Eclipsing Binary Lab

http://www.space.com/22509-binary-stars.html

Purpose

The objective of this lab was to detect the presence of an eclipsing binary pair of stars, as well as to characterize information regarding the stars' masses and radii. Beyond applying a light curve characterization technique to this eclipsing binary, there is great significance in the method used in this lab. This same light curve technique can be used to find exoplanets throughout the universe, thereby enabling the search for habitable planets.

The data being collected for this eclipsing binary pair adds to a relatively sparse database that allows us to better understand evolutionary models of the universe. Therefore, by collecting new, accurate data, this lab provides valuable information to the greater scientific community.

Black Image Portion: http://en.wikipedia.org/wiki/Variable_star#/media/File:Light_curve_of_binary_star_Kepler-16.jpg

For this experiment, we used the Clay Telescope on the roof of the Harvard Science Center to to track the movement of the low mass eclipsing binary, NSVS01031772. Incorporated with the Clay Telescope is a CCD camera that allows for capturing light data on the targeted stars. As the ecplising binary pair of stars pass in front of and behind one another, the light curve generated by the CCD images create a distinct shape. As the smaller, dimmer star passes in front of the larger, brighter star, the amount of light reaching the CCD is actually reduced, thus causing a primary dip in the light curve. Then, during a second transit, as the larger star passes in front of the smaller star, a secondary dip in the light curve occurs. This dip is smaller, because the full brightness of the larger star is still seen, rather than being partially blocked (as it is in the other primary transit).

Methods and Theory

To help determine the mass, period, velocity, and separation of the system, we used two primary datasets: Light Curve Data & Radial Velocity Data. Additionally, we leveraged several physical properties and equations to ultimately resolve the mass, radii, and separation of our two star system.

The first dataset for the light curve data was collected during the experiment using the Clay Telescope. A sample light curve for a transiting object is shown below:

From this plot, we can see that there are five primary characterized regions: Baseline 1 (T1), Ingress (T2), Transit (T3), Egress (T4), and Baseline 2 (T5). Each of these regions provide valuable information for making calculations on the system. Additionally, we can characterize an important vertical property of the plot, known as the depth of the transit, \( \delta \). This depth of transit can be used to calculate the radii of the two stars in the system. Over the course of the transit, several mathematical equations can be used to express the physical state of the system. First, to derive an expression for the the radial relationships of the two stars, we can consider what the two depths of the light curve transits tell us. While we might expect the relationship to simply be \( \delta = \frac{ {R_2}^2 } { {R_1}^2} \) as we found in an earlier worksheet, we need to account for the fact that both stars are luminous, and we actually need to utilize the difference in transit depths to create a more accurate expression of: \[\frac{ {R_2}^2 } { {R_1}^2}  = \frac{ \delta_1 } { 1 - \delta_2 } \] Now, we can create expressions for R1 and R2 using the transit times from the light curves. Since the two stars are traveling past each other, their relative velocity during the transit is the sum of their individual radial velocities, \( V_1 + V_2 \). Additionally, for the full transit to be completed, the eclipsing star must travel a distance of \( 2R_1 \) across the eclipsed star, and then an additional \( 2R_2 \) for completing the ingress and egress periods. Since we know the total time of the transit from our light curve data, and we can capture the radial velocities of each star from the spectral analysis data, we can then solve for these two radii: \[ R_1 + R_2 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 } \] To solve for the two different radii values individually, we can use our depth equation from above: \[\frac{ {R_2}^2 } { {R_1}^2}  = \frac{ \delta_1 } { 1 - \delta_2 } \] \[ R_2 =  \sqrt{ \frac{ \delta_1 } { 1 - \delta_2 }} R_1 \] Applying this to our above equation, we get:  \[ R_1 + \sqrt{ \frac{ \delta_1 } { 1 - \delta_2 }} R_1 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 } \] \[ R_1 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 (1 +  \sqrt{ \frac{ \delta_1 } { 1 - \delta_2 }}) } \] Applying the same approach to \( R_2 \), we get: \[ R_2 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 (1 +  \sqrt{ \frac{ 1 - \delta_2 } {  \delta_1 }}) } \]

The next physical quantity that we need to utilize is the center of mass of the system. This is characterized by: \[ X_{com} = \frac{ \sum m_i x_i}{m_i} \] By defining \(X_{com}\) to be zero, we will have the following relationship for our system. \[ M_1 a_1 = M_2 a_2 \] \[ \frac{M_1}{M_2} = \frac{a_2}{ a_1} \] Next, we can use the definition of period as: \[ P = \frac{ 2 \pi a }{V} \]  Then, solving for a, we get: \[ a_1 = \frac{V_1 P }{2 \pi } \] \[ a_2 = \frac{V_2 P }{2 \pi } \] Plugging these values into our mass equations from above, we get a simple expression, since the constants \( 2 \pi \, and \, P \) cancel out from each term: \[ \frac{M_1}{M_2} = \frac{V_2}{ V_1} \] This expression tells us that the larger of the two stars will be moving slower. This allows us to use the radial velocity curves obtained by Lopez and Morales and determine which radial velocity is attributed to which star.

