What is the formula for orbital period?

Kepler's third law in its Newtonian form is T = 2π·√(a³ / (G·M)), where T is the orbital period in seconds, a is the semi-major axis of the orbit in metres, M is the central body mass in kilograms, and G is the gravitational constant 6.674e-11 N·m²/kg². For example, a satellite orbiting Earth (M = 5.972e24 kg) at a 6.771e6 m radius (about 400 km up) has a period of roughly 5544 seconds, close to 92 minutes, which is why the ISS circles the planet about 16 times a day.

Why is Earth's year about 365 days from this formula?

Plug in the Sun's mass M = 1.989e30 kg and Earth's semi-major axis a = 1.496e11 m (1 AU). Then a³ is about 3.35e33, G·M is about 1.33e20, the square root of their ratio is about 5.02e6, and times 2π gives roughly 3.156e7 seconds. Divide by 86 400 seconds per day and you get about 365.25 days, the length of a year. The tool does this in one click with the Sun preset and the AU length unit.

What does Kepler's third law actually say?

It says the square of the orbital period is proportional to the cube of the semi-major axis, T² ∝ a³, for any body orbiting the same central mass. So if you double the semi-major axis, the period grows by 2^1.5, about 2.83 times, not double. Mars sits at 1.52 AU and takes 1.88 years, and 1.52^1.5 is indeed about 1.88. Type a, then 2a, into this calculator and watch the period scale by 2.83 exactly.

Can it solve for the semi-major axis from a known period?

Yes. Switch the mode tab to Semi-major axis and supply the central body mass and the measured period, and the tool computes a = ∛(G·M·T² / (4π²)). This is how you find the orbital radius from a satellite's revisit time. A geostationary satellite has a period of one sidereal day (about 86 164 s) around Earth, and the formula returns a ≈ 4.22e7 m, which after subtracting Earth's radius is the famous 35 786 km altitude.

How fast does a body move in its orbit?

For a circular orbit the speed is v = √(G·M / r), and this calculator reports it at r = a alongside every period. Earth orbits the Sun at about 29.8 km/s, a low Earth orbit satellite travels around 7.7 km/s, and a geostationary satellite moves at about 3.07 km/s. The larger the orbit, the slower the body moves, which is the same inverse relationship that makes outer planets crawl across the sky while inner ones race.

What units should I use?

The physics works in SI: kilograms for mass, metres for the semi-major axis, seconds for the period, giving G = 6.674e-11 N·m²/kg². For convenience you can enter the semi-major axis in metres, kilometres or astronomical units and read the period in seconds, days or years, and the tool converts to SI internally before computing, then converts the answer back. Scientific notation like 1.496e11 is accepted in any numeric field, which is essential for astronomical distances and masses with two dozen digits.

Work a physics homework problem and check every step

The textbook asks for the period of a satellite orbiting Earth at a semi-major axis of 6.771e6 m. Pick the Earth preset for M, type 6.771e6 for a with metres selected, and read the period in minutes by switching the time unit. The on-screen steps spell out a³ over G·M and the square root, so you can copy the working into your answer and show how you got there, not just the final number.

Reverse-solve a geostationary orbit altitude

You want the radius of a geostationary satellite, the orbit whose period matches one sidereal day. Switch to Semi-major axis mode, pick Earth, enter 86164 seconds, and the tool returns a ≈ 4.22e7 m. Subtract Earth's 6.371e6 m radius and you have the textbook 35 786 km altitude, derived rather than memorised.

Show a class why outer planets take so long

Set the central body to the Sun and the axis to 1 AU; the result is about one year. Bump the axis to 5.2 AU for Jupiter and the period jumps to about 11.9 years, because T scales as a^1.5. Share the URL with the class and every student opens the exact same problem, a one-click demonstration of T² ∝ a³ that no static slide can match.

Plan or sanity-check a satellite mission

Designing a constellation, you need the orbital speed and period at a chosen altitude. Enter Earth and your semi-major axis, and the tool reports both the period and v = √(G·M / r) in one view. If your mission software gives a period off by orders of magnitude, you have almost certainly mixed up units or confused radius with altitude.

Orbital Period Calculator — Kepler's Third Law (T = 2π√(a³/GM))

T = 2π·√(a³ / GM) · Kepler's third law · solve period or semi-major axis · orbital speed · Sun/Earth/Jupiter presets · 100% browser-only

Runs locally
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Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

What do you want to solve for?

