Orbital Velocity Calculator

What Is the Orbital Velocity Calculator?

How to use the orbital velocity calculator

1. Choose the unknown. The Solve for menu opens on Orbital speed, and the other two choices are Central mass and Orbital radius, from the centre. Any two of the three fix the third.
2. Type the central mass. It takes kg, Earth masses or solar masses and opens on 5.9722e+24 kg. It is the mass of the body being orbited, never the mass of the thing orbiting it.
3. Type the radius from the centre. It takes km, m, Earth radii or au and opens on 6771 km. Add the body’s own radius to the altitude: 6371 + 400 for the ISS.
4. Type the speed. It takes km/s, m/s, km/h or mph and opens on 7.673 km/s. It is an input only when you are solving for the mass or the radius.
5. Read the answer. The headline carries four significant figures in whichever unit that box was showing. Set the unit while the quantity is still an input, then point Solve for back at it.
6. Read the period chip. It is T = 2 pi r / v, printed in the unit the subject uses: 92.41 min at the defaults, 23.93 h at geostationary radius, 27.45 d at the distance of the Moon.
7. Read the gravity chip. It is a = v2/r = G M/r2, the gravitational field strength at that radius and the centripetal acceleration the orbit needs at once. At the defaults it reads 8.694 m/s2.
8. Read the speed chip. It reprints the speed in km/s to four figures with the trailing zeros kept, which the headline cannot do: a surface-skimming orbit is 7.910 km/s and the headline can only render that as 7.91.
9. Read the escape-speed chip. One comparison, once: the escape speed at the same radius is sqrt(2) times the orbital speed. The chip links the page that owns it.
10. Open Show working. The steps cancel the orbiting mass, list your figures in SI, rearrange, substitute one stage at a time, give the period and the field strength, and finish with what four times your radius would do to your speed.

This page exists for one step: turning a mass and a radius into the single speed that radius allows. The orbital velocity guide is the prose half of it, with the derivation set out line by line, eight worked problems and the misconceptions in full. Where the G M m / r2 on the left of the balance comes from is a different question and a different page: the law of gravitation guide is the inverse-square law this formula starts from, and nothing here derives it.

Three mistakes account for most wrong answers here. Typing an altitude into the radius box — 400 instead of 6771 — returns 31.57 km/s, more than four times the real answer, because the formula will happily work out a speed for a radius inside the planet. Entering the mass of the satellite rather than the mass of the Earth is the next, when the satellite mass cancels and has no box at all. Last comes expecting the inverse directions to hand back exactly the numbers you started from, which a four-figure speed cannot do.

Worked example: change one thing at a time

The table starts at the defaults, moves the unknown through all three quantities, then raises the orbit one step at a time from the surface out to the Moon, changes only the body, walks all three unit menus, and finally reaches the entries the calculator declines. Rows 1 to 3 are one orbit solved three ways. Every answer and chip cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 3 are one orbit with the Solve for menu in each of its three positions, and they deliberately do not agree in every digit. The forward direction turns 5.9722e24 kg at 6771 km into 7.673 km/s; feeding that four-figure speed back returns 5.973e+24 kg and 6770 km, because both inverse directions go as the square of the speed and 7.673 carries about 5 parts in 100,000 of rounding. The forward direction is the one to trust when you have a mass, and the working says so rather than hiding a fifth digit in a default to make the table tidy.

Rows 4 to 7 are the thesis of the whole page as one falling column. A surface-skimming orbit would need 7.910 km/s, 200 km up needs 7.789 km/s, a Hubble-class orbit needs 7.597 km/s, and four times the ISS radius needs only 3.836 km/s. The halving is exact in the arithmetic even though the two rounded strings are not exactly two to one, which is why the working prints the halved figure for whatever radius you typed rather than inviting you to divide the printed one.

Row 4 is a limit rather than a place. It is also the one row where the headline and the speed chip differ: the chip reads 7.910 km/s and the headline can only render those four figures as 7.91, because the engine drops a trailing zero. A real object at a radius of 6371 km would be inside the atmosphere, and the formula has no way of knowing that.

Rows 8 to 10 climb out of low orbit, and the period unit climbs with them: 11.96 h, 23.93 h, then 27.45 d. The switch from minutes to hours happens at two hours rather than one, because every source quotes a low orbit in minutes and a high one in hours. The speeds fall the whole way, from 3.875 km/s to 1.018 km/s.

