Introduction/ Purpose:
This blog post discusses and analyzes the data of an eclipsing low-mass binary star system, NSVS01031772, observed by the Astronomy 16 class at Harvard University using the optical Clay Telescope. The goal of the lab is to find the masses, radii, and semi-major axes of the stars of the double-lined spectroscopic binary using data from a radial velocity plot and a light curve.
A binary star is a system of two stars which are gravitationally bound and are orbiting around their common center of mass. When binary stars are very far away, they cannot be resolved and are seen as a single point, but because they are orbiting each other, we see a decrease in brightness as they cross over one another. This is because the unresolved point that we see has the brightness of both stars, but when one is crossing the other, it blocks the light from the other star, as demonstrated in the animation below.
A binary star is a system of two stars which are gravitationally bound and are orbiting around their common center of mass. When binary stars are very far away, they cannot be resolved and are seen as a single point, but because they are orbiting each other, we see a decrease in brightness as they cross over one another. This is because the unresolved point that we see has the brightness of both stars, but when one is crossing the other, it blocks the light from the other star, as demonstrated in the animation below.
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| Figure 1: The graph shows the dip in brightness as one of the stars transits the other. Image Credit: http://38.media.tumblr.com/fa98981e91a5ea5d13953d3a8086c2ac/tumblr_n0s10t9MS21rnq3cto1_500.gif |
For this lab, we observed the transits of the low-mass binary stars and analyzed the depths of the dips in their brightness to find information about the masses, radii, and separation between the two stars. Because these low-mass binary stars have low surface brightness, they are hard to detect, though there are many of these stars. Due to this fact, there is limited data on stars less massive than \(0.6M_\odot\). Finding information on the mass-radius relationship is important, however, since it provides constraints for evolutionary models of main sequence stars. It can also be used to infer similar information on planets orbiting these stars.
Methods:
López-Morales et al. performed an investigation on the binary star, NSVS01031772. In their paper, they included a radial velocity (RV) plot of the two stars in the system, which they found using Doppler spectroscopy. We use this plot to determine important information about the binary system. The RV plot shows the radial velocity of each star at a given time, where negative radial velocity means that the star is moving towards the observer, and positive radial velocity means that the star is moving away from the observer.
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| Figure 2: The radial velocity (RV) plot of NSVS01031772. Image Credit: López-Morales et al., 2006 |
The RV plot can be used to find the maximum radial velocities of each star, which is given by the peak radial velocities of the stars subtracted by the radial velocity of the center of mass. The center of mass has a radial velocity to due relative motion between us and the binary system and it is shown in the plot by the horizontal dotted line at a radial velocity of \(19.0\pm 1.0\) km/s. With the maximum radial velocities, one can determine the mass ratio of the system. The star with a larger maximum radial velocity is less massive than the star with smaller maximum radial velocity. We will call the star represented by a solid line in the RV plot Star 1, and the star represented by the dotted line Star 2. Star 1 has a peak radial velocity of about -120 km/s and Star 2 has a peak radial velocity of about 180 km/s. Subtracting the radial velocity of the center of mass, this gives us maximum radial velocities of around -140 km/s for Star 1 and 160 km/s for Star 2. These are poor approximations, however, since they are measured by eye along the axis of the plot. López-Morales et al. provide data with significantly better accuracy using the Southeastern Association for Research in Astronomy (SARA) data set, getting maximum radial velocities of \(143.85\pm 0.37\) km/s for Star 1 and \(156.06\pm 0.88\) km/s for Star 2. From this, we can conclude that Star 1 is more massive than Star 2.
