# Homework 12. Finding Distances With Standard Candles

 Name: ________________________ DUE 11/20 ID number: ________________________

A star whose total luminosity L can be estimated from observations can serve as a standard candle. Such stars are very useful in determining distances, because the flux F of a star is easy to measure, and knowing F and L you can work out the distance D of the star.

Stars on the main sequence can be used as standard candles, because (as we saw in Homework 10) they have the same order when listed by ``surface'' temperature T or luminosity L; in other words, there's a one-to-one relationship between T and L. It's not too difficult to measure the temperature T of a star by looking at the color of its light (or the pattern of dark lines in its spectrum). If you're sure that this star is a main-sequence star, you can translate the measured temperature T into the corresponding luminosity L.

To see how this works, imagine you want to find the distance D to a main-sequence star of known flux F and surface temperature T. Suppose you have a ``reference'' star with distance Dref and flux Fref which - by an odd coincidence - has the same surface temperature as the star you want to find the distance to. Eureka! The two stars have the same temperature T, so they must have the same luminosity L. Therefore, the ratio of their fluxes is related to the ratio of their distances:

D : Dref = Fref : F ,

and this relationship can be solved for the unknown distance D:

D = Dref × (Fref ÷ F) .

Table 1 (on the other side of this page) lists measured fluxes F, surface temperatures T, and distances D for a suitable set of main-sequence reference stars (all quite close to the Sun, so their distances can be measured accurately using parallax). In this table, the units of F are watts-per-square meter (w/m2), the units of T are degrees Kelvin (°K), and the units of D are parsecs (pc).

Next, Table 2 lists observed fluxes F and surface temperatures T for ten stars in two clusters; stars 1 to 5 are from one cluster, while stars a to e are from the other. Your job is to use the method described above to calculate distances to these stars. There's a slight catch: the temperatures of the cluster stars don't exactly match the temperatures of the reference stars in Table 1. But it turns out that you can get pretty good results by pairing each star up with a reference star which has a similar temperature.

To compute the distance to each of the cluster stars, begin by finding the reference star with the best matching temperature. Write the name of this reference star in the ``ref. star'' column of Table 2. Copy the flux and distance of the reference star to the Fref and Dref columns. Then plug these values, along with the cluster star's flux F, into the formula for D above, and write this distance in the last column of the table.

 Star F (w/m2) T (°K) D (pc) alpha Cen A 3.11 × 10-8 5800 1.35 alpha Cen B 9.66 × 10-9 5200 1.35 epsilon Eri 1.24 × 10-9 4880 3.22 61 Cyg A 4.23 × 10-10 4400 3.48 61 Cyg B 2.47 × 10-10 4120 3.51 Lal. 21185 1.13 × 10-10 3500 2.55
Table 1: reference stars

 Star F (w/m2) T (°K) ref. star Fref (w/m2) Dref (pc) D (pc) 1 1.11 × 10-12 5100 ____________ ____________ ____________ ____________ 2 9.33 × 10-13 4920 ____________ ____________ ____________ ____________ 3 5.56 × 10-13 4600 ____________ ____________ ____________ ____________ 4 2.45 × 10-13 4220 ____________ ____________ ____________ ____________ 5 1.78 × 10-13 4040 ____________ ____________ ____________ ____________ a 2.37 × 10-14 5300 ____________ ____________ ____________ ____________ b 1.26 × 10-14 4760 ____________ ____________ ____________ ____________ c 4.74 × 10-15 4280 ____________ ____________ ____________ ____________ d 6.93 × 10-14 3950 ____________ ____________ ____________ ____________ e 1.03 × 10-15 3660 ____________ ____________ ____________ ____________
Table 2: cluster stars

But there's another catch - one of the stars is not a main-sequence star, so the distance you've computed for it is way off! Which star is it, and since it's not a main-sequence star, what kind of star is it?

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Finally, what are your best estimates for the distances to the two clusters?

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Joshua E. Barnes (barnes@ifa.hawaii.edu)