The Radial Velocity Equation in the Search for Exoplanets
( The Doppler Spectroscopy or Wobble Method )

"Raffiniert ist der Herr Gott, aber Boshaft ist er nicht ( God is clever, but not dishonest - God is subtle, but he is not malicious )", Princeton University’s Fine Hall,
carved over the fireplace in the Common Room with relativity equations as motif imprinted into the leaded glass windows - Albert Einstein ( 1879 - 1955 )

The Problem

The problem is simply to identify other unseen exoplanets orbiting dimly distant host stars with the acknowledged goal of eventually determining other intelligent SETI life by searching out the bio - chemical "signatures" of life such as carbon, oxygen, phospherous and water molecules throughout the cosmos. But our immediate goal is simply to determine velocity and mass extant in such faintly distant binary, tertiary, quaternary, etc., systems. So we must first begin with the simplest of these, namely, the binary system of one planet as an orbiting companion to one other host star.

As primarily the only realistic tool available to astrophysicists to gauge the "wobbling" light spectrum emanating from a distant host star, binary to an orbiting yet invisible planet gravitationally perturbing the host star, the relativistic red - shift using doppler spectroscopy to plot the line-of-sight, radial velocity data points for the eventual determination of time period, velocity, mass, and orbital eccentricity for both the host star and its companion binary planet, has been a highly successful method among others. That is, since measurement of distances are not sufficiently precise enough, however the relativistic red - shift providing velocities along the observer's line-of-sight is fairly well accurate. Additional observations of the host star as regards brightness and color will also provide augmented estimates for the host star's mass and radial distance. It's main drawback is that it's primarily limited to line-of-sight, eclipsing binary, tertiary, etc. systems.

All of this and still yet more, including the chemical compositions of both host star and orbiting planet coming from the light spectrum of the binary system itself, is quite an amazing feat for mathematical physics! As it should really be termed the "Philosophy of Light"!

the common center of mass, and hence motion, is inside the larger host star at the red x-mark

with a line-of-sight, edge-on eclipsing binary system, it is nearly impossible to know the orbital eccentricity - i.e., near circular or elliptical? also the host star will dim when behind the eclipsing exoplanet.

yes, a binary system. however now imagine this as a larger black hole host to a smaller binary companion star, planet, etc.

An Abreviated List of the Mathematical Physics Tools Employed

The Geometry of Elliptical Orbits

The Radial Velocity Equation - Preliminary

Area of One Orbital Revolution

Combining Equations

The Radial Velocity Equation - Almost Final Derivation
( this being highly theoretical, not yet practical ! )

Deriving the Velocity Data Points

§ Deriving Red Shift

§ Deriving the velocity data points

The Radial Velocity Semi - Amplitude K of a Wobbling Host Star to a Nearly Invisible Exoplanet
( plotting host star velocity vs. time by a gravitationally effecting exoplanet )

note: is the doppler radial velocity semi - amplitude - i.e., it is both the spectroscopic doppler velocity as well as the semi - amplitude of either the host star or orbiting planet plotted along a sine curve of doppler measured light spectrum frequencies!

The Final Derivation of Phase Velocity

Therefore,

and

Assuming that the Host Star is Circularly Perturbed

If it is assumed at the outset that the host star is perturbed strictly in a circular fashion without consideration of eccentricity, then the equation for radial velocity is reduced down to a much, much simpler derivation:

And therefore,

Summary

The Philosophy of Light
( or how the human mind overcomes narrow solipsistic naïve reality )

Finally, the electromagnetic light spectrum combined with mathematical physics, a creation of the human mind, indeed allows us to pierce the dark starlite veil of the cosmos so that perhaps eventually we can as a human race intelligently communicate with other ETs in the cosmos. And all of this is totally made possible by a speculative sort of "philosophy of light" to be able to imagine beyond our immediate and extremely naïve sense of sight!

Doppler Spectroscopy

source: NASA / JPL

Radial Velocity Simulator

source: http://astro.unl.edu
source: http://astro.unl.edu/classaction/animations/extrasolarplanets/radialvelocitysimulator.html

Planet X - Beyond Pluto: 2012 VP113 a new 9th planet?