Solving for the mass ratio, we can begin with a relation derived from the center of mass equation in a previous problem:
\(a_1M_1=a_2M_2\)
\(\dfrac{M_1}{M_2}=\dfrac{a_2}{a_1}\)
Where \(a_1\) is the distance from Star 1 to the center of mass of the system, \(a_2\) is the distance from Star 2 to the center of mass, \(M_1\) is the mass of Star 1, and \(M_2\) is the mass of Star 2. Then, from the equation for the orbital period \(P=\frac{2\pi a}{v}\), we get the relation:
\(\dfrac{a_1}{a_2}=\dfrac{v_1}{v_2}\)
So we find a mass ratio of:
\(\dfrac{M_1}{M_2}=\dfrac{v_2}{v_1}=\dfrac{156.06\text{ km/s}}{143.85\text{ km/s}}\approx 1.08\)
In addition to finding the maximum radial velocities and the mass ratio, we can also use the RV plot to describe the relative motion of the stars at different points on the plot. The image below depicts what the orbit of the two stars would look like based on the RV plot. Star 1 is shown in orange, Star 2 is shown in yellow, and their center of mass is located at the red pluses.
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| Figure 3: The orbits of the two stars relative to each other. Star 1 is shown in orange and is the more massive than Star 2, shown in the yellow. |
In the first scenario, the stars are at their peak radial velocities, which means that all of their motion is directed towards and away from the observer. In the second scenario, where there is no radial velocity, the stars' motion is directed to the left and right of the observer, with no components of motion along the direction of the observer. Finally, the third scenario shows a point along the orbit where some of the motion is directed towards and away from the observer, and some is directed to the left and right, here the radial velocity is smaller than the peak radial velocity.
Another important plot of data is the light curve, which shows the flux of electromagnetic radiation of an astronomical object over time. They are generally in a particular band or frequency interval, and can sometimes be periodic, as in the case of eclipsing binary stars. Using the Clay telescope to collect optical data of the NSVS01031772 binary star system, we were able to plot the light intensity of the eclipsing binaries over time. At the points where the stars transit one another, there would be a dip in the light intensity. The frequency of these dips in intensity can tell us the period of the stars' orbit. Furthermore, the depth of the dip, its duration, and time of ingress and egress can be determined from the light curve and can be used to find the radii, masses, and semi-major axes of the stars.
To find the masses of the stars, we need to use Kepler's Third Law, the center of mass equation, the equation for period, and the Newtonian improvement to the Keplerian semi-major axis:
To find the masses of the stars, we need to use Kepler's Third Law, the center of mass equation, the equation for period, and the Newtonian improvement to the Keplerian semi-major axis:
\(P^2=\dfrac{4\pi ^2a^3}{G(M_1+M_2)}\)
\(a_1M_1=a_2M_2\)
\(P=\dfrac{2\pi a_1}{v_1}=\dfrac{2\pi a_2}{v_2}\)
\(a=a_1+a_2\)
We begin with Kepler's Third Law, substituting the Newtonian improvement to the semi-major axis for a, and substituting \(M_2=\frac{a_1}{a_2}M_1\):
\(P^2=\dfrac{4\pi ^2(a_1+a_2)^3}{G(M_1+\frac{a_1}{a_2}M_1)}\)
Then, solving the equations of the periods for the semi-major axes and using the relation \(\frac{a_1}{a_2}=\frac{v_1}{v_2}\):
\(P^2=\dfrac{4\pi ^2\left(\frac{P(a_1+a_2)}{2\pi}\right)^3}{G(M_1+\frac{a_1}{a_2}M_1)}\)
\(\Longrightarrow M_1=\dfrac{v_2P(v_1+v_2)^2}{2\pi G}\)
Repeating a similar process for the second mass, we get:
\(\Longrightarrow M_2=\dfrac{v_1P(v_1+v_2)^2}{2\pi G}\)
Now, to solve for the radii of the stars, we need to use the transit depth equation, which was derived in a previous problem, and the transit time expression, which we must derive. To do this, we must consider the Stefan-Boltzmann law, \(L\propto R^2T^4\). From here, we can see that if the temperatures are relatively similar, a star with a larger radius will have a bigger luminosity. Thus, we can conclude that the primary transit (the transit with the larger depth) occurs when the star with smaller radius passes in front of the star with larger radius, blocking a higher proportion of light; the secondary transit (with a smaller depth) would then occur when the star with a larger radius passes in front of the star with a smaller radius.
