Satellite Basics: Orbits, Periods, Elevation and Footprints

Elevation, and why the minimum matters

A user standing under the satellite sees it at 90 degrees, straight overhead. Move away and the angle falls, at first slowly and then fast, until the satellite is on the horizon and then below it.

A system does not use the whole geometric horizon. Close to the horizon the signal travels a long way through the atmosphere, it is blocked by buildings, hills and trees, and it arrives with multipath from the ground: [Multipath, Fading and the Doppler Effect] is at its worst there. So a system specifies a minimum elevation angle, commonly 10 degrees, and the footprint is the area from which the satellite is at least that high.

That minimum is expensive. The program computes what it costs: from geostationary orbit, insisting on 10 degrees rather than 0 shrinks the footprint radius from 9,050 km to 7,952 km, and insisting on 30 degrees shrinks it to 5,841 km. A handheld terminal, which cannot tolerate obstruction at all, may want 30 degrees or more, and that is one of the reasons handheld systems need many satellites.

Figure 104.1 The elevation angle is measured from the user's own horizon, not from the vertical

The four orbits

Figure 104.2 The orbit bands to scale, with the radiation belts that shape the choice

GEO, the geostationary orbit, at 35,786 km above the equator, with a period of one day so that the satellite appears fixed in the sky. MEO, medium earth orbit, a few thousand to about 20,000 km, used by navigation systems. LEO, low earth orbit, a few hundred to about 1,500 km, used by imaging satellites and by communication constellations. And HEO, a highly elliptical orbit, which is not a height at all but a shape: a low perigee and a very high apogee.

The elliptical orbit is worth the program's last section. With a perigee of 1,000 km and an apogee of 39,400 km the period is 12 hours, and because a body moves slowly when it is far away, the satellite spends 8.8 hours of every 12 above 20,000 km. It therefore hangs over one hemisphere for most of its orbit and rushes through the other half in a couple of hours. Three such satellites, spaced in time, keep one of them always high over a high latitude, which is what a geostationary satellite cannot do: from a high latitude, a satellite over the equator is always low in the sky or below the horizon.

Why not every altitude

Two things rule out most of the space between the bands.

The first is the radiation belts, regions where the earth's magnetic field traps charged particles. They are usually given as roughly 2,000 to 6,000 km and roughly 15,000 to 30,000 km, though they have no sharp edges and different books draw them differently. A satellite that spends its life inside one needs heavier shielding and radiation-hardened electronics, so a designer either keeps below the first, aims for the gap, or builds for the radiation.

The second is delay, and it is the price of altitude. The program's last table gives the round trip: 5.2 ms through a satellite at 780 km, 134.8 ms at 20,200 km, 238.7 ms at geostationary height. A conversation over a geostationary link has a noticeable pause; a protocol that waits for an acknowledgement before sending more, as [Traditional Transport Control Protocols: TCP and UDP] does, runs at a fraction of its rate over one.

How long a satellite stays up in your sky

A geostationary satellite never sets: it turns with the earth, so a dish is aimed once and bolted down. Every other orbit moves through the sky, and the program computes how long it is usable. At 780 km a satellite passing straight overhead is above 10 degrees for only 10.4 minutes out of its 100 minute orbit; at 1,200 km, for 14.6 minutes; at 20,200 km, for 264.8 minutes.

Ten minutes is the whole reason a low constellation is hard. A call must be handed from one satellite to the next every few minutes, whether or not the user has moved at all: this is the inter-satellite handover of [Localization and Handover in Satellite Systems], and it is a handover caused by the network moving rather than the user. Many satellites must be in orbit for one to be there at all times, and the ground station must track.

The bands satellites work in

Satellite bands are named by letters that came from wartime radar and stuck. In common use: L around 1 to 2 GHz and S around 2 to 4 GHz for mobile and handheld terminals, where a small antenna can work; C with an uplink near 6 GHz and a downlink near 4 GHz, the oldest fixed-service band, robust in rain; Ku with an uplink near 14 GHz and a downlink near 11 to 12 GHz, used for direct-to-home television, where the higher frequency lets a small dish have the gain that [Antennas: Radiators, Dipoles and Radiation Patterns] computes; and Ka near 30 GHz up and 20 GHz down, which gives the most bandwidth and suffers the most from rain.

