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.