Note - the data for all the interactive sessions is available for download.

# Measuring Rotation Periods for Stars in M44

##### Version 0.1

The goal of this exercise is to measure the rotation period for several young stars located in the Beehive open cluster (M44). As a nearby open cluster that is only $\sim 600$ Myr old, it serves as a good calibration point for age-rotation relations (i.e. Gyrochronology). The exercises here largely follow results from Agueros et al. 2011.

We will use light curves of known M44 members to measure their rotation periods. We will then plot these periods against both the J-H colors of the stars and their mass to see which stars (if any) have spun down after 600 Myr.

For simplicity, we have already measured relative light curves for each of the stars in this cluster, which was observed during the first year of PTF operations.

By AA Miller (c) 2014 Aug 22

## Problem 1) Measure Rotation Periods

Light curves for the M44 members are included in the $\texttt{light_curves/}$ with names: $\texttt{cluster_star??.lc}$, where the "??" indicates the number for the star.

The format of these light curves is slightly different from what you seen elsewhere during these sessions - we recommend you look at the files to see how they are formatted.

##### Part A) Measure Periods

For each of the cluster stars - measure the best-fit period using a generalized lomb-scargle periodogram.

Hint - you can adapt code that you developed during the previous interactive session.

In []:
### measure rotation periods for the cluster stars

# COMPLETE
clusterLCs = glob.glob("light_curves/"    # COMPLETE
# COMPLETE

##### Part B) Plot Phase Folded Light Curves

Now that you have measured periods for each of the cluster stars, plot up the phase folded light curves to quickly check that the best-fit periods look reasonable.

In []:
### plot phase-folded light curves

# COMPLETE
errorbar(    # COMPLETE
# COMPLETE


## Problem 2) Empirical Relations for Rotation

The headers of the light curve files contain the $J - K_s$ colors of each of the stars in question. Extract these values from the text headers, and use it to make a plot of rotation period vs. $J - K_s$.

Hint - one way to get information from the textfile header is to read-in the lines of the file using the Python function $\texttt{readlines}$:

with open(filename) as f:
ll = f.readlines()

Now, the first line from the file is stored in $\texttt{ll[0]}$, while the second line is stored in $\texttt{ll[1]}$, etc.

In []:
### make a plot of rotation period vs. J-K

with open(     # COMPLETE

# COMPLETE
# COMPLETE

plot(    # COMPLETE
# COMPLETE
yscale("log")


If you compare to Fig. 5 of Agueros et al. 2011 what are the main differences that you notice? Does this tell you anything about PTF data?

## Problem 3) Modeled Relations for Rotation

Similar to Problem 2, the light curve headers also contain the absolute magnitude of each of these stars are measured in the $K_s$ band ($M_{K_s}$). There are empirical equations for converting from $M_{K_s}$ to stellar mass ($M_\odot$), which we will use to plot rotation against mass.

Using the empirical relation from Delfosse et al. 2000():

$\log (M/M_\odot) = 10^{-3} \times ( 1.8 + 6.12 \, M_{K_s} + 13.205 \, M_{K_s}^2 - 6.2315 \, M_{K_s}^3 + 0.37529 \, M_{K_s}^4 )$

plot the best-fit period for the young stars against their mass.

In []:
### make a plot of Msun vs. J-K

with open(     # COMPLETE

# COMPLETE
# COMPLETE

plot(    # COMPLETE
# COMPLETE
yscale("log")


## Bonus Problem) Examine period significance of each of the stars

Using an algorithm of your choosing, evaluate the significance of each of the periods that you measured for the cluster members of M44. Should any of these members be excluded from the above plot due to a poorly fit light curve (i.e. are all of the best-fit periods actually reliable)

How does the plot change if you remove the unreliable periods?

In []: