Delta Cephei is one of the most visible stars in the night sky. Its brightness has periods of 5.4 days, the average brightness is 4.0 and its brightness varies by plus-or-minus 0.35. Find a formula that models the brightness, b, of Delta Cephei as a function of time, t, with t equals 0 at peak brightness
To model the brightness of Delta Cephei as a function of time, we can use a sine function, since the brightness variation follows a periodic pattern.
The general formula for a sine wave is:
b = A * sin(2π/P * t + φ) + C
where:
- A is the amplitude, which represents half of the total brightness variation (in this case, A = 0.35).
- P is the period, which represents the time it takes for the brightness to complete one full cycle (in this case, P = 5.4 days).
- φ is the phase shift, which determines the starting point of the sine wave (in this case, φ = 0 since t equals 0 at peak brightness).
- C is a constant term that represents the average brightness (in this case, C = 4.0).
Combining these values into the formula, we get:
b = 0.35 * sin((2π/5.4) * t) + 4.0
Therefore, the formula that models the brightness (b) of Delta Cephei as a function of time (t) is:
b = 0.35 * sin((2π/5.4) * t) + 4.0
To model the brightness of Delta Cephei as a function of time, we can use a sinusoidal function.
The general formula for a sinusoidal function is:
b(t) = A * sin(ωt + φ) + C
Where:
- A is the amplitude of the function, representing the maximum deviation of the brightness from the average value.
- ω is the angular frequency, representing how fast the brightness oscillates.
- φ is the phase shift, which determines the position of the function in time.
- C is the average brightness.
In this case, since Delta Cephei has a period of 5.4 days, we can determine the angular frequency, ω, using the formula:
ω = 2π / T
Where T is the period of 5.4 days.
Given that the average brightness, C, is 4.0 and the variation is plus-or-minus 0.35, the amplitude, A, can be calculated as half the difference between the maximum and minimum brightness values:
A = (max brightness - min brightness) / 2
Since the variation is plus-or-minus 0.35, the maximum brightness is the average brightness plus 0.35, and the minimum brightness is the average brightness minus 0.35.
Therefore, the formula to model the brightness of Delta Cephei as a function of time is:
b(t) = A * sin(ωt + φ) + C
Substituting the values we have, the formula becomes:
b(t) = (4.35 / 2) * sin((2π / 5.4) * t + φ) + 4.0
The only unknown in this equation is the phase shift, φ, which determines when the brightness is at its peak. To find the value of φ, we need additional information or data points.
b = 4.0 + 0.35 cos (2 pi t/T)
where T is the period 5.4
use cos because it is +1, max at t = 0