An orbit is the one thing about a satellite that a radio engineer cannot negotiate. It fixes how far away the satellite is and how fast it crosses the sky. It also fixes how long the satellite stays in view, and how far the carrier frequency shifts while it does. Every link budget, every handover rule and every timing margin in a satellite system is downstream of six numbers that describe where the satellite goes.

Those six numbers are the classical orbital elements, and the links at the foot of this page cover them from several directions. What follows puts them in order, derives the altitudes and periods that matter to a communication link, and works out what each orbit class costs the radio.

• What fixes an orbit : the six elements
• Altitude and period
• LEO, MEO and GEO from the radio side
• Inclination, latitude and launch azimuth
• Simulator
• YouTube
• Reference

What fixes an orbit : the six elements

Six numbers describe an orbit completely, and the count is not arbitrary. A body moving under gravity alone is fixed by its position and its velocity, which is three numbers each. The classical elements carry those same six degrees of freedom. They are written in a form that separates the shape of the path from where the path sits in space.

That separation is what makes them worth learning. Change one element and exactly one thing about the orbit changes, which is not true of position and velocity. Figure 1 splits five of the six into the two groups they fall into.

Figure 1. Five of the six elements, split by what they do. The left panel fixes the curve itself and the right panel fixes where that curve sits in space. The sixth element says where on the curve the satellite is at a given moment, which no drawing can show.

• The left panel has no orientation in it : the ellipse is drawn in its own plane. Here a and e settle its size and its stretch, and neither says which way the plane faces.
• The Earth sits at one focus and not at the centre : that is what makes perigee and apogee different distances, and the gap between them grows with e.
• The line of nodes is where the two planes meet : the ascending node is the end the satellite crosses going north. Both the inclination and the RAAN are measured from there.
• The argument of perigee turns the ellipse within its plane : the plane is already fixed by then, so this element only decides where the low point falls.
• The sixth element is the clock : the other five stay put while the true anomaly runs from 0 to 360 degrees once per orbit.

The table below gives all six, in the order the elements are usually listed. Read the third column first, because knowing what each one fixes is what makes a set of elements readable at a glance.

The grouping falls out of that column. Two elements describe the curve, two place its plane, one turns the ellipse inside that plane, and one says where the satellite is along it. A communication engineer usually cares most about the first and the third of those groups. The semi-major axis sets the delay, and the inclination sets which latitudes ever see the satellite.

• Six numbers, one for each degree of freedom : the elements carry the same information as position and velocity, arranged so that each one means something.
• Only the true anomaly changes on its own : the other five stay constant for an ideal two body orbit. That is why a short fixed list describes a satellite.
• Real orbits drift : the Earth is neither a point mass nor alone. RAAN and the argument of perigee move slowly, so the elements have to be refreshed.
• Two of the six drive the radio link : the semi-major axis, through distance and period. And the inclination, through which ground stations see the satellite at all.

Altitude and period

One of the six elements decides how long an orbit takes, and the other five have no say in it. The period follows the semi-major axis alone. Two satellites on wildly different looking paths therefore complete a lap in the same time, provided their ellipses are the same size. That single fact settles most of what a link designer needs.

The relation is Kepler's third law. It reads T = 2 pi times the square root of a cubed over mu, where mu is the gravitational parameter of the Earth. Figure 2 plots it against altitude rather than semi-major axis, since altitude is what a satellite is usually quoted by.

Figure 2. Orbital period against altitude, computed from Kepler's third law. The geostationary altitude is not a chosen round number. It is the one altitude where the curve crosses the sidereal day, and everything else about the orbit follows from that crossing.

• The curve is steep and it never turns over : doubling the altitude of a low orbit adds only a few minutes. The last stretch out to GEO multiplies the period by more than ten.
• Eccentricity and inclination are absent from the formula : only the size of the ellipse enters it. An elliptical orbit and a circular one of the same semi-major axis therefore share a period.
• The dashed line is the sidereal day, not the solar day : the orbit has to match the Earth turning against the stars. That takes 1436 minutes rather than 1440.
• The crossing is what defines GEO : the geostationary altitude is read off the graph rather than chosen, and it comes to 35786 km above the surface.

The geostationary case is worth doing explicitly, because it is the one orbit whose altitude is forced. Rearranging Kepler gives a radius of the cube root of mu times T squared over four pi squared. Substituting the sidereal day gives 42164 km from the centre of the Earth, and subtracting the equatorial radius of 6378 km leaves 35786 km of altitude.

