The Cepheid variables that Hubble observed in M31 had an apparent magnitude of about 18, and thus an apparent brightness relative to the Sun (b/b_Ÿ) of about 1.3 × 10^-18. What approximate periods would these Cepheids need in order to yield a distance to M31 of about 750 kpc?

Question

Quizplus · Accepted Answer

The period-luminosity relationship for Cepheid variables can be expressed as follows:
M = alog(P) + b
where M is the absolute magnitude (brightness), P is the period of the Cepheid in days, and a and b are constants that depend on the particular type of Cepheid. Using this relationship, we can determine the distance to M31 if we know the period of the Cepheids in M31 and their apparent magnitude.

For Cepheids with an apparent brightness relative to the Sun of 1.3 × 10-18, we can use the following formula to calculate their absolute magnitude:
M = m - 5log(d/10)
where m is the apparent magnitude (18 in this case) and d is the distance in parsecs. We want a distance of 750 kpc, which is equivalent to 2.44 × 1021 km. Converting this to parsecs gives us d = 7.93 × 106 parsecs. Plugging in the numbers, we get:
M = 18 - 5log(7.93 × 106/10) ≈ -20.9

For Cepheids with an absolute magnitude of -20.9, the period-luminosity relationship gives us:
-20.9 = alog(P) + b
Taking a = -3.1 and b = -1.4 (appropriate for classical Cepheids), we can solve for P to get:
P ≈ 70 days

Therefore, the approximate period that these Cepheids would need to yield a distance to M31 of about 750 kpc is 70 days.

The Cepheid Variables That Hubble Observed in M31 Had an Apparent