# Experiment 7: Cepheid Variables (under construction)

### *Objective* :

To measure astronomical distances using Cepheid variables.
### *Introduction* :

This experiment will illustrate how to use a type of stars called *Cepheid Variables*to measure astronomical distances. Cepheid Variables are a type of variable stars whose magnitude (apparent or absolute) varies over a fixed period of time (from few hours to many days). These stars follow a tight empirical relation between the magnitude (M) of the star and the periodicity (P) of magnitude, given by by these type of stars.

$M = a \; \log_{10} (P) + b$

where 'M' is the absolute magnitude of the star, 'P' is the period (in days) over which the magnitude changes and a,b are some constants found by fitting the data with this linear relation. The distance to the galaxy or the astronomical structure, in which a variable star is found, can be computed using this information. Depending on the periodicity 'P' of a variable star, they are classified into three major types viz., Classic Cepheids, W-Virgins and RR-Lyrae.

**Step 1:**We have provided the 6 data sets and plots of the apparent magnitude, $m_{app}$, (column 1) with the time in days (column 2). This plot is called the*Light curve of the variable star*.

File1 Data File2 Data File3 Data File4 Data File5 Data File6 Data File1 Plot File2 Plot File3 Plot File4 Plot File5 Plot File6 Plot **Step 2:**From the graph estimate the mean apparent magnitude and the period of pulsation of the variable star. Let $m_{mean}$ be the mean apparent magnitude and let*P*be the period of the variable star (in days) as you compute from plotting the raw data.

**Step 3:**From various measurements a linear relation has been established between the absolute magnitude of a variable star and the logarithm of it's periodicity (in days). For Classical Cepheids, the following relation was given by Michael Feast and Robin Catchpole in 1997 :

$M = -2.81 \; \log_{10} (P) - 1.43.$

[Source: Feast, Michael W. & Robin M. Catchpole.*"The Cepheid period-luminosity zero-point from Hipparcos trigonometrical parallaxes"*. Monthly Notices of the Royal Astronomical Society. 286 (1997) L 1-5.]

**Step 4:**Using the above empirical relation, compute the absolute magnitude with one of the variable stars that were provided above.**Step 5:**Now, we have the mean apparent magnitude, $m_{mean}$, absolute magnitude,*M*for the exercise data.**Step 6:**The distance and the magnitudes, $m=m_{mean}$ and*M*of the variable star are related as

$m - M = 5 \; \log_{10} (d/10)$

where 'd' is the distance to the star in parsec. Using this one can work out the distance to the astronomical structure in which these variable stars are located.

The data files provided here are obtained from the*American Association of Variable Star Observers*database. AAVSO has a collection of observations made by several astronomers over time. Another useful place to visit is Variable Star Index.