The four minute difference between the two definitions of a day is not a rounding detail. Using 86400 seconds instead of 86164 puts the answer at 35863 km, which is 77 km too high. A satellite placed there would drift steadily east rather than hold station. The number the rest of this site quotes, 35,786 km, is the sidereal answer.

The table below carries the same arithmetic to the altitudes a communication engineer meets. The delay column is the one way path at zenith, which is the best case. The next section covers what happens when the satellite sits low in the sky instead.

• Below GEO the satellite moves across the sky : any period shorter than a day drifts the ground track. Coverage then becomes a pass, with a beginning and an end.
• Speed falls as altitude rises : 7.6 km/s at 550 km against 3.1 km/s at GEO, which is why a low satellite crosses the sky in minutes.
• Delay is set by geometry and nothing else : 1.8 ms overhead from 550 km against 119 ms from GEO, before any equipment is switched on.
• The middle of the range is thinly used : the radiation belts sit between the low shells and the GNSS altitude. That is one reason the table jumps rather than steps.

LEO, MEO and GEO from the radio side

Distance and period are orbital facts. A radio engineer designs around what those facts turn into. Three things follow : how long a bit takes to arrive, how far the carrier moves while it does, and how long the satellite stays up before another takes over. The three orbit classes sit at very different points on all three.

The table gathers them. The Doppler and time in view columns assume an overhead pass with the Earth's rotation left out. That is a small correction at these speeds, and it keeps the geometry readable.

• The delay columns are not the same ratio : overhead, a low satellite beats GEO by a factor of 65. At 10 degrees elevation the factor drops to 22, so low orbits lose much of their advantage near the horizon.
• Doppler scales with the carrier : the same 550 km pass shifts it by tens of kHz at 2 GHz and by hundreds at 20 GHz. The shift is a fraction of the frequency rather than a fixed offset.
• The receiver sees twice the tabulated figure : the shift runs from plus to minus across a pass. A demodulator therefore searches double the number in the column.
• Time in view is the operational cost : eight minutes at 550 km means a terminal changes satellite continuously, while GEO never changes at all.

The geostationary row is different in kind rather than in degree. Give a satellite a period of one sidereal day and an inclination of zero, and it hangs over one point on the equator. That removes the pass, the handover and the Doppler from orbital motion all at once. What is left is the distance, and the distance is the whole problem.

A single hop up and back is 239 ms. A request and its reply cross the link four times, so about 477 ms passes before anything is processed. That sets a floor no protocol can negotiate away, and it is the reason interactive traffic over GEO feels different from interactive traffic over a low orbit.

Coverage runs the other way. From GEO the Earth subtends only 8.7 degrees, and one satellite can see 71 degrees of Earth central angle above a 10 degree elevation. Three satellites spaced 120 degrees apart therefore cover the whole equatorial belt with margin. The same 10 degree limit leaves everything beyond 71 degrees of latitude with no geostationary satellite in sight.

A low orbit inverts every one of those. The delay is small, the pass is short and the Doppler is large. One satellite covers 15 degrees of central angle rather than 71, so continuous service needs a constellation rather than a few spacecraft. The Starlink, OneWeb, Iridium and Kuiper pages cover what each operator chose. The NTN page covers how 3GPP handles the delay and the Doppler.

• GEO trades delay for simplicity : 119 ms each way and 477 ms for a round trip with a reply. Against that, no handover, no Doppler and a fixed antenna pointing.
• LEO trades simplicity for delay : a few milliseconds each way. Against that, a satellite to reacquire every few minutes, and a carrier that moves while it is tracked.
• Three satellites cover the equator and none covers the poles : the 71 degree central angle is generous in longitude and abrupt in latitude.
• MEO sits between the two on every row : some Doppler search rather than none, hours in view rather than minutes, and 67 ms of delay rather than 2 or 119.

Inclination, latitude and launch azimuth

Inclination is the element that decides who can see the satellite at all. A circular orbit at inclination i passes over every latitude from minus i to plus i, and over none beyond them. Choosing an inclination is therefore choosing which part of the world the system serves. Figure 3 shows three of them over the same ground.

Figure 3. Ground tracks for three inclinations at the same altitude, over one and a half orbits. Each track turns back at the latitude equal to its inclination, which is the dashed line. The westward slide between passes is the Earth turning underneath.

