Orbital Velocity Lab Simulator

What Is the Orbital Velocity Lab Simulator?

How to use the orbital velocity simulator

1. Read the four cards before you touch anything. They are already answering for a satellite 400 km above the Earth: Orbital speed 7.673 km/s, Orbital period 92.41 min, Orbital radius 6,771 km and Gravity at that radius 8.694 m/s2. That last figure is the most useful number on the panel, and we will come back to it.
2. Look at the third card’s subline, not just its answer. It reads = R + h = 6,371 + 400 km, and it prints that addition on purpose: the radius this formula wants is measured from the centre of the body, and feeding it an altitude is the single commonest way to get a wrong answer. The card never lets the addition out of sight.
3. Now drag Altitude h above the surface, slowly, and watch the first card fall. At 535 km it reads 7.597 km/s; at 20,180 km, 3.875 km/s; at 42,164 km of radius, 3.075 km/s. One slider, and the counter-intuitive half of the subject: going higher makes you slower. The orbital velocity guide is where that result is derived properly.
4. Watch the orbit pane while you drag, and notice what it refuses to do. The orbit is on one true linear scale fitted to the current ring, and the bottom-left caption prints that scale in kilometres per pixel. The figure follows the width of the pane as well as the orbit, so the stable thing about it is the ratio: between the opening state and the geostationary preset it grows by the same factor of a little over six that the radius did. At 400 km the ring genuinely hugs the disc, because 6,771 km really is only six per cent more than 6,371 km.
5. Keep dragging until a second, cream marker appears. It rides the 400 km orbit of whichever body is selected, and both markers are driven by one shared time compression that the caption prints beneath the scale, such as motion speeded up 6,155x. Because the compression is shared, the lapping you see is the true ratio of the two rates rather than a visual effect.
6. Read the compact row of four cells under the cards. The same speed restates the first card in car units — 27,620 km/h at the opening state — and Gravity left puts the fourth card as a percentage, 89 % of surface gravity. Compared with escape is the one glance this lab takes at its neighbour, reading Escape speed here 10.85 km/s = sqrt(2) x v, and the escape velocity calculator is where that quantity belongs.
7. Read the fourth cell, the one that is a sentence rather than a number. Verdict exists because the formula keeps returning an answer long after the physics has stopped applying. At 200 km it says Low orbit - a real satellite here decays in weeks; at 100 km, Hypothetical - this orbit is inside the atmosphere. The speed card carries on regardless, which is exactly why the sentence is there.
8. Change the body without touching the slider. Press Mars and leave the altitude at 400 km: the speed drops to 3.362 km/s, the period stretches to 118.0 min, and the radius subline rewrites itself as = R + h = 3,389.5 + 400 km. Both the mass and the radius changed, and the card shows you which.
9. Read the sentence strip under the canvas. It restates the thesis in the state you are actually in — at the opening state, Four times this radius would halve the speed to 3.836 km/s — and it grows a second clause wherever the verdict has a caveat to add. It is the only place on the panel that does arithmetic you did not ask for.
10. Use the curve pane to see where you are on the whole curve. It plots speed against radius with the radius axis logarithmic and anchored on the decades, ticks the real orbits of the selected body — on Earth at a desktop width, surface, ISS, GPS, GEO and Moon — and draws your current state as a filled point on it. The ladder stops rather than skips when the next label will not fit, so a narrow pane shows fewer of them and never an arbitrary selection. The surface tick is a limit of the formula, not a place the slider can reach.
11. Press Pause orbit when you want to look rather than watch. It cancels the animation frame outright and leaves a clean static frame, which is the right state for reading the captions and comparing the two ring radii. The button then reads Play orbit, and no reading on the panel changes either way. Reset puts you back on Earth at 400 km with the orbit running.

Dragging is the right way to see which reading answers to the slider, and a poor way to hit an exact figure. When you already have a mass and a radius and want the third quantity typed out, the orbital velocity calculator solves the same relation in all three directions with unit menus on every box. It takes a radius from the centre rather than an altitude, so the addition this panel prints for you is one you do yourself there.

Worked example: change one thing at a time

Nothing in the tables below was worked out here. Each row is a state the lab was put into on this page and then read back from, cell by cell, so every string in them is the panel’s own. The first table moves the slider and nothing else; the second changes the body button and, in two rows, the slider as well.

Read the speed column downwards and the whole page is in it. Every row is higher than the one above it and every row is slower, without a single exception, from 7.848 km/s at the bottom of the slider to 0.9904 km/s at the top. The period column runs the other way, from 86.34 min to 29.84 d.

Rows one and four are the pair that settles the arithmetic. The radius goes from 6,471 km to 6,906 km, seven per cent, and the speed moves by a little over three per cent — roughly half, which is what a square root does. Four times the radius would be exactly half the speed, and the sentence under the canvas prints that figure for whatever row you are standing on.

