## 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 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. ## Gravitational DynamicsKepler's Laws. The effects of gravity within the solar system were first presented in the 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 Second Law. A line from the star at 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 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 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 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 This illustrates that planetary orbits are possible because the orbital velocity balances the gravitational acceleration, and also suggests that 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
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
where the acceleration due to gravity (
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
so that the necessary force is now defined as:
Finally, Kepler's Third Law,
where the masses of the two orbiting bodies are The Solar Standard Formulas. Because the Earth is only about 0.0001% (one millionth) the mass of the Sun, the mass of the combined system is effectively the mass of the Sun, and the Earth's period at the Earth's average orbital radius is effectively a measure of the solar mass. This means the dimensions of the solar system can provide units of measurement that are already standardized on the gravitational constant, so it can be dropped from the equations. If solar standard units are used — the astronomical unit (AU) for the semimajor axis
( In the case where the observed orbit is too slow to yield an orbital solution, the relative mass of the two components of the system can be estimated from their apparent magnitudes. Assuming that both stars are on the main sequence (and therefore have a luminosity that corresponds to the mass), the system
where The fact that the orbital dynamics are determined by the mass of the components, and a parallax estimate of distance yields the absolute luminosity of the components, allowed the stellar mass/luminosity relation to be determined through the painstaking, century long measurement of a small number of eclipsing variable stars, spectroscopic binaries and closely orbiting visual double stars within a few hundred parsecs of the Earth. ## Building a Binary OrbitThe most effective way to understand the binary orbit is to build one — from the simplest possible to the more complex.
To summarize, binary stars can be represented in one of three ways: (1) The As the center of mass, the barycenter traces the galactic orbital trajectory of the binary system which, if it were visible, would appear as a straight line proper motion across the celestial sphere. In closely orbiting, short period binaries, the two components of the system appear to oscillate or "wiggle" back and forth around this straight line path. If the second component is too faint to be optically visible, the direction and pace in the proper motion of the primary star will appear to change periodically, and these perturbations allow the presence and mass of the secondary to be estimated. Both Sirius and Procyon were first identified as binary stars in this way. What About Triple Stars? Are binary orbits the most complex possible? What about triple, quadruple, quintuple stars? The answer is that, in nearly all cases where stable multiple systems have been identified, the orbits are dynamically segregated binary orbits. If it is a triple star, then the third (single) component orbits the binary at a much greater orbital radius than the binary, forming a "binary" of a binary and single component. If it is a quadruple star comprising two binaries, then the binaries orbit their common barycenter at much greater distances than the orbits of either binary, in effect forming a "binary" of two binary components ... and so on. The basic principle is that orbits are spaced dynamically so that the inner orbits are not perturbed by the motions of the outer components. How far apart is far enough? Observations of multiple stars in the solar neighborhood suggest the separations are 100 to 1000 times the separation inside the binary unit, and computer simulations suggest that these systems can be both stable and bound with an outer orbital radius of 100,000 AU or more. Current theories of star formation suggest that multiple stars form as a result of turbulent fragmentation inside the same collapsing cloud core, and computer simulations show that triple stars born in such close proximity will dynamically "unfold" into a binary plus single or 2+1 system by transferring angular momentum from the binary pair (making their orbit smaller) to the singleton (making its orbit larger, more energetic and typically more elliptical). There are a few arcane orbital configurations of three stars that can coexist in close orbit with each other, but it is difficult to see how these would form naturally. Instead, multiple stars that cannot reach a stable segregation of orbital energies are most likely to break apart, always by keeping the binary elements intact. ## Double Star Orbital ElementsThe orbits of binary systems can be analyzed if sufficiently accurate positional (or The diagram (above) summarizes the relationships between the absolute, relative (or "true") and apparent orbits, using the calculated orbit of iota Leonis as an example. The key constants, indicated by the dotted lines connecting the different orbits, are: (1) the angular separation or apparent distance between the components at every point in the orbital cycle (including apastron and periastron) is identical between the absolute and relative orbits; and (2) the angular width of the line of nodes (between the ascending and descending nodes) is identical between the relative and apparent orbits. Distances between the components in the apparent orbit are described in units of angular width (such as arcseconds or arcminutes), as these are the units of