The brightness of a variable star adds a component to the simple harmonic motion we have studied in this lesson. Since the brightness is variable the vertical axis may no longer be equal to zero. Also included in the variance is a phase shift. In this case, the equation for this function would be: y = a cos w(t – c ) + b.

Suppose we have a variable star whose brightness alternately increases and decreases. For this star, the time between periods of maximum brightness is 6.5 days. The average brightness (or magnitude) of the star is 5.0 and its brightness varies by + 0.25 magnitude.

1. What is the amplitude of the function for this model?

2. What is the period?

3. What is w?

4. What is the vertical shift?

5. Is there a phase shift? If so, what is it?

6. What is the function

The given model is
y a cos w(t - c) + b

1. The brightness varies by 0.25. Therefore the amplitude is
a = 0.25

2. The time between maximum brightness is 6.5 days.
Therefore the period is T = 6.5 days

3. By definition,
w = 2π/T. Therefore,
w = 2π/6.25 = 0.967

4. The average brightness is 5, therefore the vertical shift is b = 5.

5. Assume that maximum brightness occurs at the time, t = 0.
Therefore there is no phase shift so that c = 0.

6. The function is
y = 0.25 cos(0.967t) + 5

A graph of the function is shown below.