A variable star is one whose brightness alternately increases and decreases.
For the variable star Delta Cephei, the time between periods of maximum brightness varies by +/- 3.5 magnitude.
(a)Find a function that models the brightness of Delta Cephei as a function of time.
How do I do this?
I see I have the period to be 5.4 so
T = (2pi)/w
I solved for omega got 2.7/pi
I do not know were to go form here is +/- 3.5 the phase shift???
is the average brightness (or magntiude) of the star 4.0 the amplitude???
THANKS!
They probably expect you to assume that the magnitude varies sinusoidally with time, although that is NOT true for delta Cephei. Its brightess reaches maximum in about half the time it takes to return to minimum. Also, they (or you) got the amplitde of the variation wrong. It is +/- 0.35 magnitude, not 3.5. They did get the period right, it is 5.4 days.
I will leave you to figure out whether to use the false data they gave you or the true data.
Magnitude = 3.95 + A sin (2 pi t/5.4) would approximately model a sinusoidal variation with period 5.4 days, if t is in days. "A" would be the true variation amplitude (which should be 0.35). I have made use of the fact that the mean magnitude is 3.5 for that star, although they did not mention that.
My next to last sentence should have read .."the mean magnitude is 3.95 for that star, although they did not mention that."
To model the brightness of the variable star Delta Cephei as a function of time, you can use a sine function.
Given that the time between periods of maximum brightness (the period) varies by +/- 3.5 magnitude, this means that the amplitude of the sine function will be 3.5. The amplitude represents the maximum deviation from the average or mean brightness.
You mentioned that you have the period to be 5.4, which is correct. The period represents the time it takes for the star to complete one full cycle from maximum to minimum brightness and back to maximum again.
The average brightness or magnitude of the star, which you mentioned is 4.0, will be the vertical shift or displacement of the sine function. It represents the middle or equilibrium brightness level.
Combining these values, you can write the function as:
f(t) = A * sin(wt) + C
Where:
- f(t) represents the brightness (or magnitude) of the star at time t.
- A is the amplitude, which is 3.5 in this case.
- w is the angular frequency, calculated as w = (2 * pi) / T, where T is the period. In this case, T = 5.4, so w = (2 * pi) / 5.4 ≈ 1.167.
- t represents the time.
- C is the vertical shift or displacement, which is the average brightness, given as 4.0.
So, the function that models the brightness of Delta Cephei as a function of time is:
f(t) = 3.5 * sin(1.167t) + 4.0
Remember to use the appropriate units for time to ensure accurate calculations and interpretations.