Geostationary Orbits

Some satellites appear to hover motionless above a single point on the Earth's surface. Switch on a satellite TV dish and you will see that it points in a fixed direction — somewhere above the equator, steady for hours, days, years. Ask where these satellites are and you will hear the answer "geostationary orbit". What exactly is a geostationary orbit? Why must it be above the equator? Why is it at 36 000 km altitude and not some other height? And what is it used for?

This final lesson of the course answers all of these questions. It is the capstone of OCR A-Level Physics A Module 5.4 (Gravitational fields), pulling together Kepler's third law, orbital mechanics, and the physics of circular motion from earlier in the course.

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What is a Geostationary Orbit?

A geostationary orbit is a circular orbit around the Earth in which the satellite:

1. has an orbital period of exactly one sidereal day (approximately 24 hours — more precisely 23 h 56 min 4 s);
2. moves in the same direction as the Earth's rotation (eastward);
3. lies in the equatorial plane — that is, directly above the Earth's equator.

Taken together, these conditions mean that the satellite stays over the same point on the Earth's surface at all times. A ground-based observer sees it as a fixed point in the sky.

A closely related concept is the geosynchronous orbit: an orbit with a 24-hour period, but not necessarily equatorial. A geosynchronous satellite returns to the same position in the sky once per day but traces a figure-of-eight path during that day. A geostationary satellite is a special case of a geosynchronous satellite — one that happens to be equatorial.

Exam Tip: OCR questions distinguish carefully between "geostationary" and "geosynchronous". For full marks, you must state all three conditions: (i) 24-hour period, (ii) equatorial, (iii) same direction as the Earth's spin.

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Why 36 000 km?

The altitude of a geostationary orbit is fixed by Kepler's third law: given the period T = 24 h, the radius is determined.

T² = 4π² r³ / (GM)
r³ = GM T² / (4π²)
r = ∛(GM T² / (4π²))

With GM = 3.98 × 10¹⁴ m³ s⁻² and T = 86 400 s:

T² = (86 400)² = 7.46 × 10⁹ s²
GM × T² = 3.98 × 10¹⁴ × 7.46 × 10⁹ = 2.97 × 10²⁴
GM × T² / (4π²) = 2.97 × 10²⁴ / 39.48 = 7.52 × 10²²
r = ∛(7.52 × 10²²) = 4.22 × 10⁷ m

This is the radius from the centre of the Earth. The altitude above the Earth's surface is

h = r − R_E = 4.22 × 10⁷ − 6.37 × 10⁶ = 3.59 × 10⁷ m ≈ 35 900 km

Which is usually rounded to 36 000 km. There is exactly one altitude at which a satellite can be geostationary, set entirely by the mass of the Earth and the length of its day.

Orbital Speed at Geostationary Altitude

v = √(GM/r) = √(3.98 × 10¹⁴ / 4.22 × 10⁷) = √(9.43 × 10⁶) = 3.07 × 10³ m s⁻¹

About 3.07 km s⁻¹ — roughly 40% of the ISS's 7.67 km s⁻¹. Higher orbits move more slowly; this is a consequence of v = √(GM/r).

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Why the Orbit Must Be Equatorial

A non-equatorial orbit cannot be geostationary. Imagine a satellite in a 24-hour orbit inclined at 30° to the equator. Over the course of one day, as viewed from the Earth's rotating surface, the satellite appears to trace a figure-of-eight pattern ("analemma"), rather than staying over a single point. For a dish antenna on Earth to track a fixed position in the sky, the satellite must remain directly above the same longitude at every instant — and that can only happen in the equatorial plane.

Geometrically, the condition is that the satellite's orbital plane must contain the Earth's rotation axis — or, equivalently, its orbit must be perpendicular to the rotation axis. Only orbits in the equatorial plane meet this requirement.

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Why the Orbit Must Be Prograde

The satellite must orbit eastward — in the same direction as the Earth spins (prograde). A retrograde (westward) satellite at the same altitude would still have a 24-hour period, but relative to the ground it would complete two apparent passes per day, moving westward at twice the Earth's rotation rate. Only a prograde 24-hour orbit stays fixed above a single ground point.

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Applications of Geostationary Orbits

Because a geostationary satellite stays over a fixed point, it can continuously transmit to and receive from ground stations with fixed (non-tracking) antennas. This makes geostationary orbit the obvious home for:

• Satellite television (Sky, DirecTV, etc.) — a single satellite covers nearly a third of the globe, and your dish needs to point only once.
• Communications satellites carrying phone calls, internet traffic and data between continents.
• Weather satellites such as Meteosat, which continuously photograph the same hemisphere for weather forecasting.
• Early warning and military surveillance satellites for persistent coverage of a given region.
• Some navigation augmentation systems such as EGNOS and WAAS, which broadcast GPS corrections.

What geostationary orbit is not suitable for:

• High-resolution Earth imaging — at 36 000 km the ground is too far away for fine detail. Imaging satellites (e.g. Sentinel, Landsat) use much lower orbits.
• Polar observations — the view from an equatorial satellite is very oblique near the poles; the poles themselves cannot be seen at all.
• GPS-like navigation — the GPS constellation uses 12-hour "semi-synchronous" orbits at about 20 000 km altitude, not geostationary.

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Worked Example 1 — Mass of Mars from Phobos

Phobos, a moon of Mars, orbits at an average radius of 9.38 × 10⁶ m with a period of 7 h 39 min. Estimate the mass of Mars.

Convert the period to seconds:

T = (7 × 3600) + (39 × 60) = 25 200 + 2340 = 27 540 s

From Kepler's third law:

M = 4π² r³ / (G T²)
r³ = (9.38 × 10⁶)³ = 8.25 × 10²⁰ m³
T² = (27 540)² = 7.58 × 10⁸ s²
4π² = 39.48