The pattern is the one [Frequencies for Radio Transmission] sets out: higher frequency means more bandwidth and smaller antennas, and worse weather. The uplink is always the higher of a pair, because the earth station can afford the power and the satellite cannot.

Orbits, computed

# Orbits: the period, the speed, the elevation angle and how long a satellite stays up.
import math

GM = 3.986004418e14        # earth's gravitational parameter, m3/s2
R = 6378137.0              # equatorial radius, m
C = 299792458.0

def period(alt_km):
    """Kepler's third law for a circular orbit of the given altitude."""
    r = R + alt_km * 1000
    return 2 * math.pi * math.sqrt(r ** 3 / GM)

def speed(alt_km):
    return math.sqrt(GM / (R + alt_km * 1000))

# 1. Why it stays up: at the right speed, gravity supplies exactly the
#    centripetal acceleration the circle needs, and nothing is left over.
print("Why a satellite stays up: gravity is the centripetal force, no more and no less.")
for alt in (780, 20200, 35786):
    r = R + alt * 1000
    v = speed(alt)
    print("  at %6d km: gravity pulls at %.4f m/s2, a circle at %.0f m/s needs %.4f m/s2"
          % (alt, GM / r ** 2, v, v ** 2 / r))
print("  too slow and the orbit falls towards the earth; too fast and it climbs away.")

print()
print("The period and the speed of a circular orbit:")
print("  altitude        period            speed     one-way delay to the sub-satellite point")
for alt in (200, 780, 1200, 10000, 20200, 35786):
    T, v = period(alt), speed(alt)
    print("  %6d km   %8.1f min   %8.0f m/s   %6.1f ms"
          % (alt, T / 60, v, alt * 1000 / C * 1000))

# 2. The elevation angle. A user at earth-central angle gamma from the
#    sub-satellite point sees the satellite at this angle above the horizon.
def elevation(alt_km, gamma_deg):
    r = R + alt_km * 1000
    g = math.radians(gamma_deg)
    # from the plane triangle earth-centre, user, satellite
    return math.degrees(math.atan2(math.cos(g) - R / r, math.sin(g)))

print()
print("How the elevation angle falls as the user moves away from the sub-satellite point:")
print("  ground distance       elevation seen from the ground")
print("  from the sub-point      780 km    20200 km    35786 km")
for gamma in (0, 5, 10, 20, 40, 60, 71):
    arc_km = math.radians(gamma) * R / 1000
    cells = []
    for alt in (780, 20200, 35786):
        e = elevation(alt, gamma)
        cells.append("%8s" % ("%.1f deg" % e if e >= 0 else "below"))
    print("  %5.1f deg %7.0f km %s" % (gamma, arc_km, " ".join(cells)))

# 3. Footprint: the earth-central half-angle for a minimum elevation, and the
#    ground radius that gives.
def footprint(alt_km, elev_deg):
    r = R + alt_km * 1000
    e = math.radians(elev_deg)
    gamma = math.acos(R / r * math.cos(e)) - e
    return math.degrees(gamma), gamma * R / 1000

print()
print("What a minimum elevation costs in footprint (the user must see this high):")
print("  altitude    0 deg minimum       10 deg minimum      30 deg minimum")
for alt in (780, 1200, 20200, 35786):
    out = []
    for e in (0, 10, 30):
        g, radius = footprint(alt, e)
        out.append("%5.1f deg, %5.0f km" % (g, radius))
    print("  %6d km   %s" % (alt, "   ".join(out)))

# 4. How long one satellite stays visible. It sweeps its own footprint at its
#    orbital rate; the earth's rotation is a small correction, ignored here.
print()
print("How long one satellite stays above 10 degrees, passing straight overhead:")
for alt in (780, 1200, 20200):
    T = period(alt)
    g, _ = footprint(alt, 10)
    visible = T * (2 * g / 360.0)
    print("  %6d km   %5.1f min of a %6.1f min orbit" % (alt, visible / 60, T / 60))
print("   35786 km   the satellite keeps pace with the earth, so it never sets")