the visual measurements; arcseconds are also used to describe the semimajor axis of the calculated relative orbit. Distances between the components in the absolute orbit are described in terms of astronomical units (or kilometers), and separation in astronomical units can also be applied to the relative orbit, simply by multiplying the arcsecond length of the semimajor axis (a) by the distance of the system in parsecs. Note that the angular dimension of the secondary orbit major axis is always smaller in the absolute than in the relative orbit, although the eccentricity of the orbits is the same; and that the eccentricity of the orbits is generally not the same between the relative and apparent orbits. In addition, the points where a binary star apparent orbit presents the smallest and largest angular separation (green dots in diagram) are typically not the apastron and periastron of the relative orbit, and the two points typically do not lie on a line through the primary star. This means the ephemeride date of periastron passage will not indicate the time of closest visual separation. The table (below) indicates the principal orbital elements in the apparent orbit and relative orbit (sometimes called the
The orbital plane of the absolute orbit is almost never viewed in an orientation perpendicular to our line of sight from Earth. The orbital inclination ( The inclination combines two different features of the relative orbit. First, it indicates the tilt of the plane of the relative and absolute orbits as an angle between the line of sight to Earth and the plane of the relative orbit, from 0° to 180° (diagram, below). Second, the sign of the cosine of the inclination determines the direction of the secondary orbital motion as viewed from Earth: a direct (counterclockwise) orbit is coded as an angle between 0° and 90° (positive cosine), and a retrograde (clockwise) orbit is coded as an angle between 90° and 180° (negative cosine). The line of nodes is the line formed by the intersection of the two planes of the true and apparent orbits, measured in counterclockwise direction from a line to the Earth's celestial north; it always passes through the primary (brighter or more massive) star. The ascending node is the point on the line of nodes where the component star passes through the line of nodes and is moving away from Earth. In the great majority of binary stars where this cannot be determined due to an unmeasurably small orbital velocity, it is arbitrarily assigned to the position angle that is less than 180°. Thus, the inclination and ascending node in many cases represent an arbitrary rather than physical description of the binary system. Note that the periastron rather than apastron is preferred as an orbital parameter because the relative orbital velocities of the two components at that point are at maximum. Either the radial velocity or the positional parameters (or both) will change most rapidly at that point, which usually minimizes error in the estimation of the time of periastron and therefore error in the predicted future relative positions of the components. ## Diagramming a Double Star Relative OrbitThe Campbell elements can be used to diagram both the true and apparent orbits, and this is quite easy to do when working in Photoshop. The diagram below of iota Leonis provides a template. 1. Determine from the arcsecond scale of the semimajor axis the total system width and image scale. Use a large enough scale to minimize rounding errors. In the diagram, the semimajor axis equals 1.91", so the system width is about 4 arcseconds. The scale chosen for the example diagram is 120 pixels = 1 arcsecond. 2. Draw the cartesian x and y axes; the origin is the location of the system primary star. 3. Calculate 4. Scale 5. Calculate and scale 6. Using the ellipical marquee tool while holding down the "Alt" key, click on 7. Create a new layer and draw a horizontal or vertical line, then rotate the line to correspond to the angle given as the argument of the periastron ( 8. Copy the line of nodes layer, and rotate this line clockwise the PA of the line of nodes ( 9. Merge the orbit layer with the line of nodes layer, and copy this layer. Draw (copy) the line of apsides and point 10. Rotate the copied orbit layer so that the line of nodes is exactly either horizontal or vertical, then use the Transform command to reduce the scale perpendicular to the line of nodes by a percentage equal to the cosine of the angle of inclination. In the example, the line of nodes is at 35° to the horizontal axis, so the copied layer was rotated counterclockwise by the same amount ( –35°). Then 11. Rotate the copied orbit layer in the reverse direction so that the line of nodes is in its original orientation. Align the copied orbit so that the two orbits intersect at the line of nodes, and the intersection of the lines of apsides and nodes in the copied layer is at the origin of the plot. This is the apparent orbit. 12. Merge the orbits, and rotate them so that the line of right ascension is vertical, with north at the bottom. 13. If desired, use the catalog position angle for the secondary star to locate the star on the apparent orbit. Then use a line perpendicular to the line of nodes, and through the location of the secondary on the apparent orbit, to locate it on the relative orbit. 14. If necessary reduce and crop the canvas to the finished image size, and label elements as needed. Multiple stars can be plotted in the same way, provided the orbital elements are available separately for hierarchical centers of mass: first A/B, then AB/C, then ABC/D, etc. The diagram (above) of zeta Cancri (STF 1196) was created by first plotting the orbit of the AB pair, then the orbit for the AB/C "pair", rotating them both so that celestial north is at the bottom, then superimposing the primary of the AB pair on the "primary" focal point of the AB/C pair.
Last revised 10/12/17 • ©2017 Bruce MacEvoy |