Calculator Inputs
Formula Used
Mean anomaly advances linearly with time:
M = M₀ + n·Δt
Mis the mean anomaly at the target time.M₀is the mean anomaly at the chosen epoch.Δt = t − t₀is the elapsed time since that epoch.nis the mean motion in radians per second.
If you know orbital period P, then:
n = 2π / P
If you know semi-major axis a and gravitational parameter μ:
n = √( μ / a³ )
(valid for two-body Keplerian motion).
How to Use This Calculator
- Select the angle unit you prefer for mean anomaly.
- Enter
Δtand choose its time unit. - Provide
M₀, or enable periapsis mode forM₀ = 0. - Pick a method: period, mean motion, or
aandμ. - Click Calculate to view results above the form.
- Use Download CSV or Download PDF for exports.
Example Data Table
| Scenario | M₀ (deg) | Δt (days) | P (days) | Expected M (deg) |
|---|---|---|---|---|
| Earth-like year, modest shift | 30 | 50 | 365.25 | ~79.29 |
| Fast orbit, half day later | 10 | 0.5 | 1.0 | ~190.00 |
| Periapsis epoch, quarter period | 0 | 0.25 | 1.0 | ~90.00 |
| Two-day orbit, 3.2 days later | 120 | 3.2 | 2.0 | ~336.00 |
| Mean motion input (0.9856 deg/day) | 0 | 100 | — | ~98.56 |
Values are approximate and assume simple wrapping to the selected range.
Mean Anomaly in Orbital Mechanics
Mean anomaly is a time-like angle that tracks orbital phase on a Keplerian orbit. It answers: “How far through the orbit should we be now if motion were uniform?” It is not the same as true anomaly, which depends on position on the ellipse. For circular orbits, mean anomaly and true anomaly match.
The Linear Propagation Idea
The key relationship is M = M₀ + n·Δt.
Because M grows linearly, it behaves like an orbital clock and is easy to propagate.
This is useful for observation planning and quick element validation.
The calculator shows both a raw and a normalized result.
Mean Motion and Period
Mean motion n is the average angular rate of one revolution.
If the orbital period P is known, n = 2π/P.
An Earth-like year corresponds to about 0.9856 deg/day.
Short-period orbits produce larger n and faster wrapping.
Computing n from a and μ
If period is unavailable, the two-body model gives n = √(μ/a³),
where μ = GM for the central body and a is the semi-major axis.
The calculator accepts a in m, km, or AU and converts internally to SI units.
Ensure μ matches the same central body used for the orbit.
Choosing a Good Epoch
The epoch is the reference time t₀ where M₀ is defined.
Many catalogs publish M₀ at a fixed epoch such as a Julian date.
Propagating far from t₀ magnifies unit errors, so keep definitions consistent.
If you change the epoch, recompute M₀ or propagate from the original epoch.
Periapsis Mode for Clean Inputs
At periapsis passage, mean anomaly is defined as zero, so M₀ = 0 by convention.
Periapsis mode applies this directly and helps avoid sign and unit mistakes.
It is handy when the epoch is “time of periapsis.”
Use Δt to move forward from that moment.
Units and Normalization Options
You can work in degrees or radians; the calculator converts internally. Mean motion is accepted in deg/day, rev/day, or rad/s to match common sources. Normalization wraps angles to 0..360 (0..2π) or -180..180 (-π..π). Wrapping changes display only, not the underlying phase.
How Results Feed Later Steps
Mean anomaly is typically used to solve Kepler’s equation for eccentric anomaly,
then converted to true anomaly and orbital radius for position and velocity.
This calculator focuses on the propagation step, which is stable and repeatable.
Exported M, n, and P support downstream solvers and ephemeris workflows.
FAQs
1) What does mean anomaly represent?
It is a time-based angle that increases uniformly, indicating orbital phase relative to an epoch. It is convenient for propagation because it advances linearly even when real orbital speed varies on an ellipse.
2) Why is my normalized result different from the raw value?
Both values represent the same angle. The raw value can grow beyond one revolution, while the normalized value is wrapped into a chosen range such as 0..360 or -180..180 for easier interpretation.
3) When should I use periapsis mode?
Use it when your epoch time corresponds to periapsis passage. In that case, mean anomaly is defined as zero at the epoch, so the calculator sets M₀ to 0 automatically.
4) Which input method is best: period, n, or a and μ?
Use period if you have it, mean motion if a catalog provides it directly, and a with μ if you are modeling a two-body orbit and know the central body parameter and orbit size.
5) Do degrees or radians change the computation?
No. The physics is identical. Degrees and radians are just different units for the same angle. The calculator converts internally so you can input and read results in your preferred unit.
6) Can mean anomaly predict the exact position on an elliptical orbit?
Not by itself. Mean anomaly is used to solve Kepler’s equation to obtain eccentric anomaly, then true anomaly and distance. This calculator outputs the mean anomaly needed for those steps.
7) What can cause large errors in propagated mean anomaly?
Inconsistent units, incorrect epoch definition, or using a period/μ that does not match the actual orbit. Over long Δt, small input errors accumulate into large phase offsets.