How much time does a geostationary satellite, orbiting at about 35,790 km above the Earth's surface, take to complete one orbit around the Earth?

Introduction / Context:
Geostationary satellites are used for communications, weather monitoring, and broadcasting because they appear to stay fixed above one point on the Earth's equator. This apparent "standing still" occurs because the satellite orbits the Earth at the same rate that the Earth rotates. To achieve this effect, the satellite must be placed in a specific orbit and must take a specific amount of time to complete one revolution. This question asks you to identify that orbital period.

Given Data / Assumptions:

• The satellite is described as geostationary, meaning it appears stationary over one point on the equator.
• The orbital height is approximately 35,790 km above the Earth's surface.
• Geostationary orbits are circular and in the equatorial plane.
• We assume standard definitions of a day, a month, and a year.

Concept / Approach:
A geostationary satellite must have an orbital period equal to the Earth's rotational period about its axis, as measured relative to the distant stars. This period is about 23 hours 56 minutes, which is very close to the 24-hour solar day that we commonly use. By matching this period, the satellite remains above the same geographic point on the equator. If the satellite had a period of 12 hours or 30 days, it would move relative to the Earth's surface, and observers would see it drift across the sky. A period of 365 days corresponds to the Earth's orbit around the Sun and has no relation to a geostationary orbit.

Step-by-Step Solution:
Step 1: Recall that geostationary satellites appear fixed in the sky because their orbital period matches the Earth's rotational period. Step 2: Remember that the Earth rotates once about its axis in approximately 24 hours. Step 3: Understand that a satellite with a 24-hour orbit around the Earth in the equatorial plane will have an angular speed equal to the Earth's rotation. Step 4: Note that 12 hours would correspond to a satellite that circles the Earth twice per day, which would not remain stationary in the sky. Step 5: Recognise that 30 days and 365 days correspond roughly to the Moon's orbit and the Earth's orbit around the Sun, not to geostationary satellites. Step 6: Conclude that the correct orbital period for a geostationary satellite is 24 hours.

Verification / Alternative check:
Space agency documentation and physics textbooks state that geostationary satellites orbit at a radius of about 42,164 km from the Earth's centre, which corresponds to approximately 35,790 km above the surface. At this altitude, the gravitational pull and orbital speed result in a period of exactly one sidereal day, about 24 hours. Communication satellites like many television and weather satellites are placed in geostationary orbit for this reason. Observers on the ground see antenna dishes pointed at a fixed spot in the sky, confirming that the satellite's apparent position is constant, which is only possible with a 24-hour period.

Why Other Options Are Wrong:
30 days, matching approximately one lunar month, is incorrect because a satellite with this period would not remain fixed above one point; it would wander relative to the Earth's rotation.
365 days, matching one revolution of the Earth around the Sun, is wrong because such a period has nothing to do with geostationary motion around the Earth and would place the satellite far away in space.
12 hours, matching half a day, is incorrect because a satellite with a 12-hour period would pass over the same point twice a day but would not appear stationary.

Common Pitfalls:
Some students confuse the various time scales in astronomy, mixing up the length of a day, a month, and a year. Others think that any satellite orbit can be called geostationary as long as it goes around the Earth, regardless of period. To avoid confusion, remember that "geo" refers to Earth and "stationary" means staying over the same spot, which can only happen if the satellite's orbital period matches the Earth's rotation: about 24 hours.

Final Answer:
A geostationary satellite takes 24 hours, matching one rotation of the Earth on its axis to complete one orbit.