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 *D*_{ref} and flux *F*_{ref} 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:

and this relationship can be solved for the unknown distance

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/m^{2}), 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
*F*_{ref} and *D*_{ref} 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/m^{2}) |
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 |

Star | F (w/m^{2}) |
T (°K) |
ref. star | F_{ref} (w/m^{2}) |
D_{ref} (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 | ____________ | ____________ | ____________ | ____________ |

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?

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

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

Last modified: November 13, 2001

`http://www.ifa.hawaii.edu/~barnes/ast110/homework/hw12.html`