Time Since Perigee⚠ unverified

Understanding the Parameters

• Why position and time are not proportional — Kepler's second law: the radius vector sweeps equal areas in equal times, so the body moves fast near perigee and slow near apogee. Equal angles therefore take unequal times. You cannot just scale $\nu$ by the period; you must account for this non-uniform speed, which is exactly what the eccentric anomaly and Kepler's equation do.

• The three anomalies — three different angles measure position, each useful for a different job:

• True anomaly $\nu$ — the real geometric angle at the focus (what you see).

• Eccentric anomaly $E$ — an auxiliary angle on the circumscribing circle of the ellipse; it makes the geometry linear enough to integrate.

• Mean anomaly $M$ — a fictitious angle that grows uniformly with time, $M = n(t - t_p)$. It is time in angular disguise. The whole calculation is a translation $\nu \to E \to M \to t$.

• Mean motion $n = \sqrt{\mu/a^3}$ — the average angular rate, $2\pi/T$ (see Orbital Period). It is the constant that converts the mean anomaly (radians) back into time (seconds): $t = M/n$. It depends only on the orbit's size $a$.

• Kepler's equation $M = E - e\sin E$ — the transcendental heart of the subject. Going forward ($E \to M \to t$), as here, is direct. Going backward (time $\to$ position, the usual need) requires solving $M = E - e\sin E$ for $E$ — which has no closed form and must be done iteratively (Newton's method). This calculator runs the easy direction.

• The quadrant caveat — like True Anomaly From Radius, the $\arccos$ for $E$ returns $[0, 180^\circ]$, so for the outbound-vs-inbound half of the orbit the sign of $E$ (and hence $t$) needs the sign of $\sin\nu$ to resolve; a raw $\arccos$ gives the first-half-orbit answer only.

Derivation (Approaching a Proof)

The goal is $t$, the time to travel from perigee to true anomaly $\nu$. The obstacle is that angular speed varies; the classical solution introduces the eccentric anomaly $E$ to linearise the problem.

Step 1 — true anomaly to eccentric anomaly. A standard relation between the two angles is

$$\cos E = \frac{e + \cos\nu}{1 + e\cos\nu},$$

which is the equation shown in the registry latex. (It comes from projecting the elliptical position onto the circumscribing auxiliary circle.) Invert to get $E = \arccos\!\big((e+\cos\nu)/(1+e\cos\nu)\big)$.

Step 2 — eccentric anomaly to mean anomaly (Kepler's equation). Kepler's second law, integrated, gives the mean anomaly directly in terms of $E$:

$$M = E - e\sin E.$$

This is Kepler's equation. The $-e\sin E$ term is the correction for non-uniform motion: it vanishes for a circle ($e = 0$, where $M = E = \nu$ and motion is uniform) and grows with eccentricity.

Step 3 — mean anomaly to time. The mean anomaly advances uniformly at the mean motion $n$:

$$M = n(t - t_p) = n\,t \quad (\text{taking } t_p = 0 \text{ at perigee}),$$

so

$$t = \frac{M}{n} = \frac{E - e\sin E}{n}, \qquad n = \sqrt{\frac{\mu}{a^3}}.$$

Substituting $n$ gives the returned quantity:

$$t = (E - e\sin E)\sqrt{\frac{a^3}{\mu}}. \qquad\blacksquare$$

So the calculator's real output is this $t$ — the registry's displayed $E$ is only Step 1. Note the natural appearance of the period: $\sqrt{a^3/\mu} = T/2\pi$, so $t = M\,T/2\pi$, i.e. time is the fraction $M/2\pi$ of the period — as it must be, since $M$ sweeps $2\pi$ over one orbit.

Dimensional check. $E - e\sin E$ is a (dimensionless) angle in radians; $\sqrt{a^3/\mu} = \sqrt{\text{m}^3/(\text{m}^3/\text{s}^2)} = \sqrt{\text{s}^2} = \text{s}$, so $t$ is in seconds ✓.

History and Development

• Kepler's equation (1609). In Astronomia Nova, Johannes Kepler posed the problem of finding a planet's position at a given time and arrived at $M = E - e\sin E$. He could not solve it in closed form — famously writing that if someone showed him how, he would be "grateful" — and it is now proven that no closed-form (elementary) inverse exists. It launched centuries of work on iterative and series solutions.

• A magnet for mathematics. Kepler's equation has been attacked by nearly every great mathematician since: Newton (his iteration), Lagrange (series inversion), Bessel (whose functions arose partly from this problem), and Cauchy. It is often cited as the equation with the most published solution methods in all of applied mathematics.

• The forward vs inverse problem. This calculator does the forward direction (position → time), which is algebraic. The far more common inverse problem (time → position — "where will the satellite be in 20 minutes?") requires solving Kepler's equation numerically for $E$, done billions of times a day in every propagator, GPS receiver, and planetarium.

Related Concepts: True Anomaly From Radius, Orbital Period, Eccentricity Vector, Orbital Velocity, Vis Viva Energy, Circular Orbit Velocity

Notes: Registry latex shows only the intermediate eccentric anomaly $E$; the output is the time $t$ (s) via $M=E-e\sin E$ (Kepler's equation) and $t=M/n$, $n=\sqrt{\mu/a^3}$. The three anomalies: true $\nu$ (geometry) → eccentric $E$ → mean $M$ (∝ time). This is the forward Kepler problem (position→time, algebraic); the inverse (time→position) needs iteration. $\nu$ in degrees, $\mu$ in m³/s² (both mislabelled). $\arccos$ quadrant caveat as in True Anomaly From Radius.