Time Since Periapsis Calculator

1) Why time since periapsis matters

Time since periapsis tells you where an object sits along its orbit relative to closest approach. Mission designers use it to schedule burns, predict eclipse entry, and align tracking passes. In astronomy, it helps phase observations and compare orbital solutions across epochs.

2) Three anomalies, one timeline

True anomaly (ν) is the geometric angle at the focus. Eccentric anomaly (E) is a convenient auxiliary angle for elliptical motion. Mean anomaly (M) increases uniformly with time. This calculator converts your chosen input into mean anomaly, then converts M into elapsed time.

3) Core relationship used

The link is simple: Δt = M / n, where n is mean motion in rad/s. For an elliptic orbit, M = E − e·sin(E). For a hyperbolic path, M = e·sinh(H) − H. Because M is dimensionless, the time scale comes entirely from n.

4) Choosing mean motion inputs

You can enter n directly, derive it from the period P using n = 2π/P, or compute it from μ and a using n = √(μ/a³). Typical values: a low Earth orbit has P ≈ 90 minutes, so n ≈ 0.00116 rad/s. A geosynchronous orbit has P ≈ 86164 s, so n ≈ 7.2921×10⁻⁵ rad/s.

5) Useful gravitational parameters

Gravitational parameter μ depends on the central body. Common references include Earth μ ≈ 398600.4418 km³/s², Mars μ ≈ 42828.375 km³/s², and Sun μ ≈ 1.32712440018×10¹¹ km³/s². Using consistent units with a keeps n accurate.

6) Elliptic orbits and full revolutions

Elliptic motion repeats, so the same anomaly can occur on later passes. The revolutions field N adds 2πN to mean anomaly, letting you compute time since periapsis after multiple orbital cycles. For example, N = 10 on a 90‑minute orbit shifts time by about 15 hours.

7) Hyperbolic flybys and negative time

Hyperbolic trajectories do not repeat. The sign of M (and Δt) indicates whether you are before or after periapsis. Large true anomalies approach an asymptote, where the conversion to H becomes numerically sensitive. If inputs are near that limit, the calculator warns you.

8) Practical accuracy tips

Use radians if your source data is already in radian form. Keep e in the correct regime: 0 ≤ e < 1 for elliptical, e > 1 for hyperbolic. When using μ and a, match kilometers with km³/s², or meters with m³/s². Exported CSV helps document assumptions and compare scenarios.

1) What does “time since periapsis” represent?

It is the elapsed time from the most recent periapsis passage to the orbital position described by your anomaly input. Negative results mean the object is still approaching periapsis.

2) Which anomaly input should I use?

Use the anomaly you already have. True anomaly is common in geometry, mean anomaly is common in ephemerides, and eccentric/hyperbolic anomaly is common in orbital equation solving.

3) Why do I need mean motion or period?

Anomalies are angles, not time. Mean motion n (or period P) provides the time scale that converts mean anomaly into seconds, minutes, hours, and days.

4) How do I handle multiple orbital passes?

For elliptical orbits, increase the revolutions value N. Each added revolution increases time by one full period, while keeping the same geometric position in the orbit.

5) Can I use this for parabolic orbits (e = 1)?

No. Parabolic motion uses a different formulation and does not use E or H in the same way. Use a dedicated parabolic time‑of‑flight method instead.

6) What units should I use for μ and a?

Use consistent units. If a is in kilometers, use μ in km³/s². If a is in meters, use μ in m³/s². The calculator converts meters to kilometers when needed.

7) Why do hyperbolic cases sometimes warn about asymptotes?

As ν approaches the trajectory asymptote, tan(ν/2) grows rapidly and the atanh conversion can hit its domain limit. This makes results unstable, so the tool blocks impossible combinations.