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

Mystery-Solving and Astronomy

Last semester I took a class on Celestial Navigation (Astronomy 2) and loved it. I really liked learning about astronomy in a tangible context. I liked going outside every night with my compass and quadrant and tracking stars and getting to see for myself that they moved and, better yet, I could figure out where I was based on those movements.

Some of my class supplies.

People around me started to notice my enjoyment of this untraditional subject. My roommate thought it was funny enough to upload the below picture of me doing my pset to facebook.

Even my parents noticed, which led them to buy me Celestial Sleuth for Hanukkah this year, which, finally, leads me to the point of this blog post. 

The book is a great combination for me because I often straddle the line between wanting to major in astrophysics or art. But beyond that, I found an unexpected love for the Sleuth part of Celestial Sleuth.  "Forensic astronomy" - as the book calls it - is like an even nerdier Sherlock Holmes. It explores mysteries that I would have never thought of in relation to astronomy. The amazon description names a few examples: 

"Weather facts, volcano studies, topography, tides, historical letters and diaries, famous paintings, military records, and the friendly assistance of experts in related fields add variety, depth, and interest to the work. The chosen topics are selected for their wide public recognition and intrigue, involving artists such as Vincent van Gogh, Claude Monet, Edvard Munch, and Ansel Adams; historical events such as the Battle of Marathon, the death of Julius Caesar, the American Revolution, and World War II; and literary authors such as Chaucer, Shakespeare, Joyce, and Mary Shelley. This book sets out to answer these mysteries indicated with the means and expertise of astronomy, opening the door to a richer experience of human culture and its relationship with nature. "

Source

To give more specific examples, Olson and some of his students went to France to figure out exactly where and when Monet's 1883 The Cliff, Etretat, Sunset was painted.

Source

Using the altitude, relative size of the sun, and angle of the sun in relation to the rocks, they could determine when the painting took place, within a few minutes. All using simple techniques and tools I used in my navigation class.

Alternatively, the book spends an entire chapter on the civil war and finds that a critical turning point in the battle at Chancellorsville in 1863. The book uses first hand accounts and calculations about phases of the moon to find that the time of moonrise might have influenced the wounding of Stonewall Jackson. 

Source

Overall, its a great book and I would recommend looking into forensic astronomy if you are interested in practical applications of astronomy.  

Worksheet 14.2: Tides

Note: Once again, Latex is only working where it wants to work, for no conceivable reason and I am very open to suggestions on how to fix it.

1. Draw a circle representing the Earth (mass M‘), with 8 equally-spaced point masses, m, placed around the circumference. Also draw the Moon with mass MK to the side of the Earth. In the following, do each item pictorially, with vectors showing the relative strengths of various forces at each point. Don’t worry about the exact geometry, trig and algebra. I just want you to think about and draw force vectors qualitatively, at least initially.

1. (a)  What is the gravitational force due to the Moon, F⃗K,cen, on a point at the center of the Earth? Recall that vectors have both a magnitude (arrow length) and a direction (arrow head).
   
2. 
   
   
   
   
3. (b)  What is the force vector on each point mass, F⃗K, due to the Moon? Draw these vectors at each point.
   
4. 
   (c)  What is the force difference, ∆F⃗, between each point and Earth’s center? This is the tidal force.
   
5. 
   (d)  What will this do to the ocean located at each point?
   
6. 
   
   
7. (e)  How many tides are experienced each day at a given location located along the Moon’s orbital plane?
   
8. 
   

   In a 24 hour period, there would be 2 high tides and 2 low tides.

   
   
9. (f)  Okay, now we will use some math. For the two points located at the nearest and farthest points from the Moon, which are separated by a distance ∆r compared to the Earth-Moon distance r, show that the force difference is given by \[ \Delta F = \frac{2GmM_{\circ} \Delta r}{r^3} \] Using \[ \lim_{x\to\ 0} \frac{f(x + \Delta x ) - f(x) }{\Delta x} = \frac{d}{dx} f(x) \approx \frac{ \Delta f(x) }{\Delta x} \]  \[ F_g (r) = \frac{GmM_{\circ}}{r^2} \]  \[ F (r + \Delta r) = \frac{GmM_{\circ}}{(r + \Delta r)^2} \] Because we know  \[ \lim_{x\to\ 0} \frac{f(x + \Delta x ) - f(x) }{\Delta x} = \frac{d}{dx} f(x) \approx \frac{ \Delta f(x) }{\Delta x} \] We can take the derivative of F to find: \[ \frac{d}{dx} F = -\frac{2GmM_{\circ} \Delta r}{r^3} \]

7. (g)  Compare the magnitude of the tidal force ∆FK caused by the Moon to ∆F@ caused by the Sun. Which is stronger and by how much? What happens when the Moon and the Sun are on the same side of the Earth?
   