# 5. An elliptical orbit, the kind used to serve high latitudes. Kepler's
#    equation gives the time spent above any chosen radius.
def time_from_perigee(a, e, r0):
    """Seconds from perigee to the point where the radius reaches r0."""
    cosE = (1 - r0 / a) / e
    cosE = max(-1.0, min(1.0, cosE))
    E = math.acos(cosE)
    return math.sqrt(a ** 3 / GM) * (E - e * math.sin(E))

perigee_km, apogee_km = 1000, 39400
a = R + (perigee_km + apogee_km) / 2.0 * 1000
e = ((apogee_km - perigee_km) * 1000 / 2.0) / a
T = 2 * math.pi * math.sqrt(a ** 3 / GM)
print()
print("An elliptical orbit for high latitudes, perigee %d km and apogee %d km:"
      % (perigee_km, apogee_km))
print("  semi-major axis %.0f km, eccentricity %.3f, period %.1f h"
      % (a / 1000, e, T / 3600))
for above_km in (20000, 30000, 35786):
    t = time_from_perigee(a, e, R + above_km * 1000)
    frac = 1 - 2 * t / T
    print("  it is above %5d km for %.1f h of each orbit, which is %.0f%% of the period"
          % (above_km, frac * T / 3600, frac * 100))
print("  near apogee it moves slowly and hangs over one hemisphere, which is the point:")
print("  a few such satellites serve high latitudes that a geostationary one cannot reach.")

# 6. The delay, and the radiation belts that shape the choice of altitude.
print()
print("Round-trip delay through a satellite, and the altitude bands in use:")
rows = (("LEO", 780, "below the inner radiation belt"),
        ("MEO", 20200, "above the inner belt, in the lower outer belt region"),
        ("GEO", 35786, "above the strongest part of the outer belt, over the equator"))
for name, alt, note in rows:
    hop = 2 * alt * 1000 / C * 1000
    print("  %-4s %6d km   %6.1f ms up and down   %s" % (name, alt, hop, note))
print("  the inner belt is usually given as roughly 2,000 to 6,000 km and the outer as")
print("  roughly 15,000 to 30,000 km; the bands in use are chosen around them, and a")
print("  satellite that must sit inside one is built to withstand the radiation.")

Read the tables against each other and the design of every satellite system is visible in them. Period and speed fall with altitude. Elevation falls away from the sub-satellite point, fast for a low satellite and slowly for a high one: from 780 km a user 2,226 km away sees the satellite at only 8.1 degrees, while from geostationary height the same user sees it at 66.5 degrees. Footprint grows with altitude and shrinks with the minimum elevation demanded. Visibility is minutes for a low satellite and forever for a geostationary one. And delay grows with altitude, which is the bill.

Distinctions

What it does not mean

A satellite is not weightless because gravity is absent. Gravity at geostationary height is still about 0.22 metres per second squared, as the program prints; the satellite is in free fall, which is a different thing.

The period does not depend on the satellite's mass. Two satellites at the same altitude, one a cubesat and one a broadcasting platform, keep exactly the same period.

Elevation is not measured from the vertical. It is measured up from the user's own horizon, so straight overhead is 90 degrees, not 0.

The footprint is not a fixed property of the satellite. It depends on the minimum elevation the system insists on, and the same satellite has a larger footprint for a dish on a roof than for a handset in a street.

A geostationary satellite is not stationary. It travels at about 3,075 metres a second; it only appears fixed because the earth turns underneath it at the same angular rate.

HEO is not a fourth altitude. It is a shape: an ellipse whose apogee is used and whose perigee is passed through quickly.

Quick revision

• Why it stays up: gravity supplies the centripetal acceleration. Speed is the square root of GM over r; period T equals two pi times the square root of r cubed over GM, Kepler's third law. r is measured from the earth's centre.
• Numbers: 780 km gives 7,462 m/s and about 100 min; 20,200 km gives 3,873 m/s and about 719 min; 35,786 km gives 3,075 m/s and 1,436 min, which is 23 hours and 56 minutes.
• Elevation is measured from the user's own horizon; systems set a minimum, often 10 degrees, because low angles mean atmosphere, obstruction and multipath.
• Footprint radius at 10 degrees: 2,077 km from 780 km, 7,384 km from 20,200 km, 7,952 km from geostationary orbit. Demanding 30 degrees instead cuts the geostationary figure to 5,841 km.
• Visibility: 10.4 min at 780 km, 14.6 min at 1,200 km, 264.8 min at 20,200 km, never-setting at geostationary height.
• Delay, round trip: 5.2 ms, 134.8 ms, 238.7 ms for LEO, MEO and GEO.
• Four orbits: GEO, MEO, LEO, HEO. Belts roughly 2,000 to 6,000 km and 15,000 to 30,000 km.
• Bands: L and S for handhelds, C at 6 up and 4 down, Ku at 14 up and 11 down, Ka at 30 up and 20 down. Uplink is always the higher.
• Vocabulary: apogee, perigee, semi-major axis, eccentricity, inclination, sub-satellite point, elevation, footprint, line of sight, uplink, downlink, transponder, inter-satellite link, gateway link, mobile user link.

