it directly to his definition of force as exerted in the simplest case of a circular orbit that will have a constant radius and orbital velocity:
F = ma = mv2/r
where the acceleration due to gravity (a) is measured as the constant orbital velocity squared (v2, in meters per second) divided by the orbital radius (r, in meters). Because the force is the gravitational constant G = 6.674 x 1011 kg1 / m3 / sec2, the measured radius and velocity create a ratio with the gravitational constant that reveals the system mass (m, in kilograms):
m = rv2/G
For rapidly orbiting spectroscopic binaries, the orbital velocity can be measured directly from the maximum observed Doppler shift in the spectral lines of the individual stars, with a correction applied for the tilt of the orbit to our line of sight.
For orbital velocities that are too slow or tilted too far to the line of sight to provide a measurable velocity, the period can be estimated from an orbital solution based on the changing position of the components measured across years or decades and a parallax estimate of the system distance, which yields the orbital radius. Then:
v2/r = 2πr/P
so that the necessary force is now defined as:
G = 4π2mr/P2
Finally, Kepler's Third Law, P r3/2, generalizes to elliptical orbits, and gives
G = 4π2r3