Orbits

Where a satellite flies decides how often it sees a place, how close it gets and at what time of day it passes.

The basics

Most Earth observation satellites fly a few hundred kilometres up, roughly the distance from London to Edinburgh if you could drive straight into the sky. At that height one lap of the planet takes about an hour and a half, and the Earth turns underneath while the satellite goes round. Each lap therefore looks down on a new strip of ground, and the strips add up to the whole planet over days or weeks. A few satellites sit far higher, about 36 000 km up, where a lap takes exactly as long as a day. They hang over one spot, which suits weather and communications but is too far away for fine detail. Most imaging satellites use a tilted orbit that carries them over each place at the same local time on every pass, so shadows and lighting match from one visit to the next.

Why it matters

The orbit sets how often a satellite can look at your area and at what time of day. A low orbit brings the camera close enough for fine detail, but one satellite returns to the same place only every several days, so operators fly two or more in the same orbit to look more often. The sun-synchronous orbit most imaging satellites use means that pictures of one place taken weeks apart have the same lighting, which makes them easy to compare.

Each lap looks down on a new strip, because the Earth turns underneath the satellite. Over days the strips add up to the whole planet.

In depth

A satellite in a circular orbit of radius $a$ about the Earth's centre has a period set by gravity alone. With the Earth's gravitational parameter $\mu = 3.986\times 10^{14}\ \mathrm{m^3\,s^{-2}}$ , Kepler's third law gives the period and the speed.

T = 2\pi\sqrt{\frac{a^{3}}{\mu}}

(1)

where $T$ is the orbital period, in $\mathrm{s}$ ; $a$ is the radius of the orbit measured from the Earth's centre, in $\mathrm{m}$ , or the semi-major axis of an elliptical orbit; and $\mu$ is the Earth's gravitational parameter, the product of the gravitational constant and the Earth's mass, $3.986\times 10^{14}\ \mathrm{m^{3}\,s^{-2}}$ .

v = \sqrt{\frac{\mu}{a}}

(2)

where $v$ is the speed on a circular orbit, in $\mathrm{m\,s^{-1}}$ ; $\mu$ is the Earth's gravitational parameter; and $a$ is the orbit radius, in $\mathrm{m}$ .

For an altitude of 700 km the radius is 7078 km, so the period is 98.8 minutes, the speed is 7.50 km per second, and the satellite completes 14.6 orbits in a day. At a radius of 42 164 km, which is an altitude of 35 786 km, the period is 23 hours 56 minutes, one rotation of the Earth relative to the stars, so a satellite on the equator stays over one longitude. Figure 1 draws these altitudes to scale. Between consecutive orbits the Earth turns by about 25 degrees, which is why the successive ground tracks in the second panel sit side by side [1, 2].

(a) Altitudes, to scale

(b) Five consecutive northbound passes, swath exaggerated

Figure 1. Orbit altitudes and ground tracks. Left, altitudes drawn to scale, with the Earth's radius as the unit. Right, a 180 degree window of longitude showing five consecutive northbound passes of a satellite at 700 km, numbered in order, with the swath drawn wider than it is.

The Earth is slightly flattened, and the equatorial bulge makes the plane of a tilted orbit turn slowly about the polar axis. For a circular orbit of radius $a$ and inclination $i$ the rate of turn is given by equation 3, where $J_2 = 1.0826\times 10^{-3}$ describes the flattening, $R_E = 6378\ \mathrm{km}$ is the equatorial radius and $n = \sqrt{\mu/a^{3}}$ is the mean motion.

\dot{\Omega} = -\frac{3}{2}\,J_2\left(\frac{R_E}{a}\right)^{2} n\,\cos\, i

(3)

where $\dot{\Omega}$ is the rate at which the orbit plane turns about the polar axis, the drift of the right ascension of the ascending node, in $\mathrm{rad\,s^{-1}}$ ; $J_2$ is the Earth's oblateness coefficient, $1.083\times 10^{-3}$ without unit; $R_E$ is the Earth's equatorial radius, $6.378\times 10^{6}\ \mathrm{m}$ ; $a$ is the orbit radius, in $\mathrm{m}$ ; $n$ is the mean motion, in $\mathrm{rad\,s^{-1}}$ ; and $i$ is the inclination, the angle between the orbit plane and the equator.

If the plane turns at the rate the Earth moves round the Sun, 360 degrees in 365.24 days or 0.9856 degrees per day, the angle between the orbit plane and the Sun stays fixed. The condition needs a negative $\cos\, i$ , that is an inclination above 90 degrees, and at 700 km it gives 98.2 degrees. Such a sun-synchronous orbit crosses a given latitude at the same local solar time on every pass, as Figure 2 sets out [1, 2].

(a) Plane fixed in space

The angle between the plane and the Sun changes through the year, so the local time of each crossing drifts.

(b) Plane turns once a year, keeping pace with the Sun

The plane keeps one angle to the Sun, so the satellite crosses each latitude at the same local solar time on every pass.

Figure 2. Why a sun-synchronous orbit gives constant lighting. Left, an orbit plane fixed in space meets the Sun at a different angle through the year. Right, a plane that turns with the Earth keeps one angle to the Sun. The orbit sizes are exaggerated.

Revisit time, the interval between looks at the same place, then depends on the swath width, the latitude, how far the satellite can tilt its view, and how many satellites share the orbit. The twin Sentinel-2 satellites, with a 290 km swath, observe the same point on the equator every five days, against ten days for one satellite alone [3].

References

Montenbruck, O. and Gill, E. (2000) Satellite Orbits. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-58351-3
Capderou, M. (2014) Handbook of Satellite Orbits. Springer International Publishing. https://doi.org/10.1007/978-3-319-03416-4
Drusch, M., Del Bello, U., Carlier, S. et al. (2012) Sentinel-2: ESA's Optical High-Resolution Mission for GMES Operational Services. Remote Sensing of Environment, 120, 25-36. https://doi.org/10.1016/j.rse.2011.11.026