How to check a statement about orbital angles of exoplanets?

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SUMMARY

The discussion centers on the statistical implications of inclination angles in exoplanet detection using the Radial Velocity (RV) technique, as articulated by Christophe Lovis and Debra Fischer. The authors assert that inclination angles are randomly distributed, leading to a higher frequency of edge-on systems compared to pole-on configurations. This results in an average value of sin(i) equal to π/4 (approximately 0.79) and an 87% probability that sin(i) exceeds 0.5. The analogy of a six-sided die is used to illustrate the distribution of angles, emphasizing the greater likelihood of edge-on observations.

PREREQUISITES
  • Understanding of Radial Velocity technique in exoplanet detection
  • Familiarity with statistical distributions and probability
  • Basic knowledge of orbital mechanics and inclination angles
  • Concept of geometric probability using three-dimensional shapes
NEXT STEPS
  • Research the mathematical foundations of the Radial Velocity technique
  • Study the statistical distribution of angles in celestial mechanics
  • Explore the implications of inclination angles on exoplanet mass estimation
  • Investigate geometric probability using spheres and other three-dimensional shapes
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Students and researchers in astrophysics, particularly those focusing on exoplanet detection methods, as well as anyone interested in the statistical analysis of orbital mechanics.

Fabioonier
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Hi, everybody. Mi name is Fabio Onier Osorio Pelaez and I'm from Colombia.

I hope to be finishing my Bachelor´s degree in Physics at University of Antioquia by next August. I'm doing my final project on the detection of planets by the Radial Velocity technique and I have a question about an statement of Christophe Lovis and Debra Fischer in a paragraph in the third page or their article (page 29 in the book Exoplanets). I quote the complete paragraph emphasizing the statement in which I am interested:

"The unknown inclination angle ##i## prevents us from measuring the true mass of the companion ##m_2##. While this is an important limitation of the RV technique for individual systems, this fact does not have a large impact on statistical studies of exoplanet populations. Because inclination angles are randomly distributed in space, angles close to 90° (edge-on system) are much more frequent than pole-on configurations. Indeed, the distribution function for ## i ## is given by ## f(i)di=\sin(i)di ##. As a consequence, the average value of ##\sin(i)## is equal to ##\pi/4## (0.79). Moreover, the a priori probability that ##\sin(i)## is larger than 0.5 is 87%."

I wondered if you may tell me how did they conclude that. I'll be thankful if you can help me with that information. It will be very useful for my work.
 
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Welcome to PF;
The authors tell you: "Because the inclination angles are randomly distributed..."
 
To give you another hint, imagine a six-sided die (i.e. a cube) being thrown. Pick anyone pair of opposite sides - these represent the poles of the orbital arrangement. The remaining four sides represent the edges. As you can already see, there's more possibilities of an edge-on result (4 of 6) than of a pole-on result (2 of 6).
Generalise the cube to a sphere and see if you can get the same result the authors got.