The geostationary row is the one that has to agree with another page on this site, and it does. A radius of 42,164 km gives 3.075 km/s and 23.93 h, and the geostationary orbit guide reaches the same radius and the same period from a starting point that never uses this formula, with a speed agreeing to all four figures. The altitude reads 35,793 km here because this lab subtracts Earth’s mean radius of 6,371 km, which is the radius its own addition prints.

Rows one and two carry a caveat the speed card does not. At 100 km and 200 km the arithmetic is perfectly good and the orbit is not: the verdict column says so in both rows, in two different sentences. Every other row on Earth sits inside the plain string, because nothing there is trying to fly through air.

The same four cards, four different bodies

The first two rows below share an altitude of 400 km and differ only in which body button is pressed. That is the pair worth pressing yourself, because it is the one that shows the radius card rewriting its own addition.

The third column is the point of the table. At the same 400 km the left-hand term changes from 6,371 to 3,389.5, and the speed changes with it — not because the altitude moved, which it did not, but because both the mass and the radius under it did. An altitude on its own is not an orbit.

The two Moon rows are the other pair, and they share a body. One slider move takes the speed from 1.634 km/s to 0.1105 km/s and the period from 117.8 min to 264.5 d, and the verdict stops saying the orbit is inside anything. The Moon’s pull only dominates out to about 60,000 km, which the top of its slider is well past.

There is a coincidence in the fourth column worth noticing. Mars at 400 km reads 118.0 min and the Moon at 100 km reads 117.8 min, from two bodies whose masses differ by nearly a factor of ten. The period depends on the radius and the mass together, never on either alone, and those two states happen to land in the same place.

Formula and symbol reference

The lab prints its four relations on itself, in the fixed line under the canvas. Written out in the plain-digit form the panel uses:

v = sqrt(GM/r), then T = 2 pi r / v, then a = v2/r = GM/r2, and the one that is easiest to forget, r = R + h measured from the centre. The sublines on the four cards carry the first three of those, each over the reading it produced.

The quantity that never appears is the satellite’s own mass. The panel has no box for it, no slider for it and no card reporting it, and that is a statement rather than an omission: it cancels out of the balance between gravity and the circle, which is why a loose bolt and a space station on the same ring travel at exactly the same speed. The law of gravitation guide is where the inverse-square pull that one half of that balance comes from is set out.

Nine rows follow and only one of them is something you drag. The rest are cards, cells, buttons or quantities the panel uses without showing. For the slider the right-hand column gives the end stops, the step and the published positions; for everything else it gives the strings the lab prints and the span it prints them over.

If you read one row twice, make it h. It is the only entry you can move, its DOM value is a logarithm rather than a distance, and it is the only quantity on the panel that is not part of the physics at all — the formula never sees it, because the radius card adds it to the body radius first.

The physics: why gravity sets one speed per radius

A circular orbit is gravity supplying exactly the force the circle needs, no more and no less. Pick a radius and that condition has one solution, so there is nothing to choose: at 6,771 km round the Earth the answer is 7.673 km/s, and going at anything else means the path is not that circle. The panel has one slider for that reason.

Because the pull falls off with the square of the radius while the circle’s demand falls off with the radius itself, the surviving speed goes as one over the square root of the radius. That is why the first card drops rather than climbs, and why the lab can promise the same thing at every setting: four times the radius, half the speed, exactly. The circular motion guide is where the demand side of that balance is set out on its own terms.

The fourth card is the one that answers the commonest misconception. At the opening state gravity at that radius is 8.694 m/s2, which the compact row restates as 89 % of surface gravity. There is no shortage of gravity 400 km up; an astronaut floats because the whole station is falling, and the orbit is simply a fall that keeps missing.

The two markers are the most valuable thing the drawing does. One shared time compression drives both, so the inner one laps the outer one at the true ratio of their angular rates rather than at a convenient one. Measured off the running lab rather than read off a card: that ratio is 15.54 to one at the geostationary preset, so the cream marker goes round fifteen and a half times for every single circuit of the gold one.

Where the orbital velocity formula breaks down

Every slider position this lab can reach is answered, and answered to the full precision of a double. The arithmetic is not where the panel can let you down; the model behind it is. Six limits sit under every reading on the page, and the verdict cell compresses four of them into one sentence because they belong here, at length.

Below roughly 150 km there is still air, and the orbit does not last

The bottom of the slider, 100 km, returns 7.848 km/s and 86.34 min, and those figures are arithmetically right. They are also useless: the verdict cell reads Hypothetical - this orbit is inside the atmosphere, and the sentence strip adds that at that height the air is thick enough to pull a satellite down in hours to days. The threshold behind that sentence is an approximate figure rather than a published one.

A little higher it lasts, but not without an engine

At 200 km the verdict changes to Low orbit - a real satellite here decays in weeks, which is a different claim from the one below it and deliberately so. Nothing in the four cards notices the difference — 7.789 km/s and 88.35 min are perfectly ordinary readings. Only the sentence knows, and it is approximate too.

