Sketching a satellite orbit and predicting its period
Using a scale drawing to predict the time for a satellite close to the Earth. The results can later be compared with mathematical calculations based on v2/R.
Apparatus and materials
Brown paper sheet about 1.5 m long, 15 to 20 cm wide - or a roll of long paper
Thin wire, strong, about 3.5 m
Large mass or hook to anchor one end of the wire on the floor or near it
Metre rule with cm and mm markings
Health & Safety and Technical notes
Be careful with the thin wire, and avoid getting cuts to the hand.
This works best on the floor, or the side bench of a laboratory, with small groups of students.
The arc drawn is twice as long as needed for 2 minutes' travel, but the double length enables a symmetrical drawing which will yield a good estimate more easily.
a Using a thin wire anchored on the floor at one end and held taut with a pencil at the other end, draw an arc about 1.3 m long and with a radius of 3.3 m. The centre of the circle is not on the paper but arrange the paper so that the arc is. This represents part of a circular orbit for a satellite at an altitude of 200 km. (A scale drawing with 0.5 mm = 1 km - see Teaching Notes.)
b On that arc, XAY, mark the mid point A and part of the radius to A. Drop down along that radius the calculated fall 72 km (to scale) AM. (On the same scale 0.5 x 72 = 36 mm.)
c Draw the tangent at A, symmetry helping. And draw a chord XMY parallel to the tangent, with mid point at M.
d Transfer the fall AM out to the place where it should be shown as a fall from the tangent to the orbit, a fall NY. Measure the travel distance AY, which is covered in their chosen time 120 seconds (973 km).
e Calculate the time for a complete orbit, knowing the total travel distance 2πR, which is 2π x 6,600 metres.
1 A scale of 0.5 mm to 1 km is best for the drawing - a smaller drawing would be difficult to measure. On this scale, R = 6,400 + 200 km is represented by a radius of 0.5 x 6,600 = 3,300 mm = 3.3 m.
2 Assume the satellite is always falling inward with acceleration g (≈ 10 m/s/s). First calculate how far the satellite will fall from the tangent in 1 second (5 m). It is clearly difficult to draw a scale diagram with 5 m drawn on it and the radius of the orbit, 6,600 km, and so a longer time needs to be chosen. the satellite's free fall from the tangent to its orbit in a time of 120 seconds will be s = 1/2 gt2 = 72 km.
This will give a travel distance AY of about 973 km. The time for a complete orbit, T, will then be 120/T=973 / 2π x 6,600, giving a value of T of about 85 minutes. Students are generally very surprised to find that near-Earth satellites orbit with such short periods. Such satellites are used for astronomy, earth-observation (mapping and spying) and weather-forecasting. (Note that communications satellites must remain geostationary, and so orbit at much greater distances from the Earth.)
It helps to provide illustrative date about a variety of near-Earth satellites. For example, the first artificial satellite, Sputnik 1, had an orbit period of 96.2 minutes. NASA has a website that offers real-time tracking of satellites, including the Hubble Space Telescope and the International Space Station. Likewise the European Space Agency has a satellites in orbit webpage, where you can find orbit information for its Earth Observation missions. [Note that ESA Space Science missions do not have near-earth orbits.]
3 You could go on to calculate the Moon's orbital time, using the same scale. See the guidance note Estimating the Moon's orbit time.