In this lab, we observed a transiting binary star system using the Harvard Clay telescope. Since the system is far too far away to dirtily resolve wither equipment, we measured the observed luminosity of the system as a function of time. With this data, we can effectively find the masses, radii, and separation distance of the two stars.
This is scientifically relevant because this system was recently discovered by Professor John Johnson (of AY16) and it is a great experience for undergraduates to do real research and data collection.
These binary systems are star systems in which two stars orbit one another. This causes a very distinctive light curve when observed form Earth if one star passes in front of the other form our line of sight.
Here is a diagram of the system. The two stars orbit one another around the dotted axis a distance a apart. Imagine that the Earth is to the very far right side of your computer screen, then add a few lightyears.
This is what we would see from Earth (with an insanely powerful telescope) . At this point in time, the smaller star is in front of the larger star. This would cause a decrease in brightness rom the system. Stars are great at emitting light, but they are also good at blocking it.
A little bit later, we would see the system back at full brightness as the Stars no longer overlap.
Next, we would see a second dip in brightness as the small star passes behind the large one. As it moves behind, the light from the system will decrease over time.
As the small star begins to reemerge, the opposite of its disappearance will occur, there will be a steady increase in system brightness.
Next, the stars will not overlap, resulting in the system's return to full brightness.
Next we will see the start of the first decrease again as the small star transits the large star. Then our cycle continues.
Methods/Theory
From a transit light-curve and a Doppler shift light curve of each body, we can calculate the distance between the two stars, their masses, the time far a transit, their radii, and the period of the system.
The period is easy, since we just need to wait for a full cycle to elapse and check the time.
Transit time is also, easy, by looking at our transit light curve, we can easily tell when the transit start and end.
The relative radii of the two stars can be calculated using the depth of the transit. When the smaller star passes in front of the larger, it will block a portion of the light from the star based on visible area. If \(r_*\) is the smaller star's radius, and \(R_*\) is the larger's radius, then:
\[D = \frac{A_1}{A_2}=\frac{\pi r_*^2}{\pi R_*^2}=\frac{r_*^2}{R_*^2}\]
We can use Kepler's third law of motion and the known period to calculate the distance between the stars, \(a\).
\[P^2=\frac{4\pi^2 a^3}{G(M_*+m_*)}\]
\[a^3=\frac{P^2 G(M_*+m_*)}{4\pi^2}\]
Using \(p\) and \(a\) we can find orbital velocity. With orbital velocity and the period of a transit, we can find the exact radii of both stars by relating this to our earlier relation.
\[t_{transit}=\frac{R_*+r_*}{v}\]
\[R_*+r_*=v t_{transit}\]
This makes sense, because when the stars are "tangent" from our point of view, the smaller needs to pass all the way across the larger, and then through it's own radius to fully expose the larger. This is the total distance traveled, the total time for this is the transit duration, and the velocity is the orbital velocity of the transiting star.
To find the star's masses, we will use doppler data from the provided lab handout. When a star is moving toward us, it's light will be blue-shifted, when it is moving away, its light will be redshifted. This data lets us solve for the speed of each star, which we can use to find the true barycenter, and therefore their masses.
Observations
Observations were made using the Harvard Clay telescope on a series of (thankfully) clear nights. Cambridge, MA is not the best place for precise observations due to the massive amount of Bostonian light and air pollution, as well as its frequent cloudy weather.
We took data in the red spectrum of light for about an hour before, and an hour after the eclipse such that we could establish a baseline luminosity. After our data was collected, we used sky flats to reduce disturbances from dust in the telescope. The data was then analyzed using MaximDL as shown below.
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| http://www.fas.harvard.edu/~astrolab/object_field.png |
We used the period time to fold the normalized data into overlapping light curves to find the best fits of each transit.
This doppler data was provided in the lab manual.
Analysis
Radial Velocities:
Given from the doppler data: \(v_*=180 km/s \text{ }V_*=170 km/s\)
Transit Duration:
Using the light curve, each transit lasts approximately 1.5 hours.
Period:
From the light curve, the period appears to be 8.84 hours.
Primary Transit Depth:
\[D=\frac{\Delta f}{f}=\frac{..66}{1.33}=0.5\]
Secondary Transit Depth:
\[D=\frac{f_1}{f_2}=\frac{..54}{1.33}=0.4\]
Radius Ratio:
\[D=\frac{r_*^2}{R_*^2}\]
\[0.45=\frac{r_*^2}{R_*^2}\]
\[r_*=0.67R_*\]
Using the period, we can find separation distance.
\[2\pi a_*= Pv_*\]
\[2\pi a_* = 32000sec \times 170km/s\]
\[a_*=8.6\times10^{10} cm\]
\[2\pi a_*= PV_*\]
\[2\pi A_* = 32000sec \times 180km/s\]
\[a_*=9.3\times10^{10} cm\]
\[a=a_*+A_*=8.6\times10^{10} +9.3\times 10^{10}=1.8\times 10^{11} cm\]
Mass Ratio:
This can be found using center of mass with velocities.
\[MV=mv\]
\[M170km/s=m180km/sec\]
\[M=1.06m\]
Masses:
Using the ratio and Kepler's Third Law, the individual masses can be solved for.
\[a^3=\frac{P^2 G(M_*+m_*)}{4\pi^2}\]
\[M+m=\frac{4 \pi^2 a^3}{P^2 G}\]
\[2.06m=\frac{4 \pi^2 (1.8/times 10^{11}cm)^3}{32000sec \times 6.7\times10^{-8}}\]
\[m=1.6\times 10^{33}g\]
\[M=1.7\times 10^{33}g\]
Radii:
\[R_*+r_*=v t_{transit}\]
\[R_*+r_*=350km/sec \times 1.5 hours\]
\[R_*+r_*=1.89\times10^{11}cm\]
\[r_*=0.67R_*\]
\[r_*=7.5\times 10^{10}cm\]
\[R_*=1.1\times 10^{11}cm\]
Results
\[m_*=1.6\times 10^{33}g=0.8M_{\odot}\]
Actual: \[m=0.50M_{\odot}\]
\[M_*=1.7\times 10^{33}g=0.85M_{\odot}\]
Actual: \[M=0.54M_{\odot}\]
\[r_*=7.5\times 10^{10}cm=1.07M_{\odot}\]
Actual: \[r=0.51R_{\odot}\]
\[R_*=1.1\times 10^{11}cm=1.57M_{\odot}\]
Actual: \[r=0.54R_{\odot}\]
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