Central body mass M

Semi-major axis a

Orbital period T

G = 6.674e-11 N·m²/kg²

Result

Orbital period0.9999 years

Circular orbital speed (at r = a)29,790 m/s

Formula steps

T = 2π·√(a³ / (G·M))
T = 2π·√((1.496e+11)³ / (6.674e-11 × 1.989e+30))
T = 2π·√(2.522e+13)
T = 3.155e+7 s

What this tool does

Free orbital period calculator built on Kepler's third law and Newtonian gravity, T = 2π·√(a³ / (G·M)). Enter the central body mass M in kilograms and the orbital semi-major axis a in metres, kilometres or astronomical units, and the tool returns the period in seconds, days or years, with every step of the formula shown so you can check the algebra by hand. It also runs the relationship backwards: give it the central mass and a measured period, and it reverse-solves the semi-major axis from a = ∛(G·M·T² / (4π²)). Alongside the period it reports the circular orbital speed at that radius from v = √(G·M / r), so you can see how fast a satellite or planet has to travel to stay in orbit. Built-in central body presets for the Sun, Earth, Jupiter, Mars and the Moon set up textbook and mission problems in one click. Earth at 1 AU around the Sun returns about one year, exactly as the law predicts. The gravitational constant used is G = 6.674e-11 N·m²/kg². Everything runs in your browser; one-click copy and a shareable URL reproduce your exact problem. 100% client-side, no upload, no account.

Tool details

Input: Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Orbital Period Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Work a physics homework problem and check every step
The textbook asks for the period of a satellite orbiting Earth at a semi-major axis of 6.771e6 m. Pick the Earth preset for M, type 6.771e6 for a with metres selected, and read the period in minutes by switching the time unit. The on-screen steps spell out a³ over G·M and the square root, so you can copy the working into your answer and show how you got there, not just the final number.
Reverse-solve a geostationary orbit altitude
You want the radius of a geostationary satellite, the orbit whose period matches one sidereal day. Switch to Semi-major axis mode, pick Earth, enter 86164 seconds, and the tool returns a ≈ 4.22e7 m. Subtract Earth's 6.371e6 m radius and you have the textbook 35 786 km altitude, derived rather than memorised.
Show a class why outer planets take so long
Set the central body to the Sun and the axis to 1 AU; the result is about one year. Bump the axis to 5.2 AU for Jupiter and the period jumps to about 11.9 years, because T scales as a^1.5. Share the URL with the class and every student opens the exact same problem, a one-click demonstration of T² ∝ a³ that no static slide can match.
Plan or sanity-check a satellite mission
Designing a constellation, you need the orbital speed and period at a chosen altitude. Enter Earth and your semi-major axis, and the tool reports both the period and v = √(G·M / r) in one view. If your mission software gives a period off by orders of magnitude, you have almost certainly mixed up units or confused radius with altitude.

Common pitfalls

Cubing the wrong quantity. The law uses a³ inside the square root, not a². Forgetting the cube, or squaring instead, throws the period off by large factors. This tool always cubes the semi-major axis for you and shows the value in the steps so you can confirm it.
Confusing altitude with semi-major axis. For an orbit around Earth, a is measured from Earth's centre, so it equals the altitude plus Earth's radius (about 6.371e6 m). Feeding the altitude alone instead of the full radius gives a period that is far too short, especially for low orbits.
Mixing units instead of staying in SI. The constant G = 6.674e-11 is defined for kilograms, metres and seconds. Entering kilometres or days straight into the formula without converting throws the answer off by factors of thousands. Use the unit pickers so every input is converted to SI before the math runs.

Privacy

Every calculation here, the period, the reverse-solve for the semi-major axis, the orbital speed, the unit conversions and the formula steps, is plain JavaScript that runs in your browser tab. No masses, distances, periods or results ever leave the page, and nothing you type is logged. The one caveat: the shareable URL encodes your inputs in the query string, so a share link pasted into chat records those numbers in the recipient server's access log. For ordinary physics problems that is harmless; if your figures are sensitive, use the copy button and paste the text instead of the URL.

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Orbital Period Calculator — Kepler's Third Law (T = 2π√(a³/GM))

Result

Formula steps

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Orbital Period Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Work a physics homework problem and check every step

Reverse-solve a geostationary orbit altitude

Show a class why outer planets take so long

Plan or sanity-check a satellite mission

Common pitfalls

Privacy

FAQ

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