SME
Remote Sensing · Problem type
Orbital period of a satellite
Use this for “An imaging satellite circles 705 km above the ground. How long does one orbit take, and how many orbits does it make in a day?”
Diagram
Orbital
speed:
v
=
√
GM/r
=
√(6.67
×
10
-11
×
5.97 ×
10
24
/(7.08
×
10
6
))
=
7,510
m/s
Period: T = 2πr/v = 2π × 7.08 × 10⁶/7,510 = 5,920 s (98.7 min)
Gravity is the centripetal force: GMm/r² = mv²/r, so the satellite’s own mass cancels.
It orbits 705,000 m above the surface (r − R, with R = 6.37 × 10⁶ m).
This is an example. Type your own values to start a new problem.
H
Altitude
example
r
Orbit radius
calculated
T
Period
calculated
n
Orbits a day
calculated
Formulas
r
=
6,371,000
+
H
7,076,000 = 6,371,000 + 705,000
T
=
2π
×
√(
r
3
3.986 ×
10
14
)
5,923.6987
=
2π
×
√(
7,076,
000
3
3.986 ×
10
14
)
n
=
86,400
T
14.5855
=
86,400
5,923.6987
Formulas are worked in m, s. The step-by-step shows the conversions.
Assumptions
•
The orbit is a circle, and Earth is a sphere of radius R = 6,371 km pulling as if all its mass were at its centre.
•
GM = 3.986 × 10¹⁴ m³/s² for Earth, so T = 2π√(r³ ÷ GM) with r in metres and T in seconds.
•
A day is 86,400 s; Earth’s own turn under the orbit is left out.
Step-by-step
WE KNOW
H = 705 km
FIND
orbit radius (r), period (T), orbits a day (n)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m, s).
H = 705 km = 705,000 m (1 km = 1000 m)
Step 2 · Find orbit radius (r)
Use r = 6,371,000 + H
The orbit is measured from Earth’s centre: its radius, 6,371 km, plus the altitude.
r
=
6,371,000
+
705,000
r
=
7,076,000
m
Step 3 · Find period (T)
Use T = 2π × √(r³ ÷ (3.986 × 10¹⁴))
Gravity supplies the pull that keeps the satellite on its circle, which fixes the time for one turn at each radius.
T
=
2π
×
√(
7,076,
000
3
3.986 ×
10
14
)
(7.076 ×
10
6
)
3
=
3.543 ×
10
20
m
3
3.543 ×
10
20
3.986 ×
10
14
=
8.888 ×
10
5
s
2
√
8.888 ×
10
5
=
942.8
s
2π × 942.8 = 5,923.7 s
T
=
5,923.6987
s
Step 4 · Find orbits a day (n)
Use n = 86,400 ÷ T
The seconds in a day over the seconds in one orbit.
n
=
86,400
5,923.699
n
=
14.5855
Step 5 · Convert the answers
Change each answer back to the unit you chose.
r = 7,076,000 m = 7,076 km (1 km = 1000 m)
T = 5,923.6987 s = 98.7283 min (1 min = 60 s)
Check
Put the numbers back in. Both sides should match.
7,076,000 = 6,371,000 + 705,000 ✓
5,923.699
=
2π
×
√(
7,076,
000
3
3.986 ×
10
14
)
✓
14.5855
=
86,400
5,923.699
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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