Rows 11 to 13 change only the body. Four hundred kilometres above Mars is 3.362 km/s where the same altitude above Earth is 7.673 km/s; a hundred kilometres above the Moon is 1.634 km/s; and a thousand kilometres above the 1-bar level of Jupiter — which has no surface, so that is the level meant — is 42.27 km/s. Rows 11 and 12 have periods within a quarter of a minute of one another, 118.0 min against 117.8 min, from two bodies that share nothing: a coincidence, and a reminder that the period depends on the body only through G M and r.

Rows 15 to 18 change nothing but the units, and every one returns the same strings. One Earth mass is the 5.9722e24 kg in the box, 6771000 m is the 6771 km, and one solar mass at one astronomical unit reproduces row 14 exactly. Row 17 is the neatest of them: a radius of exactly one Earth radius is the surface-skimming orbit of row 4.

Rows 19 and 20 are one answer in different clothes. The same orbit reads 7673 m/s and 17160 mph, and the headline shows whichever unit the speed box was set to before it became the unknown. Choose the unit while the quantity is still an input, then point Solve for back at it.

Rows 21 and 22 are the two inverse directions doing real work. The distance and speed of the Moon weigh the Earth at 5.969e+24 kg, which is its mass to three significant figures from two quantities an astronomer can measure; and a tracked speed of 3.875 km/s places a satellite 26550 km from the centre with a period of 11.96 h. Neither recovers the fourth figure, for the reason rows 1 to 3 showed.

Rows 23 to 29 sit outside the domain and are declined there rather than answered. A zero or negative mass is not a body, a zero radius is a division by zero, a negative radius is not a distance, and a zero or negative speed has no circular orbit behind it. The last of them is anything that parses to infinity, which has to be tested for explicitly: Infinity > 0 is true, so a sign test on its own would publish it.

The no answer rows are what the widget actually does with those entries: the result area prints That combination has no valid solution. Check your inputs.

No chips are drawn beside that message, which is why the last two columns of those rows are hyphens rather than figures. It is a refusal rather than a crash, and nothing on this page ever returns a NaN or an infinity.

One thing the table cannot show is the entry the page does not refuse. A radius smaller than the central body still returns a speed, because there is no radius-of-the-body input for it to be compared against. The formula does not know where any surface is, and recognising that is the reader’s job.

Formula and symbol reference

One rearrangement does all the work. Gravity pulls with G M m / r2 and a circle of radius r travelled at speed v demands m v2 / r; setting them equal cancels the orbiting mass m and leaves v = sqrt(G M / r). The other two positions of the Solve for menu are that same line rearranged: M = v2 r / G and r = G M / v2.

Two quantities follow from the answer rather than needing their own formula. The period is the circumference divided by the speed, T = 2 pi r / v — substitute the speed into it and you get T2 = 4 pi2 r3 / (G M), which is Kepler’s third law for a circle, and the Kepler’s third law calculator is where the general form with a semi-major axis belongs. The field strength is a = v2/r = G M/r2, which is the gravitational acceleration at that radius and the centripetal acceleration the orbit needs, equal because the balance says so; the centripetal force calculator is the force side of that same balance for any circular motion.

One comparison with the neighbour, and then nothing more about it. At the same radius the escape speed is exactly sqrt(2) times the orbital speed, so the 7.673 km/s of the default state sits beneath an escape speed of 10.85 km/s — the escape velocity guide and the escape velocity calculator are where that quantity is derived and worked, and this page does neither.

The physics: why gravity sets one speed per radius

A circular orbit is not a balance between gravity and some outward force; there is no outward force. It is gravity being exactly the right size to bend a straight line into a circle of that radius at that speed, and nothing else. Write the two expressions down and set them equal, and the whole of this page falls out.

Gravity pulls the orbiting body inwards with G M m / r2, from the inverse-square law that the law of gravitation guide sets out. Travelling a circle of radius r at speed v requires a centre-pointing force of m v2 / r, which is what the circular motion guide derives. Setting G M m / r2 = m v2 / r and dividing both sides by m leaves G M / r2 = v2 / r, and one more step gives v = sqrt(G M / r).

That cancellation is the most important line on the page. The m that disappeared is the mass of the satellite, so the formula that remains cannot possibly depend on it: a loose bolt and a space station on the same circle travel at exactly the same speed. It is also why the calculator has no box for the orbiting mass, and why the answer is a property of the orbit rather than of the thing in it.