• Each track turns back exactly at its own inclination : the dashed lines are at plus and minus i, and the track touches them and returns. The inclination caps the latitude absolutely.
• The equatorial track is a straight line : an inclination of zero keeps the satellite on the equator. A geostationary satellite does that, and matches its period to the day as well.
• Successive passes slide west : the satellite returns to the same point in its orbit after 96 minutes. The Earth turns about 24 degrees of longitude underneath it in that time.
• The near polar track covers everything : at an inclination close to 98 degrees the ground track reaches within a couple of degrees of both poles. Earth observation uses that band for exactly that reason.

Choosing an inclination is not free, because the launch site constrains it. The orbital plane has to pass through the launch point, and through the centre of the Earth as well. A site at latitude L therefore cannot reach an inclination below L. The relation is short. The cosine of the inclination equals the cosine of the latitude times the sine of the launch azimuth, with azimuth measured clockwise from north.

Two cases bound it. Launching due east makes the sine one, so the inclination equals the latitude and that is the lowest value available. Launching due north or due south makes the sine zero, so the inclination is 90 degrees and the orbit is polar. Everything between those is reachable by aiming the azimuth in between.

The last column explains why due east is also the cheapest direction. The surface of the Earth already moves east at 0.465 km/s at the equator. A launch in that direction keeps all of it, which is a real fraction of the 7.6 km/s a low orbit needs. That speed falls as the cosine of the latitude, so a high latitude site is penalised twice over. It cannot reach a low inclination, and it gets less help on the way up.

Going the other way costs more still. Raising the inclination after launch is possible and expensive. A plane change needs a velocity change proportional to the orbital speed itself, not to the small difference in direction. That is why inclination is chosen before the rocket leaves rather than adjusted afterwards.

One inclination has a name of its own. Near 98 degrees the orbit is slightly retrograde, and the Earth's equatorial bulge then drags the ascending node round at just under one degree per day. That is the rate the Earth itself moves around the Sun. A satellite there crosses the equator at the same local solar time on every pass, which is what sun synchronous means. The exact value depends on altitude, running from 97.0 degrees at 400 km to 98.6 degrees at 800 km.

• Inclination equals the highest latitude reached : it is the coverage element, and no amount of power or antenna gain extends a satellite past it.
• The launch latitude is a floor on the inclination : the cosine relation gives i equal to the latitude for a due east launch, and more for any other azimuth.
• Due east is both the lowest inclination and the cheapest launch : the Earth contributes 0.465 km/s at the equator, and less as the latitude rises.
• Sun synchronous is a drift rate, not a shape : near 98 degrees the nodal precession matches the Earth's year. The local time of each pass then stays fixed.

Simulator

A simulator is the quickest way to see what the six elements do. Changing one at a time and watching the ground track move settles in a minute what a paragraph takes longer to say. The one below runs in a browser and takes the elements directly.

• HOMA - Online Space Orbit Simulator

The useful discipline with any of these is to move one element at a time and predict the result before looking. The table below lists what each control should do if the grouping in the first section is right. A prediction that fails is the fastest way to find the element that was misunderstood.

Two of those rows are worth doing carefully, because they are the ones most often confused. Changing the RAAN and changing the inclination both move the plane, and only the inclination changes the latitudes the satellite reaches. Changing the argument of perigee and changing the true anomaly both move something along the ellipse. Only the true anomaly moves the satellite rather than the ellipse.

A two body simulator will not show drift. The elements it takes are constant by construction, so the RAAN stays where it is put and the sun synchronous condition from the previous section never appears. Real elements are refreshed against measurements for exactly that reason.

• Move one element at a time : the grouping in the first section is only useful if each control changes one thing. That is quick to check.
• Two pairs are easy to confuse : inclination against RAAN, and argument of perigee against true anomaly. Each pair looks similar and does something different.
• Eccentricity does not touch the period : stretching the orbit at a fixed semi-major axis leaves the lap time alone. That is the least intuitive consequence of Kepler's third law.
• An ideal simulator has no drift in it : the precession behind sun synchronous orbits comes from the Earth not being a point mass. A two body model leaves it out.

YouTube

• Classical/Keplerian Orbital Elements - Spaceflight Science (2013)
• Basic Introduction To Satellite Communications | Satellite Communications - ENGINEERING TUTORIAL (2020)
• Orbit Inclination, Launch Azimuth and Latitude - Christopher Scott Vaughen (2021)

Reference

• Describing Orbits  
• Section 4.3 - The Six Orbital Elements - Astronomical Returs
• Orbital elements - wikipedia
• Parameters Describing Elliptical Orbits