8. 
   
9. The only pertinent differences in the equation are M and r so we can compare:\[ \Delta F_k = \frac{M }{r^3} \] \[ \frac{M_{\odot} }{r^3} vs = \frac{M_{\circ} }{r^3} \] \[ \frac{2 \times 10^{33} }{(1.5 \times 10^{13})^3} vs \frac{7.3 \times 10^{25}}{(3.8 \times 10^{10})^3} \] \[5.9 \times 10^{-7} \: vs \: 1.3 \times 10^{-6} \] So the moon would have a stronger pull. If they are on the same side, both forces would pull together and add for a stronger tidal force. 
10. 
    
11. (h)  How does the magnitude of ∆F caused by the Moon compare to the tidal force caused by Jupiter during its closest approach to the Earth (r « 4 AU)? \[ \frac{M_{\circ} }{r^3} vs \frac{M_{jup}}{r^3} \] \[ \frac{7.3 \times 10^{25}}{(3.8 \times 10^{10})^3} vs  \frac{1.9 \times 10^{30} }{(4 \times 1.5 \times 10^{13})^3} \] \[ 1.3 \times 10^{-6} \: vs \: 8.8 \times 10^{-12} \] So the moon would once again have a greater tidal force. 

I worked with Sean, Barra, and April on this problem.

Worksheet 14.1, Problem 2: White Dwarfs

1. A white dwarf can be considered a gravitationally bound system of massive particles. 
   
   1. (a)  Express the kinetic energy of a particle of mass m in terms of its momentum p instead of the
      usual notation using its speed v.
      
   2. \[ \frac{1}{2}mv^2 = \frac{p^2}{2m} \]
   3. (b)  What is the relationship between the total kinetic energy of the electrons that are supplying
      the pressure in a white dwarf, and the total gravitational energy of the WD?
      
   4. We know \[ K = - \frac{1}{2} U \] and \[ U = \frac{GM^2}{r} \] and \[ K = \frac{p^2}{2 m_e} \] Combining these gives: \[ \frac{GM^2}{r} = \frac{p^2 N}{2m_e} \] Where \[ N = \frac{M_{\star}}{m_p + m_e} = \frac{M_{\star}}{m_p \left( 1 + \frac{m_e}{m_p} \right) } \approx \frac{M_{\star}}{m_p} \]  So \[ \[ \frac{GM^2}{r} = \frac{p^2 M_{\star}}{2m_e m_p} \] 

   1. (c)  According to the Heisenberg uncertainty Principle, one cannot know both the momentum and
      position of an election such that ∆p∆x > \(\frac{h}{4\pi} \) . Use this to express the relationship between the 4π  kinetic energy of electrons and their number density ne (Hint: what is the relationship between an object’s kinetic energy and its momentum? From here, assume p = ∆p and then use the Uncertainty Principle to relate momentum to the volume occupied by an electron assuming Volume ~ (∆x)3.)  \[ V \sim x^3 \sim  n_e^{-1}  \]  \[ px = \frac{ h}{4 \pi} \] \[ p = \frac{h}{ 4 \pi x } = \frac{h n_e^{ \frac{1}{3}}}{4 \pi } \] For 1 electron \[ K = \frac{p^2}{2m_e} = \frac{h n_e^{\frac{2}{3}}}{32 \pi^2 m_e } \] For many \[ K = \frac{h^2 n_e^{\frac{2}{3}}}{32 \pi^2 m_e } \cdot \frac{M}{m_p} \]

   
   

   1. (d)  Substitute back into your Virial energy statement. What is the relationship between ne and the mass M and radius R of a WD?  \[ K = - \frac{1}{2} U \] \[\frac{h^2 n_e^{\frac{2}{3}}}{32 \pi^2 M_e } \cdot \frac{M}{m_p} = \frac{GM^2}{2r} \] \[ \frac{16M}{R} = \frac{h^2 n_e^{\frac{2}{3}}}{\pi^2 m_e m_p } \]

   
   

   1. (e)  Now, aggressively yet carefully drop constants, and relate the mass and radius of a WD. \[ h^{\frac{2}{3}} = \frac{M}{R} \] Where \[ n = \frac{m}{V} = \frac{m}{\frac{4}{3} \pi R^3 }\]  \[ \left( \frac{M}{R} \right)^{\frac{2}{3}} = \frac{M}{R} \] \[ R = \frac{1}{M^{\frac{1}{3}} } \]

   
   

   1. (f)  What would happen to the radius of a white dwarf if you add mass to it? 
      
