LEO and MEO

Handover, constantly. Because a satellite is usable for at best ten minutes, a thirty minute call on a 780 km constellation is handed to the next satellite at least three times, and more when the pass is not overhead. This is a handover with no counterpart in a terrestrial network: the user has not moved at all, the network has. [Localization and Handover in Satellite Systems] gives it a name and a procedure.

Doppler, constantly. A satellite closing at 6,548 metres a second shifts a 1,620 MHz carrier by 35.4 kHz, and the shift reverses sign as the satellite passes over. A receiver must search a wide band to find the carrier at all and then track it as it slides, which is real work and real power: [Multipath, Fading and the Doppler Effect] sets out the mechanism, and a fixed dish on a geostationary satellite never meets it.

The systems. The classification in common use divides them by what they carry. Little LEO systems work below 1 GHz at low data rates, for messaging, tracking and telemetry. Big LEO systems carry voice: Iridium, 66 satellites in near-polar orbit at about 780 km, notable for carrying traffic between satellites rather than dropping it to the ground at every hop, which [Routing in Satellite Systems] takes up; and Globalstar, 48 satellites at about 1,414 km in inclined orbits, which does the opposite and relays each call straight down to a gateway, so a call only works where a gateway is also in view. Broadband LEO constellations fly lower still, around 550 km, with thousands of satellites and steered beams, trading an enormous constellation for low delay and high capacity.

Medium earth orbit

MEO is the band between the belts and below geostationary orbit, and in practice it means about 19,000 to 23,000 km, where the navigation systems are. From the program, GPS at 20,200 km has a period of 718.7 minutes, which is close to half a day, and a satellite is usable for 264.8 minutes, over four hours at a time. One satellite sees 29.9 per cent of the earth.

That last number is the point of MEO, and it is a different point from LEO's. A navigation receiver does not want one satellite in view; it wants four, for the reason [Applications of Satellite Systems] gives, three coordinates and the receiver's own clock error. Spread 24 satellites evenly and the expected number above ten degrees is, by the program, 7.2. That is a comfortable margin over four, which is why GPS is 24 satellites and not 12: the requirement is not coverage but redundancy of geometry, everywhere, all the time.

The systems: GPS, nominally 24 satellites in six planes at about 20,200 km; GLONASS at about 19,100 km; Galileo at about 23,222 km, where the program gives 7.5 satellites in view on average; BeiDou, which mixes medium, geostationary and inclined geosynchronous orbits; and India's NavIC, which is regional and uses geostationary and inclined geosynchronous satellites to serve India and its neighbourhood rather than the world. A medium orbit is also used for broadband by a small constellation at about 8,000 km, which sits between the delay of geostationary orbit and the constellation size of a low one.

The orbits, computed

# Low and medium orbits: what they buy, what they cost, and how many it takes.
import math

GM, R = 3.986004418e14, 6378137.0
C = 299792458.0

def period(alt_km):
    return 2 * math.pi * math.sqrt((R + alt_km * 1000) ** 3 / GM)

def cap(alt_km, elev_deg=10.0):
    """Earth-central half-angle, and the fraction of the earth, in view."""
    r = R + alt_km * 1000
    e = math.radians(elev_deg)
    g = math.acos(R / r * math.cos(e)) - e
    return g, (1 - math.cos(g)) / 2.0

def slant(alt_km, gamma):
    r = R + alt_km * 1000
    return math.sqrt(R * R + r * r - 2 * R * r * math.cos(gamma))

# 1. The orbits themselves.
print("The orbits, with a ten degree minimum elevation:")
print("  system          altitude   period    visible per pass   share of the earth in view")
for name, alt in (("Iridium", 780), ("Globalstar", 1414), ("a broadband LEO", 550),
                  ("GPS", 20200), ("GLONASS", 19100), ("Galileo", 23222)):
    T = period(alt)
    g, frac = cap(alt)
    print("  %-15s %6d km %7.1f min %10.1f min %14.1f%%"
          % (name, alt, T / 60, T * (2 * math.degrees(g) / 360.0) / 60, 100 * frac))

# 2. How many are overhead. Spread N satellites evenly over the sphere and the
#    expected number in view is N times the share in view. Real constellations
#    are not evenly spread, so this is an average, not a guarantee.
print()
print("How many satellites a receiver can expect to see at once, above 10 degrees:")
for name, alt, n in (("Iridium, 66 satellites", 780, 66),
                     ("Globalstar, 48 satellites", 1414, 48),
                     ("GPS, 24 satellites", 20200, 24),
                     ("Galileo, 24 satellites", 23222, 24)):
    frac = cap(alt)[1]
    print("  %-26s %5.1f on average, and a navigation fix needs four"
          % (name, n * frac))
print("  a low constellation gives about one satellite at a time, so a call is handed on;")
print("  a medium one gives several at once, which is what a position fix requires.")