The radial velocity was obtained through spectral analysis, and the data is shown below:

http://www.fas.harvard.edu/~astrolab/rvcurve.pdf

From this plot, we can extract some valuable data about the system. However, to first better understand what is going on with this radial velocity plot, we will overlay an illustration of what the two stars are doing during each orbital phase below.

To interpret this graph, it is important to note that the positive radial velocities correspond to a star moving away from us, while a negative radial velocity corresponds to a star moving towards us. Additionally, we can see from this plot that the maximum radial velocities occur when the stars are side-by-side, with one moving towards us, and one moving away. This allows us to use the doppler effect, where the wavelength of the light will either be increased or decreased depending on the direction that the relevant star is moving. Since we know that the slower velocity is attributed to the more massive star, we can extract this radial velocity data as follows: \[ V_1 = 143.85 \, km/sec \] \[ V_2 = 156.06 \, km/sec \] From this, we can now determine the mass ratio of the stars with the expression: \[ M_1 = \frac{V_2}{V_1} M_2 \] However, to determine an actual value for each mass, we have some more work to do. To achieve this, we can now apply Kepler's law to begin solving our system of equations: \[ P^2 = \frac{ 4 \pi^2 (a_1 + a_2)^3}{G(M_1 + M_2)} \] Solving for mass, we get: \[ M_1 + M_2 = \frac{ 4 \pi^2 (a_1 + a_2)^3}{G P^2} \] Substituting in the expressions for a from above into Kepler's equation, we get: \[ M_1 + M_2 = \frac{ 4 \pi^2 (\frac{P V_1}{2 \pi} + \frac{P V_2}{2 \pi})^3}{G P^2} = \frac{ P (V_1 + V_2)^3 }{2 \pi G}  \] Now, since \[ M1 = \frac{V_2}{V_1} M_2 \] We can write a complete expression as: \[ M_1 + \frac{V_1}{V_2} M_1 = \frac{ P (V_1 + V_2)^3 }{2 \pi G} \] \[ M_1 = \frac{ P(V_1 +V_2)^3}{ 2 \pi G ( 1 + \frac{ V_1}{V_2} ) } \] \[ M_2 = \frac{ P(V_1 +V_2)^3}{ 2 \pi G ( 1 + \frac{ V_2}{V_1} ) } \] 

With these expressions, we now have all of the equations needed to solve for each of our quantities.

Separation:
\[ a_1 = \frac{V_1 P }{2 \pi } \] \[ a_2 = \frac{V_2 P }{2 \pi } \]

Radii:
\[ R_1 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 (1 +  \sqrt{ \frac{ \delta_1 } { 1 - \delta_2 }}) } \] \[ R_2 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 (1 +  \sqrt{ \frac{ 1 - \delta_2 } {  \delta_1 }}) } \]

Mass:
\[ M_1 = \frac{ P(V_1 +V_2)^3}{ 2 \pi G ( 1 + \frac{ V_1}{V_2} ) } \] \[ M_2 = \frac{ P(V_1 +V_2)^3}{ 2 \pi G ( 1 + \frac{ V_2}{V_1} ) } \]
Observations

The light curves generated for this lab were created through data collection via the Harvard Clay Telescopes and an Apogee Alta U47 CCD to collect luminosity measurements.

Overall, there were a total of six datasets collected between March 24th and April 12th on nights with mostly clear visibility. We observed using the R-band filter with the telescope directed at our target of NSVS01031772 at RA, DEC = 13:45:35 + 79:23:48. To ensure proper images were taken, we adjusted the exposure length of the CCD to around 30,000, thereby avoiding saturation (~65,000). To then track our target stars during the observation, we selected guide stars using the control software.

Finally, we set the system to automatically take exposures for the duration of the transit and beyond to allow for a baseline to be established before and after on our light curve.