   2. 
      
   3. The relationship is inverted, so it would decrease. 
   4. 
      

   UPDATE: I know my latex is not compiling in some places; I have no idea why. No matter how many different ways I type it, it won't compile (but just for some?). Do you have any suggestions?

I worked with Barra, April, and Sean on this problem.

Atrobite: On and On They Spin

This astrobite discusses the mysterious case of stars that rotate much faster than they should. The paper focuses of supergiants, bright giants, normal giants, and sub giants of luminosity class K and G (which are based on temperature).

Source

Theoretically, these stars' rotation should slow down with age due to a loss of moment of inertia due to stellar winds. And that is true for most of these stars; however, some of these stars actually spin faster. This has been found frequently enough to be significant:

Source. This graphs demonstrates the large range of velocities found in the studied stars.

One of the mysteries surrounding these stars (literally) is a cloud of warm dust around the stars little is known about this dust or its function, but is is being studied as a possible influence on the rotational speed of the stars. Hypotheses include that the dust is due to magnetic stellar winds or comes from collisions of planets.

Source- The large offsets in this data might indicate excess emissions.

The authors proposed two causes for the rapidly rotating stars. One explanation could be interactions with a rotating low mass star (like a brown dwarf) or tidal interactions with a certain type of planet called a "hot Jupiter." These two possibilities would produce different chemical abundances. Going forward, the authors of the paper propose to search for differences in chemical abundances to distinguish between these two hypotheses.

Freeform post: Lyman-Alpha emitting glalxies

For this freeform post I am going to talk about one of my past experiences that definitely helped me decide to go into astrophysics. The summer before my junior year of high school, I participated in a summer program at UCSC called COSMOS (California State School for Math and Science). It was half a summer camp, half research experience, and a ton of fun. My group - or "cluster" - focused on Astronomy and Oceanography and for my final research project, my group worked on trying to find distant Lyman-alpha emitting galaxies. Lyman-alpha emitting galaxies are host, dusty, star-producing galaxies from the beginning of the universe. Lyman-alpha "blobs" are some of the largest objects in the universe, some are more than 300,00 light years across. They are called Lyman-alpha because newly forming stars emit high amounts of UV radiation which ionize hydrogen atoms. The hydrogen ions neutralize in an excited state, with the electron on a high energy level. Lyman Alpha photons are emitted when the electron returns to the first energy level. Those wavelengths are far enough away that they have been extremely red-shifted. Amazingly, they are red-shifted enough that they can be detected in visible light. We actually got to remotely control the Keck Telescope in Hawaii in order to take "masks" of the sky that look like this:

We then got to comb through hundreds of these masks, marking interesting objects as shown above. There were some very interesting objects such as binary stars and quasars detected (shown below).

But unfortunately, finding Lyman-alphas are very rare. They look like the image below:

We collectively looked through almost 500 masks and found only 4 Lyman-alpha galaxies (and I am proud to say I found the oldest and rarest one, though that was entirely luck).  We then analyzed the redshift of each galaxy to figure out how old they were and the results were amazing.

The one I found was 12.8 billion years old, which is pretty significant when compared to a 14 billion year-old-universe.  The galaxy's redshift of 6.4 didn't even fit on the accepted collective graph of all discovered Lyman-alpha emitting galaxies.

Ultimately, our research helped the leading professor (Dr. Raja Thakurta) congregate data on Lyman-alphas in order to have a more normalized understanding of these mysterious early galaxies from the beginning of our universe.

This experience caused me to take a position in an astronomy lab the next summer researching black holes, but I will talk about that in my next blog post.