Far enough out, the body you chose stops being the only thing pulling

Take the Moon to the top of its slider and the verdict reads Beyond this body’s gravitational domain - a real orbit here would be perturbed away, with 0.1105 km/s still printed beside it. The Moon holds sway out to about 60,000 km and the slider reaches 400,000 km, so the formula is answering a question that has stopped being about the Moon. That figure is an order of magnitude, not a measurement.

Every orbit on this panel is a circle, and most real ones are not

There is one ring in the drawing and there will not be an ellipse. On a real elliptical orbit the speed changes continuously — fastest at the closest approach, slowest at the furthest — so no single figure describes it and this panel would have nothing to print. That case belongs to the guide to Kepler’s laws, and the Kepler’s laws simulator is where it is actually drawn.

Four significant figures is the ceiling, and a fifth digit would be noise

The gravitational constant is known to about five significant figures and the four masses behind the body buttons to four or five, so the product that drives every reading cannot support more. Trailing zeros are kept rather than dropped for the same reason: the surface limit of the curve is 7.910 km/s, and writing it as 7.91 would publish a four-figure result as three. Only the altitude escapes the ceiling, because the radius card prints it inside an addition.

Jupiter has no surface, so its altitudes are measured from a pressure level

Press Jupiter and the body line adds the qualification itself: R 69,911 km (1-bar level, not a surface). Everything the panel prints for that body is correct about the radius; it is the word altitude that is doing something unusual, because there is nothing underneath to stand on. The radius card still prints the addition, = R + h = 69,911 + 1,000 km, and the speed there, 42.27 km/s, is the fastest reading in either table above.

Where orbital velocity is actually used

Knowing how long a low-orbit pass lasts

The ISS preset is the state the lab opens in, and the period card is the reason the station comes back over the same part of the sky so often: 92.41 min a lap, at 7.673 km/s, or 27,620 km/h if that helps. Drag up to Hubble at 535 km and the lap stretches to 95.19 min — three minutes bought with 135 km of altitude.

Why a navigation constellation sits where it does

Press the GPS preset and the period reads 11.96 h, which is close to half a sidereal day, so the same satellites come round to the same place about twice a day. The altitude on that button, 20,180 km, is the nominal figure the constellation is described by rather than an exact one, and the panel is honest about the speed it gives: 3.875 km/s, a little over half the ISS’s.

Why a television dish can be bolted down and left

At a radius of 42,164 km the period card reads 23.93 h and the speed card 3.075 km/s, which is the combination a satellite needs if it is to stay over one spot. The geostationary orbit guide is where that match is argued out; this panel only shows you that the radius and the speed fall out of the same slider as every other row. The geostationary orbit simulator draws the station-keeping itself.

Planning an orbit round something that is not the Earth

The body buttons are there because the formula does not care which body it is given. A 400 km parking orbit round Mars needs 3.362 km/s rather than 7.673 km/s, and a 100 km lunar orbit needs 1.634 km/s, which is why a lander leaving the Moon has so much less to undo. The guide to weight on other planets is the same mass-and-radius pair asked a different question.

Reading a radius back out of a measured speed

Track something, get a speed, and the curve pane turns it into a radius: drag until the first card matches what you measured and read the third. It works because the curve is one-to-one — one speed, one radius, no ambiguity — which is the practical payoff of there being a single solution at each radius. The Kepler’s third law calculator is the route for anyone holding a period instead.

Settling an argument about gravity in space

The fourth card and the gravity cell are the quickest answer there is. At the ISS state they read 8.694 m/s2 and 89 % of surface gravity; even at the geostationary radius there is still 0.2242 m/s2, which is 2.3 % of surface gravity rather than none. The guide to free fall is what being in orbit actually is.

Where to go next

A panel can show you which reading the slider is allowed to move, and that is the end of what a panel is for. Orbital Velocity: Formula and Worked Examples is where the subject itself is written out: the balance set up and solved, the symbol table, the worked problems and the misconceptions. The calculator that takes these figures typed heads the list of related tools below.

Two neighbouring speeds have pages of their own and are not argued here. The escape velocity guide is the speed that leaves rather than circles, which the compact row compares with in a single cell. The law of gravitation guide is where the pull itself comes from, and it is the page this one is built on top of.

Further out, the guide to Kepler’s laws is the elliptical case this lab cannot draw, the centripetal force guide is the demand side of the balance in newtons, and the gravitational potential energy guide is what climbing to a higher orbit actually costs — which is the question this panel does not answer.

Nearer home, the universal gravitation lab is the inverse-square pull before any orbit is asked of it, the circular motion lab is a circle held by a string rather than by gravity, and the escape velocity lab is the neighbouring speed drawn as a launch. Beyond that there is the simulation library to work through, an index of calculators for when typing beats dragging, and the blog for everything written out at length.