To account for any blemishes or noise due to the CCD or Telescope, we also took sky flats that were used to normalize our data from each night of observation.

To clean and reduce the data, averages were taken over multiple data points to remove random variance and smooth the final light curve.

Analysis

To analyze the collected data, we used MaximDL to generate our light curve. First, to do this, we delineated which objects in our images were reference stars, and which objects were our targets. The reference stars were used to provide a 'reference' to compare the varying luminosity of our eclipsing binary pair.

http://www.fas.harvard.edu/~astrolab/object_field.png

With these objects identified, we then created light curves for each of the six observation nights. This allowed us to not only compare observations and generate a more reliable light curve, it also allowed us to understand the period of the system. To determine the period, we adjusted the different observation days by an offset until they were aligned. This occurred with a period of 8 hours, 50 minutes, and 8 seconds (8.84 hours). The aligned light curves are shown below:

From this curve we could then establish the depth and overall time of both the primary and secondary transit. To calculate the depth of the transits, we subtracted the lowest point of each dip on the curve from the baseline level. With a baseline level of ~1.35 and minimums of 0.68 and 0.75 respectively, we found that \( \delta_1 = 0.67 \) \( \delta_2 = 0.6 \) However, we then normalized the data by dividing by the baseline, giving us normalized values of \[ \delta_1 = 0.496 \] \[ \delta_2 = 0.444 \] Next, we used the light curve to determine the transit time of each dip. For the primary transit, this produced a time of about 1.3 hours, while the secondary transit was slightly shorter at around 1.2 hours.

With this data processed, we are now able to calculate final values for each of our desired characteristics of the system.

Results

From the Methods section above, we can recall equations we will use to calculate each respective value:

Separation:
\[ a_1 = \frac{V_1 P }{2 \pi } \] \[ a_1 = \frac{ (143.85 \, km/s) (8.84 \, hours)(3600 \, s/hr )}{2 \pi } = 7.286 \times 10^{10} \, km \sim 1.047 R \odot \]

\[ a_2 = \frac{V_2 P }{2 \pi } \]\[ a_2 = \frac{ (156.06 \, km/s) (8.84 \, hours)(3600 \, s/hr )}{2 \pi } = 7.9 \times 10^{10} \, km \sim 1.14 R \odot \]

\[ a_1 + a_2 = 1=2.187 R \odot \]

Radii:
\[ R_1 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 (1 +  \sqrt{ \frac{ \delta_1 } { 1 - \delta_2 }}) } \]

\[ R_1 = \frac{ ( 143.85 \, km/s + 156.06 \, km/s) \times 1.3 \, hours \times 3600 \, s/hr } { 2 (1 +  \sqrt{ \frac{ 0.496 } { 1 - 0.444 }}) } = 3.61 \times 10^5 km \sim 0.52 R \odot \]

\[ R_2 = \frac{ ( V_1 + V_2 ) \times t_{transit} } { 2 (1 +  \sqrt{ \frac{ 1 - \delta_2 } {  \delta_1 }}) } \]

\[ R_2 = \frac{ ( 143.85 \, km/s + 156.06 \, km/s) \times 1.3 \, hours \times 3600 \, s/hr } { 2 (1 +  \sqrt{ \frac{ 1 - 0.444 } {  0.496 }}) } = 3.54 \times 10^5 \, km \sim 0.508 R \odot \]

Mass:
\[ M_1 = \frac{ P(V_1 +V_2)^3}{ 2 \pi G ( 1 + \frac{ V_1}{V_2} ) } \]
\[ M_1 = \frac{ (8.84 \, hours)( 3600 s/hr)(143.85 \, km/s + 156.06 \, km/s)^3}{ 2 \pi G ( 1 + \frac{ 143.85 \, km/s}{156.06 \, km/s} ) } = 1.08 \times 10^{33} \, g \sim 0.53 M \odot \]

\[ M_2 = \frac{ P(V_1 +V_2)^3}{ 2 \pi G ( 1 + \frac{ V_2}{V_1} ) } \]
\[ M_2 = \frac{ (8.84 \, hours)( 3600 s/hr)(143.85 \, km/s + 156.06 \, km/s)^3}{ 2 \pi G ( 1 + \frac{ 156.06 \, km/s}{ 143.85 \, km/s} ) } = 9.77 \times 10^{32} \, g \sim 0.48 M \odot \]

Conclusion:

These values match reasonably well with the results obtained by Lopez-Morales, with a slight degree of error. Given that observations were taken in the city environment of Cambridge, it's impressive that our results matched so closely with the previous results!