Worksheet 13, Problem 4: Applying our knowledge to actual data

The problem: What are the periods, velocity amplitudes and planet masses corresponding to the two radial velocity time series below? The star 18 Del has \(M_{\star} = 2.3 M_{\odot} \), and HD 167042 has  \(M_{\star} = 1.5 M_{\odot} \) Notes: Each data point is a radial velocity measured from an observation of the star’s spectrum, and the dashed line is the best-fitting orbit model. Prof. Johnson found the planet around HD 167042 when he was a grad student, and each data point represents a trip from Berkeley, CA to Mt. Hamilton and a long night at the telescope. “Trend removed” just means that in addition to the sinusoidal variations, there was also a constant acceleration. What would cause such a “trend?”

Source

Graph 1: 

• Period: about 3 years
• Amplitude: about 125 m/s 
• Planet mass: We can find this from the equation for planet velocity: \[V_{\star}^3 = \frac{2 \pi G M_P^3}{P M_{\star}^2} \] Or \[M_p^3 = \frac{P M_{\star}^2  V_{\star}^3}{2 \pi G }\] \[M_p^3 = \frac{P (2.3 M_{\odot})^2  V_{\star}^3}{2 \pi G }\]  \[M_p^3 = \frac{(9.5 \cdot 10^7) (2.3 \cdot 2 \cdot 10^{33})^2  (1.25 \cdot 10^4)^3}{2 \pi 6.67 \cdot 10^{-8} }\]  \[M_p = 2.11 \times 10^{31} g\]

 Graph 2:

• Period: about 1.3 years
• Amplitude: about 30 m/s 
• Planet mass: We can find this from the equation for planet velocity: \[V_{\star}^3 = \frac{2 \pi G M_P^3}{P M_{\star}^2} \] Or \[M_p^3 = \frac{P M_{\star}^2  V_{\star}^3}{2 \pi G }\] \[M_p^3 = \frac{P (2.3 M_{\odot})^2  V_{\star}^3}{2 \pi G }\] \[M_p^3 = \frac{4.10 \cdot 10^7 (2.3 \cdot 2 \cdot 10^{33})^2  3000^3}{2 \pi 6.67 \cdot 10^{-8} }\]   \[M_p = 3.8 \times 10^{30} g\]

Part B: What is up with the radial velocity time series below? Sketch the orbit of the planet that caused these variations. (HINT: There’s only one planet orbiting a single star)

Source

A graph like this could come from a planet with a very elliptical orbit. A more circular orbit would result in the two graphs above, with a somewhat standard sinusoid. 

But to produce the graph above, the orbit would have to look more like this:

Acknowledgements: I worked with Barra on this problem.

Worksheet 13.2, Problem 2: Transiting Planets

The Problem: Now draw the star projected on the sky, with a dark planet passing in front of the star along the star’s equator.

Part A: How does the depth of the transit depend on the stellar and planetary physical properties?

For this problem we will be trying to find \( \Delta F \) which we can find using \( \frac{F_T}{F_{\star}} \) \[F_{\star}= L_{\star} A_{\star} = L_{\star} \pi R_{\star}^2 \] \[F_T = L_{\star} (A_{\star} - A_{Planet}) = L_{\star}(\pi R_{\star}^2 - \pi R_P^2) \] Combining these gives: \[ \frac{F_T}{F_{\star}}  = \frac{L_{\star}(\pi R_{\star}^2 - \pi R_P^2)}{ L_{\star} \pi R_{\star}^2 } = 1 - \frac{ R_P^2}{R_{\star}^2 } \]
What is the depth of a Jupiter-sized planet transiting a Sun-like star?

We know \( \frac{R_J}{R_{\odot} } \approx \frac{1}{10} \) so: \[ \frac{F_T}{F_{\odot}} \approx 1 - \frac{ 1}{10^2}  \approx \frac{99}{100}\] Which means that the flux of the Sun when Jupiter is transiting is about 99% of its normal flux.

Part B: In terms of the physical properties of the planetary system, what is the transit duration, defined as the time for the planet’s center to pass from one limb of the star to the other?
\[ t = \frac{d}{v} \] So we need distance and velocity. Distance would just be twice the radius of the star and velocity would be: \[ v_p = \frac{2 \pi a }{P} \] Combining those we get: \[t = \frac{2R_{\star}}{\frac{2 \pi a }{P} } = \frac{PR_{\star}}{\pi a } \]

Part C: What is the duration of “ingress” and “egress” in terms of the physical parameters of the planetary system?

Once again, we can use: \[ t = \frac{d}{v} =  \frac{2R_{P}}{\frac{2 \pi a }{P} } = \frac{PR_{P}}{\pi a } \] 

Acknowledgements: I worked with Sean, April, and Barra on this problem.