From Figure 4, we can see that in the frame of the bigger star, the smaller star has a velocity of \(v_1+v_2\) and must travel a distance of \(2R_1+2R_2\). The transit time, \(t_{trans}\), is given by:
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| Figure 4: The transit of the smaller star across the bigger star. |
From Figure 4, we can see that in the frame of the bigger star, the smaller star has a velocity of \(v_1+v_2\) and must travel a distance of \(2R_1+2R_2\). The transit time, \(t_{trans}\), is given by:
\(t_{trans}=\dfrac{2(R_1+R_2)}{v_1+v_2}\)
We also know the transit depth equation from a previous result:
\(\delta =\left( \dfrac{R_2}{R_1}\right)^2\)
This equation, however, was derived under the assumption that the transiting object was opaque. In this case, the transiting star is emitting its own light. We can correct for this by subtracting the luminosity of the transiting star from the light curve. Figure 5 gives and example of what this would look like.
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| Figure 5: The dotted line represents the light curve with luminosity of the smaller star subtracted. \(\delta_1\) is the depth of the primary transit, and \(\delta_2\) is the depth of the secondary transit. |
Having subtracted the depth of the secondary transit from the light curve, our transit depth equation for the primary transit now becomes:
\(\left( \dfrac{R_2}{R_1}\right)^2=\dfrac{\delta_1}{1-\delta_2}\)
Solving for \(R_2\) and plugging into the transit time equation, we get:
\(R_1+\sqrt{\dfrac{\delta_1}{1-\delta_2}}R_1=\frac{1}{2}(v_1+v_2)t_{trans}\)
\(\Longrightarrow R_1=\frac{1}{2}(v_1+v_2)t_{trans}\left(\dfrac{1}{1+\sqrt{\frac{\delta_1}{1-\delta_2}}}\right)\)
And for \(R_2\), we get:
\(\Longrightarrow R_2=\frac{1}{2}(v_1+v_2)t_{trans}\left(\dfrac{1}{1+\sqrt{\frac{1-\delta_2}{\delta_1}}}\right)\)
Observations:
The optical observations of the binary NSVS01031772 were done using the Clay Telescope, which is a 0.4 m DFM engineered telescope, located on the roof of the Harvard Science Center. The Clay Telescope has a DFM filter wheel with Bessel filters and an Apogee Alta U47 imaging CCD with a 13' x 13' field of view. The CCD contains \(1024\times 1024\) pixels, which means that each pixel has an angular diameter of approximately 0.76'. Furthermore, the telescope is controlled using a Telescope Control System and also can be controlled using The Sky software, which provides a map of the sky with coordinates to many astronomical objects preprogrammed. This allows one to easily locate and slew the telescope to a particular object. Also, the telescope and the dome have auto-tracking features which follow a target while compensating for the rotation of the Earth. A guide star can also be used for more accurate tracking.
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| Figure 6: The Clay Telescope Image Credit: http://isites.harvard.edu/fs/docs/icb.topic207662.files/Clay_Telescope_3.jpg |
The observing process was broken up into groups that gathered data separately on different days beginning from March 24th until April 12th. I was observing on the night of April 11th, 2015. The weather was fairly clear and mostly cloudless, with a temperature of \(50^\circ\) F and a relative humidity of 32%.
We observed the binary NSVS01031772 at a Right Ascension of 13:45:35, and a Declination of +79:23:48. The images were taken as 60 second exposures in the R-band filter since the stars were M-type dwarfs with low enough temperatures that they were emitting significantly in the red wavelengths.
Possible error in our data could result from the slight cloud coverage, light pollution from Cambridge and Boston area, and some difficulties with the MaxIm DL software which caused stoppages in data collection on some of the nights.
Analysis:
The reductions on the raw data were done using MaxIm DL. MaxIm DL was set to automatically take dark frames and biases. Dark frames are exposures taken with the aperture lens closed in order to account for dark current from the CCD. Dark current is a small electric current running through the CCD that creates noise in images. Bias frames are images that are obtained without any exposure time and contain noise from the internal electronics. Additionally, flat frames were taken shortly after sunset, which are images of the uniformly lit sky. To account for any non-uniformities, the telescope is slewed a little and other flat frames are taken. All of these flat frames are then averaged and they are used to account for blemishes such as dust or smudges on the lens and any varying sensitivity of different regions of the CCD.
