Orbit shapes, periods and orbital scale

An ellipse has a centre and two foci

An orbit is a trajectory described by a dynamical model and initial conditions. For isolated Newtonian inverse-square two-body motion, a bound orbit with nonzero angular momentum has an elliptical relative trajectory. Actual astronomical systems can depart from that model; a fixed ellipse is not a universal description of every observed path.

An ellipse is the set of points whose distances to two fixed foci have a constant sum. Its major axis is the longest diameter through its centre, and the semimajor axis a is half that length. The semiminor axis b is half the perpendicular shorter diameter.

Eccentricity e measures the focus offset from the centre divided by a. For a nondegenerate ellipse, 0 ≤ e < 1. The relation b = a√(1 − e²) connects its two axes. At e = 0 the foci coincide with the centre and the ellipse is a circle.

The relative orbit describes the separation vector from one body to the other. Its origin lies at a focus in the two-body ellipse model. Choosing one body as that coordinate origin does not mean it remains fixed in an inertial frame; the next lesson examines both bodies around their common centre of mass.

Apsis distances define an arithmetic mean, not a time average

Periapsis is the nearest point of an elliptical relative orbit to its focus; apoapsis is the farthest. Their separations are rₚ = a(1 − e) and rₐ = a(1 + e). The subscript p labels periapsis and a labels apoapsis in these distance symbols.

Adding those relations gives a = (rₚ + rₐ)/2. Subtracting them gives e = (rₐ − rₚ)/(rₐ + rₚ). The semimajor axis is therefore the arithmetic mean of the two extreme separations, a geometric property of the ellipse.

It is not generally the separation averaged over elapsed time. A body spends unequal times in different parts of an eccentric orbit. Calling a an average distance without specifying the averaging rule can therefore produce a mistaken physical interpretation.

For a circle, nearest and farthest separations are the same radius, so a equals that radius. Radial collision trajectories and limiting conics need separate treatment; e = 1 is not an ordinary nondegenerate ellipse.

Timing follows swept area about the focus

Draw the relative separation line from the focus to the moving point. In the isolated central-force model, this line sweeps equal areas in equal times. The swept region follows the curved orbital arc and the two separation lines; it is not generally just the triangle joining three points.

Near periapsis the separation is shorter and the motion is faster; near apoapsis the separation is longer and motion is slower. Equal-time positions therefore need not be equally spaced in angle around the focus, in angle around the geometric centre, or in distance along the curve.

The deeper conserved quantity is angular momentum. For the relative central-force motion, r times the tangential velocity component remains constant. The tangential component alone need not remain constant as the distance changes.

This distinction matters when drawing a teaching orbit or interpreting timed observations. A series of evenly spaced dots around an ellipse can show geometric locations, but cannot be labelled equal times without checking the dynamical timing.

The period law needs a length, a mass sum and consistent units

The period T is the time for one complete bound orbital cycle in the isolated model. Let μg = G(m₁ + m₂), the gravitational parameter formed from the two masses and the gravitational constant. Its dimensions are length cubed divided by time squared.

For the physical relative semimajor axis a, the period relation is T² = 4π²a³/μg. The length is the full relative orbit’s semimajor axis, not the semimajor axis of just one component’s path around the common centre.

If one mass is negligible compared with a dominant mass M, μg ≈ GM is a useful approximation. State that condition rather than silently using it for an equal-mass pair. Nor is T² = a³ a universal unit-free identity: numerical simplifications require declared units and mass assumptions.

For the same mass sum, period ratios follow T₂/T₁ = (a₂/a₁)^(3/2). Semimajor-axis ratios 1, 2 and 4 therefore correspond to period ratios 1, 2√2 ≈ 2.828 and 8. If the mass sum changes, this comparison needs its additional gravitational-parameter factor.

An original model separates focus, centre, geometry and time

Use arbitrary common units L₀ for length and T₀ for time, with μg = 1 L₀³/T₀². Choose relative semimajor axis a = 2 L₀ and eccentricity e = 0.6. The semiminor axis is 2√(1 − 0.36) = 1.6 L₀.

Place the relative origin, a focus, at (0, 0) and the centre at (−1.2, 0), in L₀ units. Periapsis is at (0.8, 0), and apoapsis at (−3.2, 0). Thus the nearest and farthest separations are 0.8 and 3.2 L₀, whose arithmetic mean is 2 L₀.

The period is 2π√(2³/1) = 4π√2 T₀, about 17.7715 T₀. Four equal quarter-period intervals last about 4.44288 T₀ each. Each focus-swept area is one quarter of πab, or 0.8π L₀², about 2.51327 L₀².

The figure’s quarter-time positions follow this same model rather than uniform angular spacing. All coordinates and intervals are synthetic. The focus identifies the relative coordinate origin, not the size, surface or stationary inertial location of a real star.

A projected ellipse is not the full dynamical solution

A telescope measures directions and other signals from an observing location. Viewing orientation can change the apparent shape of a physical orbit, and conversion from angular separation to physical length needs an appropriate distance and geometry.

A sky-projected ellipse therefore cannot simply be inserted into the period formula as if its measured angular semimajor axis were the physical relative length. Time coverage, coordinate conventions, calibration and uncertainty also affect an orbital inference.

The two-body laws provide a testable starting model. Additional bodies, extended or nonspherical mass distributions and other forces can change the trajectory and the useful meaning of fitted orbital elements. Later lessons examine these departures explicitly.

Before using a period or orbital scale, identify which motion it describes. The common-centre lesson now distinguishes the relative ellipse from the two component orbits and explains why both bodies can move even when one appears much less displaced.

Worked examples

Try the reasoning

1. Is the focus always the geometric centre of an ellipse?

   Show the answer and reasoning

   No.

   They coincide for a circle; an eccentric ellipse has displaced foci.

2. Do equally spaced angles around an eccentric ellipse automatically mark equal times?

   Show the answer and reasoning

   No.

   Equal focus-swept areas, rather than equal angular increments, describe the central-force timing.

3. Does a = (rₚ + rₐ)/2 establish that a equals the time-averaged separation?

   Show the answer and reasoning

   No.

   It is the arithmetic mean of the two extreme separations, not an average weighted by elapsed time.

4. Which semimajor axis enters T² = 4π²a³/[G(m₁ + m₂)]?

   Show the answer and reasoning

   The physical relative semimajor axis.

   One component’s barycentric semimajor axis is a different length; the next lesson relates them.

What to carry forward

• Ellipse geometry distinguishes a focus, the centre, axis lengths and apsis separations.
• Equal areas in equal times do not imply equal angles or constant speed.
• The two-body period law uses physical relative scale, the mass sum and consistent units.