# 3. The floor on constellation size, and why the real number is larger.
print()
print("The fewest satellites that could cover the earth, if footprints tiled perfectly:")
for name, alt, real in (("Iridium", 780, 66), ("Globalstar", 1414, 48), ("GPS", 20200, 24)):
    frac = cap(alt)[1]
    print("  %-11s floor %3d, actually flies %2d, which is %.1f times the floor"
          % (name, math.ceil(1 / frac), real, real / math.ceil(1 / frac)))
print("  circles do not tile a sphere, orbits are planes rather than a free scatter, and")
print("  coverage must hold at every moment, so the real number is always the larger one.")

# 4. Handover: how often, over a call.
print()
print("How often a call must be handed to the next satellite:")
for name, alt in (("Iridium", 780), ("Globalstar", 1414)):
    T = period(alt)
    g, _ = cap(alt)
    best = T * (2 * math.degrees(g) / 360.0) / 60
    print("  %-11s at best %4.1f min on one satellite, so a 30 min call is handed on at"
          % (name, best))
    print("              least %d times, and more when the pass is not overhead"
          % math.ceil(30.0 / best))
print("  and that is for a user standing still: the network moves, not the caller.")

# 5. What the shorter distance is worth. Free space loss, Recommendation
#    ITU-R P.525: 32.44 dB plus 20 log10 of MHz plus 20 log10 of km.
def fsl(f_mhz, d_km):
    return 32.44 + 20 * math.log10(f_mhz) + 20 * math.log10(d_km)

print()
print("Why a handheld terminal needs a low orbit, at 1600 MHz:")
for name, alt in (("LEO at 780 km", 780), ("LEO at 1414 km", 1414),
                  ("MEO at 20200 km", 20200), ("GEO at 35786 km", 35786)):
    g, _ = cap(alt)
    d_over = alt
    d_edge = slant(alt, g) / 1000
    print("  %-16s overhead %8.1f dB, at the footprint edge %8.1f dB"
          % (name, fsl(1600, d_over), fsl(1600, d_edge)))
gain = fsl(1600, 35786) - fsl(1600, 780)
print("  the low orbit is %.1f dB easier than geostationary overhead, a factor of %.0f,"
      % (gain, 10 ** (gain / 10)))
print("  which is the whole of the difference between a dish and a telephone in a hand.")

# 6. Doppler, the price of a moving satellite. The satellite sweeps the central
#    angle at its orbital rate; the range changes fastest at the horizon.
print()
print("Doppler shift, for a satellite passing overhead:")
for name, alt, f_mhz in (("Iridium, 1620 MHz", 780, 1620.0),
                         ("Globalstar, 1610 MHz", 1414, 1610.0),
                         ("GPS L1, 1575 MHz", 20200, 1575.42)):
    r = R + alt * 1000
    g, _ = cap(alt)
    rate = 2 * math.pi / period(alt)                  # radians per second of central angle
    d = slant(alt, g)
    closing = R * r * math.sin(g) / d * rate          # metres per second, at the edge
    print("  %-22s closes at %5.0f m/s, so the carrier shifts by %6.1f kHz"
          % (name, closing, closing / C * f_mhz * 1000))
print("  the receiver must search for the carrier and track it as it slides, and the shift")
print("  reverses sign as the satellite passes over: that is work a fixed dish never does.")

# 7. The three orbits, side by side.
print()
print("The three orbits, side by side:")
print("  %-22s %10s %10s %10s" % ("", "LEO", "MEO", "GEO"))
ALTS = (780, 20200, 35786)
def row(label, fn, fmt=" %10.1f"):
    print(("  %-22s" + fmt * 3) % ((label,) + tuple(fn(a) for a in ALTS)))
row("altitude, km", lambda a: float(a), " %10.0f")
row("period, minutes", lambda a: period(a) / 60)
row("earth in view, per cent", lambda a: 100 * cap(a)[1])
row("round trip delay, ms", lambda a: 2 * a * 1000 / C * 1000)
row("loss at 1600 MHz, dB", lambda a: fsl(1600, a))
print("  %-22s %10.1f %10.1f %10s"
      % ("visible for, minutes", period(780) * (2 * math.degrees(cap(780)[0]) / 360.0) / 60,
         period(20200) * (2 * math.degrees(cap(20200)[0]) / 360.0) / 60, "always"))

Distinctions

What it does not mean

LEO is not simply a cheaper GEO. It is a different design with different problems: the satellites are cheaper to launch but there must be dozens, they must be replaced far more often, and the ground segment and the handover machinery are far more complex.

A big constellation is not needed because space is crowded. It is needed because a low satellite sees very little of the earth and does not stay put.

MEO is not chosen for its delay. At about 135 ms round trip it is much closer to GEO than to LEO. It is chosen because several satellites are visible at once from everywhere.