Using MaxIm DL, the raw data was calibrated with the dark frames, biases, and flats. Then, photometric analysis was done on the images, which is a process that measures the electromagnetic flux, or the photon count, on selected points on the images.
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| Figure 7: This is the field of view of the observations. "Obj1" refers to the object we are observing, binary system NSVS01031772, and there are four reference stars selected for photometric purposes. When taking images, we had to make sure all stars were centered in the image. Image Credit: http://www.fas.harvard.edu/~astrolab/object_field.png |
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| Figure 8: The light curve for the binary system, NSVS01031772, with data taken from six different nights of observing. The curves from the various nights of observations were aligned on the plot by using an orbital period of 8.84 fractional hours. The dip in the curve on the left is the primary transit, which occurs when the smaller star passes in front of the bigger star. The dip on the left is the secondary transit, which occurs when the larger star passes in front of the smaller star. |
On the Excel spreadsheet, the period had to be found adjusted until the curves for the six different nights of observation were aligned. The resulting value would be the orbital period of the binary system, which we found to be 8.84 fractional hours, confirmed by López-Morales et al. This value for the period has a negligible uncertainty, and was treated as a fixed constant by López-Morales et al. paper. We can also see by visual inspection of the light curve that the baseline (the flux of both stars before transit) is about 1.30. The depth of the primary transit is about \(0.60\pm 0.03\) (found by subtracting 1.3-0.7), and the depth of the secondary transit is about \(0.51\pm 0.02\) (found by subtracting 1.3-0.79).
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| Figure 9: The top panel is a magnified view of the primary transit and the bottom panel is magnified view of the secondary transit. The horizontal dotted lines are the estimated errors for the determined values of the transit depths. The estimated values for the depths are \(0.60\pm 0.03\) for the primary transit, and \(0.51\pm 0.02\) for the secondary transit. |
It is also important to note that the flux is normalized to 1.3. In our calculations, we need the light curve to be normalized to 1. In order to do this, we must divide the values of the graph by 1.3. This changes our transit depths to \(0.46\pm 0.02\) for the primary transit, and \(0.39\pm 0.02\) for the secondary transit. We can also see from the curves in Figure 9 that the transit times are approximately \(1.30\pm 0.20\) hours.
Results:
With our knowledge of the light curve, we can again find the motion of the two stars relative to each other over time.
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| Figure 10: The relative motion of the stars at different points in time along the light curve. The direction of the orbit is shown only for the sake of consistency, however, since the direction of the orbit is unable to be directly concluded from the information provided by the light curve. |
Based on our knowledge of primary and secondary transits, we know that the smaller star will be directly in front of the larger star at the bottom of the primary transit, and it will be directly behind the larger star at the bottom of the secondary transit. On the baseline, neither of the stars are covering each other, so we get the luminosity of both stars. The difference between this method and the radial velocity plot method is that in this case, we do not know the direction of the orbit from the information provided by the light curve.