The second thing the formula says is the one most people have backwards. Because r sits under a square root in a denominator, the speed falls as the orbit rises: four times the radius is exactly half the speed, which the working prints for whatever radius you typed. Press The ISS, 400 km up and then The Moon, 384,400 km out and the headline drops from 7.673 km/s to 1.018 km/s while the period stretches from 92.41 min to 27.45 d. Getting to the higher orbit still costs more energy, because the climb has to be paid for; the orbit you arrive on is simply slower.

The third thing is hiding in the gravity chip. Rearranging the same balance gives a = v2/r = G M/r2, so the number the chip prints is both the acceleration the circle needs and the gravitational field strength at that radius — they are equal by construction. At the default state that is 8.694 m/s2, and an astronaut floats in it because the whole station is falling with them.

Where the orbital velocity calculator breaks down

The arithmetic here is exact to the last digit a double can hold. What goes wrong is physical: a circle the formula assumes and a real orbit only approximates, a single central body that stops being the only thing that matters a long way out, and an atmosphere the formula has never heard of.

Circles only, and a real orbit is an ellipse

Every answer here assumes a circular orbit, where the speed is constant and gravity is always at right angles to the motion. A real orbit is an ellipse: the body moves fastest at its closest approach and slowest at its furthest point, and no single speed describes it. The guide to Kepler’s laws is where that case belongs. For a nearly circular orbit this formula at the mean radius is a good approximation and not an exact answer, which is why the Moon row comes out at 1.018 km/s against a measured mean of about 1.022.

Nothing you can type tells the page there is an atmosphere

The surface-skimming row of the table is a textbook limit, not a place to put a satellite: 7.910 km/s at a radius of 6371 km is the right answer to the question the formula was asked, and a real object there would burn. Below roughly 150 to 200 km a circular orbit decays in days to weeks. Those altitudes are order-of-magnitude statements rather than measured thresholds — treat them as “about” — and there is no box for atmospheric drag.

One central body, which stops being true a long way out

The formula knows about exactly one mass. Far enough from the central body, other bodies matter more than it does, and the number the page returns stops describing anything: Earth keeps gravitational control out to roughly 1.5 million km, but the Moon only to roughly 60,000 km, so a “lunar orbit” at 400,000 km is not an orbit at all. Both of those figures are order-of-magnitude statements this build did not fetch from a primary source. Verify before use.

Four significant figures, and the fifth is not available

The gravitational constant is known to about five significant figures and the masses on this page to four or five, so their product cannot support more than four. That is why every speed here stops at four figures: 7.673 km/s is honest and 7672.6 m/s claims a precision the inputs cannot carry. It is a cap on the inputs rather than on the arithmetic, which is exact.

The inverse directions square the speed, so they lose the fourth figure

Solve for the mass or the radius and the rounding in the speed is amplified: four significant figures carry between 5 parts in 100,000 and 5 parts in 10,000 depending on the leading digit, and both the mass and the radius go as the square of that. Type 5.9722e24 kg at 6771 km and the speed comes back 7.673 km/s; type that speed back and the mass reads 5.973e+24 kg and the radius 6770 km. Nothing is wrong: both figures are right to the precision a four-figure speed carries.

A radius inside the body is answered, not refused

The only radius the page declines is zero or negative. Type a radius smaller than the body and it will still return a speed, because the formula has no idea where the surface is — there is no radius-of-the-body input, so there is nothing for it to compare against. An orbit below a surface is not physical, and recognising that is the reader’s job rather than the widget’s.

The radius is from the centre, and nothing on the page can check that you did it

There is no altitude box, deliberately: a second input for the body’s own radius would be needed in all three solve directions. So r = R + h is an addition you do before you type. The label, the note under the widget, the Earth radii unit and the presets all push you towards it, but a radius of 400 km is a perfectly valid entry and the page answers it with 31.57 km/s, which is 4.11 times the real figure.

No orbital energy, no transfer, no burn

This page answers what speed a given circle needs. It does not say how to get there, what it costs, or what happens between two orbits, and the energy of an orbit is not computed anywhere here. A satellite that fires its engine does not simply speed up in place; it changes orbit, and the new circle is slower if it is higher. That is a different calculation on a different page.