Test yourself

1. Why does a satellite stay in orbit, and what decides its period? It is in free fall around the earth. At the right speed for its radius the earth's gravity supplies exactly the centripetal acceleration a circular path of that radius needs, so the satellite keeps turning at a constant distance. Equating the two gives a speed equal to the square root of GM divided by r, and therefore a period equal to two pi times the square root of r cubed divided by GM, which is Kepler's third law. The period depends only on the orbit's radius, measured from the earth's centre, and not at all on the satellite's mass.

2. What is the elevation angle, and why does a system set a minimum for it? It is the angle between the user's own horizon and the direction of the satellite, so a satellite straight overhead is at 90 degrees and one on the horizon is at 0. A system sets a minimum, commonly 10 degrees, because at low angles the signal travels a long way through the atmosphere, is easily blocked by buildings, hills and trees, and picks up multipath from the ground. Below that angle the link is not reliable, so it is treated as unavailable.

3. What is a footprint, and what makes it larger or smaller? It is the part of the earth from which the satellite can be seen at or above the chosen minimum elevation, and therefore the area the satellite can serve. It grows with altitude, because a higher satellite sees more of the earth, and it shrinks as the minimum elevation is raised. From geostationary orbit the footprint radius is about 9,050 km if any elevation above the horizon will do, about 7,952 km at a 10 degree minimum, and about 5,841 km at 30 degrees.

4. Compare LEO, MEO and GEO. LEO lies a few hundred to about 1,500 kilometres up, with a period of roughly 90 to 110 minutes, a round trip delay of about 5 milliseconds, and about 10 to 15 minutes of visibility per pass, so it needs many tens of satellites and constant handover; it suits handheld terminals and imaging. MEO reaches up to about 20,000 kilometres, with a period of several hours, about 135 milliseconds of delay and hours of visibility, and about ten to thirty satellites give global coverage; it suits navigation. GEO sits at 35,786 kilometres over the equator with a period of 23 hours 56 minutes, so the satellite appears fixed and a dish never moves, three satellites cover the inhabited world, but the round trip costs about 239 milliseconds and the poles are not covered.

5. What is a highly elliptical orbit for? For serving high latitudes, which a geostationary satellite cannot do because from a high latitude a satellite over the equator is low in the sky or below the horizon. An orbit with a low perigee and a very high apogee has a period that can be chosen, for example twelve hours, and a body moves slowly when it is far from the earth: the chapter's example, with a perigee of 1,000 km and an apogee of 39,400 km, is above 20,000 km for 8.8 hours of each 12 hour orbit. The satellite therefore hangs over one hemisphere for most of its orbit, and a few such satellites keep one always high over the region served.

6. Why are some altitudes avoided? Because of the radiation belts, regions where the earth's magnetic field traps charged particles, usually given as roughly 2,000 to 6,000 km and roughly 15,000 to 30,000 km. A satellite that spends its life inside a belt needs heavier shielding and radiation-hardened parts, which cost mass and money, so designers stay below the first belt, aim for the gap between them, or deliberately build the satellite to withstand the radiation where the orbit is worth it.

7. Why is the uplink frequency always higher than the downlink frequency in a band pair? Because the loss grows with frequency and the two ends are not equally able to pay for it. The earth station can use a large antenna and as much transmitter power as it needs, so it can afford the higher frequency and its greater path loss; the satellite is limited in mass, in antenna size and above all in the power its solar panels provide, so the easier, lower frequency is given to the direction it has to transmit in.