Eclipsing Binary Star Lab

Introduction:

In this lab we will be analyzing the eclipsing binary star system NSVS01031772. This system happens to eclipse directly in our line of sight, making it perfect for observing. There has been little data collected on systems like this (with stellar masses less than 0.6\(M_{\odot}\)) due to the dim nature of the stars. This specific system was discovered by Mercedes Lopez-Morales, Jerome A. Orosz, J. Scott Shaw, Lauren Havelka , Maria Jesus Arevalo, Travis McIntyre, and Carlos Lazaro, who published their results "NSVS01031772: A New 0.50+0.54 M⊙ Detached Eclipsing Binary." For this lab, we will be taking data on this binary star system over the course of three weeks and thus we will add three weeks of data to the existing pool of knowledge on binary star systems. By finding the masses and radii of these stars, we can better understand crucial mass-radius relationships for stars and obtain information that is vital for figuring out the masses and radii of any exoplanets potentially orbiting the stars.

(A binary star system Source)

In order to find the masses, radii, and separation between the two stars, we will analyze the changing brightness of the system. The two stars are so far away, they appear as one star. The only reason we know there are two stars is because there is a dip in the brightness, or flux, of the entire system when one star passes in front of the other. The largest change in flux comes when the smaller star passes in front of the larger star, blocking some of its light. The second dip, or transit, comes when the smaller star passes behind the larger one and we can only observe the light from the larger star. 

Source

The graph of flux over time is called a light curve. Once we have this light curve, we can extrapolate the total period, and time of each transit in order to calculate mass, radius, and distance.

Theory 

In order to analyze the binary star system, we will do our calculations in the frame of the center of mass of the system for simplicity's sake. Defining it as such gives us two distances, instead of the usual one between the two bodies of a gravitational system.

As discussed in worksheet 13.2, we can learn a lot from the radial velocity of a star in a binary system. We know where the star is in it's orbit based on the relative radial velocity of the star. For example, if the star is at maximum radial velocity, it is in the point of its orbit where all of the velocity is the radial because the star is pointing towards us.  

From this, we can extrapolate the actual velocity of each star, K, as the maximum radial velocity of each star. For the purposes of this lab, we will use the data from Lopez-Morales in order to get K for each star.

\[K_1= 155 \pm 5 km/s = 1.55 \times 10^{7} \pm .05  cm/s\]

\[K_2= 145 \pm 5 km/s = 1.45 \times 10^{7} \pm .05 cm/s\]

This is the radial velocity chart for the two stars, which corresponds to the previously mentioned radial velocities of each star.

Once we have velocity, all we need is period in order to figure out the distance between each star and the center of mass (See this blog post for a derivation of the following equations, the blog post discusses exoplanets, but the same principals apply to a binary star system).
\[a_1 = \frac{Pv_1}{2 \pi} \]
\[a_2 = \frac{Pv_2}{2 \pi} \]

We can find the period of rotation from the light curve of the data collected. The period is the amount of time one "cycle" of the light curve takes.

Once we have the distances, we have everything we need to get the mass of the stars. Whenever we have the period of the system, we know we will probably be using Kepler's third law.
\[P^2 = \frac{4 \pi^2 a^3}{GM}\]
Where \[a_1 + a_2 = a \] So
\[ P^2 = \frac{4 \pi^2 (a_1 + a_2)^3}{G(M_1 + M_2)} \]
Rearranging this equation for mass gives:
\[ M_1 + M_2 = \frac{4 \pi^2 (a_1 + a_2)^3}{GP^2} \]
Because we know the center of mass is zero, we know the relationship:
\[M_1 a_1 = M_2 a_2 \]
\[M_2 = \frac{M_1 a_1}{a_2} \]
We can plug this in to Kepler's law in order to get an expression for each mass:
\[ M_1 + \frac{M_1 a_1}{a_2} = \frac{4 \pi^2 (a_1 + a_2)^3}{GP^2} \]
\[ M_1  = \frac{4 \pi^2 (a_1 + a_2)^3}{GP^2} \cdot \frac{1}{1 + \frac{a_1}{a_2}} \]
\[ 1 + \frac{a_1}{a_2} = \frac{a}{a_2}\]
\[ M_1 = \frac{4 \pi^2 a^2 a_2}{GP^2} \]
This expression can be used for both masses once we go to analyze data.

To get the radius of one star, we can use a simple, distance over time formula. We can extrapolate the time of the first transit from the light curve.
\[t_{transit1} = \frac{D}{v} = \frac{2R_2}{v_1} \] \[R_2 = \frac{t_{transit1}v_1}{2} \]
This works for the first transit, because we know the smaller star is passing in front of the bigger star when in the first, larger dip.

In order to find the second radius, we can once again turn to our light curve. The light curve tells us the change in flux of the star system. From worksheet 13.2 we found that the change in flux can give us the relationship between the radii of the two stars. \[ \frac{F_T}{F_{\star}}  = \frac{L_{\star}(\pi R_{\star}^2 - \pi R_P^2)}{ L_{\star} \pi R_{\star}^2 } = 1 - \frac{ R_P^2}{R_{\star}^2 } \]

So we know \[ \Delta F = \frac{ R_2^2}{R_1^2} \] \[ R_1 = \sqrt{ \frac{R_2^2}{\Delta F}} \]

Observations:

To observe the stars, we will be using the 16" Clay Telescope on top of the Science Center.  We started taking data at the end of March and took data pretty much whenever there was a clear night and ended up with six nights worth of data. My lab group observed on day 3, which was March 29, the night started clear, but we had interspersed clouds throughout the night. The clouds and light pollution could potentially present errors. Additionally, at some point in the night, someone went into the dome and turned on the red light. Because we were observing in the red filter, the variability of the data increased dramatically in that time period.