Six entries are declined in words rather than answered

A zero or negative central mass and a zero or negative radius in the speed direction; a zero or negative speed in the mass and radius directions. Each would produce an infinity, a negative length or a speed for something that is not a body, so the page says so instead. An entry large enough to parse as infinity is refused as well, which makes seven refusals in the table rather than six.

The calculator measures nothing — you supply the numbers

Two quantities go in, one balance is rearranged, and the answer describes the circular orbit you described. The preset names are shorthand for the two or three figures behind them and are not claims about any particular spacecraft, mission or constellation.

Where the orbital velocity calculator is actually used

Sizing a low Earth orbit before anything else is decided

Almost every question about a satellite starts with how fast it has to go and how long a lap takes. Two hundred kilometres up is 7.789 km/s and 88.35 min; four hundred, where the space station flies, is 7.673 km/s and 92.41 min; a Hubble-class orbit, about 535 km up, is 7.597 km/s and 95.19 min. The speeds barely move across that band while the lap times do, which is why a low orbit is quoted by its period.

Checking a quoted altitude against a quoted period

The two are not independent, so one checks the other. Press The geostationary circle and the period comes back 23.93 h, which is one sidereal day — the geostationary orbit guide is the page that explains why that is the number a television dish needs rather than 24 hours. If a figure you have been given for an altitude and a figure for a period disagree here, one of them is wrong.

Weighing a planet from the outside

Point Solve for at the central mass and the question inverts into the measurement that actually gets made. Nobody weighs the Earth; they watch the Moon go round it. A radius of 384,400 km and a speed of 1.018 km/s return 5.969e+24 kg, which is the mass of the Earth to three significant figures from two numbers an astronomer can measure and the constant G.

Identifying an orbit from a measured speed

Point Solve for at the radius and a tracked speed becomes an altitude. A satellite measured at 3.875 km/s about the Earth is at 26550 km from the centre, which is a little over 20,000 km up: navigation-constellation territory, where the period of 11.96 h is close to half a sidereal day so the pattern of satellites repeats about twice a day. The nominal figures are close to exact and this page does not claim they are exact.

Taking the same question to another body

Only the central mass changes. Four hundred kilometres above Mars is 3.362 km/s, where four hundred above Earth is 7.673 km/s, because Mars has about a tenth of the mass and a smaller radius to add. A hundred kilometres above the Moon gives 1.634 km/s and a period of 117.8 min — within a quarter of a minute of the Martian orbit above, from two wildly different bodies, which is a coincidence worth noticing and not a result.

Settling the “no gravity in space” argument with a number

The gravity chip is the fastest answer to it. At the radius of the space station the field is 8.694 m/s2, about nine tenths of the value at the ground, and at geostationary radius it is still 0.2242 m/s2 rather than zero. Weightlessness is free fall, not the absence of gravity, and the chip puts a figure on it at whatever radius you are looking at.

Sizing the next quantity on a neighbouring tool

This page stops at the speed a circle needs. The escape velocity calculator is the speed that leaves instead of circling, and the physics lab library has the interactive versions of all of this. The two calculators listed at the foot of the page take the same orbit on to its period and to the pull in newtons.

Where to go next

For the derivation line by line, eight worked problems of rising difficulty and the misconceptions in full, read the orbital velocity guide, which this tool is the arithmetic half of. The law of gravitation guide is the inverse-square law the derivation starts from, the circular motion guide is the other half of the balance, and the escape velocity guide is the speed that leaves rather than circles.

Six more posts pick up what this page deliberately leaves out. The guide to Kepler’s laws is the elliptical case, the geostationary orbit guide is the one radius whose period is a sidereal day, the centripetal force guide is the force this balance sets equal to gravity, and the gravitational potential energy guide is what it costs to climb to a higher orbit in the first place. The free fall guide is what an orbiting astronaut is actually doing, and the weight and mass guide is why the gravity chip is not zero.

Four tools sit next to this one. The escape velocity calculator is the speed that escapes; the Kepler’s third law calculator relates a period to a semi-major axis; the centripetal force calculator and the circular motion calculator handle the circle itself once you already have a speed.

The escape velocity simulator draws the two speeds side by side, the geostationary orbit simulator is the one radius that stays over a point on the ground, the Kepler’s laws simulator shows an ellipse changing speed as it goes round, and the universal gravitation simulator is the inverse-square law on its own.

The full physics lab library and calculator index are open too.