Question 14.6: An astronomer observes a faint object close to a star. Conti...
An astronomer observes a faint object close to a star. Continued observations show the object apparently moving in a straight line through the star as shown in the diagram
The astronomer is able to estimate the apparent distance of the object from the centre of the star and records this at 30-day intervals, resulting in the following table.
| Day, t | 0 | 30 | 60 | 90 | 120 |
| Distance (10^{11}m), x | 1.0 | 1.5 | 1.9 | 2.0 | 1.9 |
The astronomer thinks that the object is a planet moving around the star in a circular orbit and that she is observing it from a point in the plane of the orbit. She decides to model the apparent distance from the centre by the SHM equation:
x = 2 sin (ωt + ε).
i) Use the values of x for t = 0 and t = 90 to find values for the constants ω and ε and verify that the other values of x are consistent with this model.
ii) Assuming the model is correct, find
a) the radius of the orbit
b) the speed of the planet
c) the number of earth days the planet takes to go round its star.
When t = 0, x = 1 so 1 = 2 sin ε
⇒ ε = \frac{π}{6}.
When t = 90, x = 2 so 2 = 2 sin \left(90ω + \frac{π}{6} \right)
⇒ sin \left(90ω + \frac{π}{6} \right) = 1
⇒ 90ω + \frac{π}{6} = \frac{π}{2}
⇒ ω = \frac{π}{270}.
So the model is x = 2 sin \left(\frac{πt}{270} + \frac{π}{6} \right)
For the other values of t:
| t | observed displacement | model’s prediction |
| 30 | 1.5 | 2 sin \left(\frac{π}{9} + \frac{π}{6}\right) = 1.53 |
| 60 | 1.9 | 2 sin \left(\frac{2π}{9} + \frac{π}{6}\right) = 1.88 |
| 120 | 1.9 | 2 sin \left(\frac{4π}{9} + \frac{π}{6}\right) = 1.88 |
This shows that the model is a very good predictor of the actual position of the object.
ii) a) The radius of the orbit is the amplitude a of the motion:
radius = 2 × 10^{11} m.
b) The speed of the planet is
aω = 2 × 10^{11} × \frac{π}{270} = 2.3 × 10^{9} ms^{-1}.
c) The total time for one orbit is the period of the SHM i.e. \frac{2π}{ω}
Number of days = 2π ÷ \frac{π}{270}
= 540 days.
Historical note
In 1822, the French mathematician Jean Baptiste Fourier (1768–1830) showed that any function of t can be written as a sum of sines of multiples of t and this is now called a Fourier series. It follows that any vibration which can be written as a function of t can be reproduced by adding simple harmonic vibrations. Fourier accompanied Napoleon to Egypt in 1798 and was made a baron ten years later. He discovered this theorem while working on the flow of heat.