Once up at the Clay Telescope, we learned how to operate the dome, move the telescope, and switch between the eyepiece and CCD camera. To move the telescope we used a computer program called The Sky, which gives a map of the sky which gives the RA and DEC of celestial objects. Before we got to work, we had some fun looking at the moon, Jupiter, and the black eye galaxy.

Once we had the coordinates, we could slew the telescope to the stars. In order to make sure the telescope tracked everything properly, we set a reference star in the field of view for the telescope to follow. We used a red filter to observe because the stars emit the most light at the red end of the spectrum. We took 1004 90 second exposures over 6 nights in order to get the data that will be analyzed below.

Analysis 

In order to analyze our data we used photometry. This is the process of analyzing the flux or brightness of an object. The field of view of our images looks something like the image below, with the binary stars in the center. 

Once we have the images, we can analyze the brightness of our object in comparison to other reference stars in the field of view. This allows us to make a light curve with flux over time.

We processed our images using flat fields in Maxim DL. Flat field corrections allow us to take clearer images by editing out imperfections in the images, such as dust or dents in the mirror.

Results 

The period of the orbit is 31805 seconds.

\[K_1= 155 \pm 5 km/s = 1.55  \pm .05 \times 10^{7}  cm/s \]

\[K_2= 145 \pm 5 km/s = 1.45 \times  \pm .05 10^{7}  cm/s \]

\[t_{transit1} = 1.5 \pm .1 hours = 5.4 \times 10^3 seconds \]
\[t_{transit1} = 1.4 \pm .1 hours = 5.04 \times 10^3 seconds \]
\[\Delta F_1 = 0.65 \]
\[\Delta F_2  = 0.55 \]

Now we can analyze our result to find the distances.

\[a_1 = \frac{Pv_1}{2 \pi} = \frac{(3.18 \times 10^4) (1.55 \times 10^{7})}{2 \pi} = 7.84 \times 10^{10} cm \] \[a_2 = \frac{Pv_2}{2 \pi} = \frac{(3.18 \times 10^4) (1.45 \times 10^{7})}{2 \pi} = 7.34 \times 10^{10} cm \] \[a = a_1 + a_2 = 7.84 \times 10^{10} + 7.34 \times 10^{10} = 1.52 \times 10^{11} cm \] Now we can calculate the masses of each star:\[ M_1 = \frac{4 \pi^2 a^2 a_2}{GP^2} \] \[ M_1 = \frac{4 \pi^2 (1.52 \times 10^{11})^2 (7.34 \times 10^{10})}{(6.67 \times 10^{-8})(3.18 \times 10^4)^2} = 9.93 \times 10^{32} grams\] Which is equivalent to \(0.499 M_{\odot} \) \[ M_2 = \frac{4 \pi^2 a^2 a_1}{GP^2} \] \[ M_2 = \frac{4 \pi^2 (1.52 \times 10^{11})^2 (7.84 \times 10^{10})}{(6.67 \times 10^{-8})(3.18 \times 10^4)^2} = 1.06 \times 10^{33} grams\] Which is equivalent to \(0.533 M_{\odot} \)

Finally, we can calculate the radius of each star \[R_2 = \frac{t_{transit1}v_1}{2} \] \[R_2 = \frac{5.4 \times 10^3  \cdot 1.55 \times 10^{7}}{2} = 4.19 \times 10^{10} cm \] Which is \(.602 R_{\odot} \)

To get the second radius we can plug in the this first radius and the change in flux in the first transit to he equation we found earlier: \[ R_1 = \sqrt{ \frac{R_2^2}{\Delta F}} \] \[ R_1 = \sqrt{ \frac{(4.19 \times 10^{10})^2}{0.65}} = 5.19 \times 10^{10} cm \] This is \(0.75R_{\odot} \)

Result	Astronomy 16	Lopez-Morales	Percent Error
M1	0.499M	0.4982M	0.16%
M2	0.533M	0.5428M	1.81%
R1	0.75R	0.5260R	42.59%
R2	0.602R	0.5088R	18.32%

There are obviously quite a few discrepancies between our results and theirs. This could have come from error in how we collected our data and error in how we analyzed the data. Error in the data collection could have come from the intermittent clouds, light pollution, and the mysterious red light that turned on during the night. Error in data analysis would have come from many sources, the most prominent of which is rounding and estimation of values visually from the data rather than extrapolating accurate numbers. This causes a lot of uncertainty in the values we analyzed, and even more uncertainty in the final results. The percent error of the masses versus the radii demonstrates this well. When calculating mass, all we needed was the velocity of each star, which we extrapolated from the given graph and there are very small percent errors in those calculations. For the masses we used data from our lab and had much larger errors. For calculating transit time, we used the first transit, which only had two nights worth of data and that could have caused some error as well. To improve the accuracy of this lab, I would ideally fit some sort of curve to the data that would allow me to extrapolate more accurate results.

Acknowledgements: I worked with Barra on this lab.

Citations:
I used our lab guidelines, and the Lopez-Morales paper.
López-Morales, M., Orosz, J. A., Staw, J. S., Havelka, L. et al., 2006, astro-ph/0610225v1, LM06