We can now use the expressions derived in the methods section to calculate the masses, radii, and separation of the two stars. We know that \(v_1\approx 143.85\pm 0.37\) km/s, \(v_2\approx 156.06\pm 0.88\) km/s, and \(P=8.84\) hours. Plugging these values into our mass equations:
\(M_1=\dfrac{v_2P(v_1+v_2)^2}{2\pi G}\)
\(M_1=\dfrac{(156.06\text{ km/s})(8.84\text{ hr}\times 3600\text{ s hr}^{-1})(143.85\text{ km/s}+156.06\text{ km/s})^2}{2\pi (6.67\times 10^{-8}\text{ cm}^3\text{ g}^{-1}\text{ s}^{-1})}\)
\(M_1\approx 1.07\times 10^{33}\) g
And similarly for \(M_2\):
\(M_2=\dfrac{v_1P(v_1+v_2)^2}{2\pi G}\)
\(M_2=\dfrac{(143.85\text{ km/s})(8.84\text{ hr}\times 3600\text{ s hr}^{-1})(143.85\text{ km/s}+156.06\text{ km/s})^2}{2\pi (6.67\times 10^{-8}\text{ cm}^3\text{ g}^{-1}\text{ s}^{-1})}\)
\(M_2\approx 9.83\times 10^{32}\) g
Now, to solve for the radii, we know that the transit time is \(t_{trans}=1.3\pm 0.20\) hrs, and the depths are \(\delta_1 =0.46\pm 0.02\) and \(\delta_2 =0.39\pm 0.02\). Plugging these values into our expressions, we get:
\(R_1=\frac{1}{2}(v_1+v_2)t_{trans}\left(\dfrac{1}{1+\sqrt{\frac{\delta_1}{1-\delta_2}}}\right)\)
\(R_1=\frac{1}{2}(143.85\text{ km/s}+156.06\text{ km/s})(1.3\text{ hr}\times 3600\text{ s hr}^{-1})\left(\dfrac{1}{1+\sqrt{\frac{0.46}{1-0.39}}}\right)\)
\(R_1\approx 3.76\times 10^{10}\text{ cm}\approx 0.54 R_\odot\)
Similarly for \(R_2\), we get:
\(R_2=\frac{1}{2}(v_1+v_2)t_{trans}\left(\dfrac{1}{1+\sqrt{\frac{1-\delta_2}{\delta_1}}}\right)\)
\(R_2=\frac{1}{2}(143.85\text{ km/s}+156.06\text{ km/s})(1.3\text{ hr}\times 3600\text{ s hr}^{-1})\left(\dfrac{1}{1+\sqrt{\frac{1-0.39}{0.46}}}\right)\)
\(R_2\approx 3.26\times 10^{10}\text{ cm}\approx 0.47R_\odot\)
Finally, to solve for the separation between the two stars, we only need to use the Newtonian improvement to the Keplerian semi-major axis and the equations for period:
\(P=\dfrac{2\pi a_1}{v_1}=\dfrac{2\pi a_2}{v_2}\)
\(a_1=\dfrac{P v_1}{2\pi}\) and \(a_2=\dfrac{Pv_2}{2\pi}\)
\(\Longrightarrow a=\dfrac{P(v_1+v_2)}{2\pi}\)
Plugging in values, we get:
\(a=\dfrac{(8.84\text{ hr}\times 3600\text{ s hr}^{-1})(143.85\text{ km/s}+156.06\text{ km/s})}{2\pi}\)
\(a\approx 1.52\times 10^{11}\text{ cm}\approx 2.19R_\odot\)
Conclusion:
In this lab, we observed the eclipse of the binary NSVS01031772 between March 24th and April 12th using the Harvard Clay telescope. From our optical observations, we were able to create a light curve. Drawing information from the light curve, we found the period of the stars' orbit, their radii, masses, and separation. When comparing our calculations to those done in the López-Morales et al. paper, our calculations are fairly accurate. The table below compares our results to those of López-Morales et al.
In this lab, we observed the eclipse of the binary NSVS01031772 between March 24th and April 12th using the Harvard Clay telescope. From our optical observations, we were able to create a light curve. Drawing information from the light curve, we found the period of the stars' orbit, their radii, masses, and separation. When comparing our calculations to those done in the López-Morales et al. paper, our calculations are fairly accurate. The table below compares our results to those of López-Morales et al.
The uncertainties of our measurements were done by eye. These uncertainties can be propagated through our calculations to find the uncertainties of our results. However, due to the fact that the uncertainties were done in a heuristic manner, it is unnecessary to do so for the purposes of this experiment.
Possible sources of error might come from noise due to light pollution from Boston area. Also, on some of the observing nights, groups had problems with MaxIm DL, which crashed and lost a few data points, though the effects of this on accuracy are probably negligible. Furthermore, in our calculations, we assumed perfectly circular orbits and zero inclination of the orbit. It is unlikely that these assumptions would hold perfectly true, though the extent to which they might affect our results is unknown. Also, some of the values used in the calculations were visually determined, which allows room for human error. Despite these possible sources of error, our results agreed to a large extent with the results of the paper by López-Morales et al. Thus, we can suppose that our results will have small uncertainties.
I would like to thank Charles Law, Andy Mayo, Daniel Chen, and the 3 TFs for their help in completing this lab and in performing the necessary analysis of the data.
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