Doppler is not a detail. Tens of kilohertz at 1.6 GHz forces the receiver to search and track, and a system designed as if the satellite were fixed would not acquire the carrier at all.

Globalstar's lack of links between satellites is not an oversight. It is a deliberate trade: simpler, cheaper satellites, in exchange for needing a gateway within the same footprint as the user.

NavIC is not a smaller GPS. It is regional by design, and it uses geostationary and inclined geosynchronous orbits rather than the medium orbits GPS uses.

Quick revision

• LEO: about 500 to 1,500 km. At 780 km, period 100.5 min, best pass 10.4 min, 2.6 per cent of the earth in view, round trip 5.2 ms, loss at 1600 MHz overhead 154.4 dB.
• MEO: about 19,000 to 23,000 km. At 20,200 km, period 718.7 min, pass 264.8 min, 29.9 per cent in view, round trip 134.8 ms, loss 182.6 dB.
• Why low: 33.2 dB easier than geostationary at 1600 MHz, a factor of about 2,100, which is what a handheld needs; and the poles are reachable.
• Why many: one low satellite sees 2.6 per cent, so the floor for global coverage is 39 and Iridium flies 66.
• Why 24 for GPS: the need is four in view with good geometry, everywhere; 24 gives 7.2 on average.
• Handover: at best every 10.4 min in LEO, so a 30 min call is handed on at least three times, caused by the network moving.
• Doppler: 35.4 kHz at 1620 MHz in LEO, 4.8 kHz at GPS L1, effectively none at GEO.
• Systems: Iridium 66 at 780 km with inter-satellite links; Globalstar 48 at 1,414 km without them; broadband constellations near 550 km; GPS, GLONASS, Galileo, BeiDou; NavIC regional.

Test yourself

1. Why does a handheld satellite telephone need a low orbit? Because of the link budget. At 1,600 MHz a satellite directly overhead at 780 kilometres costs 154.4 dB of free space loss, while a geostationary satellite costs 187.6 dB. The difference of 33.2 dB is a factor of about 2,100, and a handset with a stub antenna and a small battery has no way to find it. The low orbit supplies that margin by being close, which no amount of design in the handset could do.

2. Why does a low constellation need so many satellites? Because one satellite at 780 kilometres sees only 2.6 per cent of the earth and does not stay over any part of it. Even if footprints tiled a sphere perfectly, 39 would be needed for continuous global coverage, and since circles do not tile a sphere, orbits are confined to planes, and the coverage must hold at every instant rather than on average, the real number is larger: Iridium flies 66.

3. What kind of handover does a low constellation force, and why is it unusual? Inter-satellite handover, caused by the satellite setting rather than by the user moving. A satellite at 780 kilometres is above ten degrees for at most 10.4 minutes, so a thirty minute call is passed to the next satellite at least three times even if the caller stands perfectly still. In a terrestrial network handover means the user has moved; here the network moves and the user need not.

4. Why is MEO the orbit for navigation? Because a navigation fix needs at least four satellites in view at once, for three position coordinates and the receiver's clock error, and it needs them with good geometry everywhere on earth at all times. One satellite at 20,200 kilometres sees 29.9 per cent of the earth, so a constellation of 24 puts 7.2 in view on average, a comfortable margin over four. A low orbit would need hundreds of satellites to give the same simultaneous visibility, and a geostationary ring could not serve high latitudes at all.

5. Why is Doppler shift a problem in LEO and not in GEO? Because the shift is proportional to how fast the distance is changing. A satellite at 780 kilometres closes on a user at up to about 6,548 metres a second, which shifts a 1,620 MHz carrier by 35.4 kHz and reverses sign as the satellite passes overhead, so the receiver must search a wide band to acquire the carrier and then track it continuously. A geostationary satellite does not move relative to a fixed dish, so the shift is effectively zero and the receiver can be tuned and left.

6. Compare Iridium and Globalstar. Both are big LEO voice systems, but they make opposite choices. Iridium flies 66 satellites in near-polar orbits at about 780 kilometres and carries traffic from satellite to satellite, so a call can be routed across the constellation and delivered far from where it started, including over oceans and the poles where no gateway exists; the price is complex satellites and a routing problem in the sky. Globalstar flies 48 satellites at about 1,414 kilometres in inclined orbits with no links between them, so every call is relayed straight down to a gateway; the satellites are simpler and cheaper, but a call only works where a gateway shares the footprint, and the high latitudes are not served.

7. The two costs of altitude cross at about 2,000 km. Why does nothing fly there? Because the number of satellites needed falls with altitude while the delay rises with it, and around 2,000 kilometres the two are equal, which would look like the natural compromise. But that is also where the inner radiation belt begins, and a satellite that spends its life inside it needs heavy shielding and radiation-hardened parts. The belt pushes designs to either side of the crossing, which is why the orbits in use form three separate bands rather than a smooth range.