   
This animation shows the motion of object 2012 VP₁₁₃ over 5 hours as recorded in its discovery images. The field of view is about 1 arc-minute wide. This object is currently about 83 astronomical units (7.7 billion miles) from the Sun — nearly as close as it ever gets. By Scott S. Sheppard / Carnegie Inst. of Science.

ESOcast 87: Planet found around closest Star Proxima Centauri to Earth

Proxima b is 1.3 light years away; is 1.3 times size of Eart; orbits Proxima Centauri star every 11.2 days in a habitable zone for water; and orbits closer to its star than Mercury orbits to our Sun being only 5% of the distance between Earth and the Sun.

§ References:

1. "The Derivation of the Radial Velocity Equation", Publications of the Astronomical Society of the Pacific Vol. 25, No. 149 ( August, 1913 ), pp. 208 - 211, by Dr. George Frederick ( G. F. ) Paddock ( born Providence, Rhode Island, August 9, 1879 - died Providence, Rhode Island, August 15, 1955 ), Phd. University of Virginia, Assistant Astronomer Emeritus Lick Observatory, Mt. Wilson, as well as a long - time Lick Observatory ( LO ) assistant to William Wallace ( W.W. ) Campbell; published mostly in the Lick Observatory Bulletin Series or in the Publications of the Astronomical Society of the Pacific; primarily involved in the radial velocity problems of the Lick Observatory and its program for determining the Radial Velocity Equation as his primary astronomical interest lay almost wholly in the field of astronomical spectroscopy. His was a very kindly personality and diffident as a person. Never married.
   
   U. of Virginia - Department of Astronomy Ph.D. Dissertations: "Some adaptations and criticisms of spectroscopic orbit formulae and an application to nu 4th Eridani AGC 4821X", Paddock, George Frederic ( 1912 )


2. "How Pluto's Orbit was Figured Out", November, 1930, by Henry Norris Russell, Phd. ( 1877 - 1957 ), Department of Astronomy and Director of the Observatory at Princeton University.


3. "The Radial Velocity Equation - A detailed derivation", by Kelsey I. Clubb, Department of Physics & Astronomy, San Francisco State University, 2008; now presently research associate in the UC Berkeley Astronomy Department. Much of the inspiration for this web page derivation is owed to the paper by Ms. Kelsey I. Clubb.


4. "Principles of Astrophysics - Using Gravity and Stellar Physics to Explore the Cosmos"© 2014, by Charles Keeton, Department of Physics and Astronomy, Rutger University


5. "A Sedna-like body with a perihelion of 80 astronomical units", by Chadwick A. Trujllo & Scott S. Sheppard, in Nature, Vol. 507, pgs. 471-474, March 27, 2014. This key discovery, termed '2012 VP₁₁₃', or "Biden" after U.S. Vice President Joe Biden, is the closest yet to Planet X - The Ninth Planet in our solar system. This particular Sedna-like exoplanet has a perihelion of 80 AUs and an aphelion of approximately 446 AU and an eccentric orbital period of 4,300 years.


6. "Evidence for a Distant Giant Planet in the Solar System", by Konstantin Batygin and Michael E. Brown, Division of Geological and Planetary Sciences, California Institute of Technology, The Astronomical Journal, February, 2016


7. "False Positive Probabilities for All Kepler Objects of Interest: 1,284 Newly Validated Planets and 428 Likely False Positives,", by Timothy D. Morton et al., 2016 May 10, Astrophysical Journal


8. "Solar Obliquity Induced by Planet Nine", by Elizabeth Bailey, Konstantin Batygin and Michael E. Brown, Division of Geological and Planetary Sciences, California Institute of Technology, The Astronomical Journal, July 21, 2016, mathematically demonstrating that the 6-degree obliquity of the plane of the sun's spin axis is giving an appearance for this unusual tilt of the Sun where, further, "... the solar system has no choice but to slowly twist out of alignment" due to "an external torque [ which ] has misaligned the angular momentum vectors of the sun and the planets", says Elizabeth Bailey, graduate student and lead author of this study.


9. "'New evidence in support of the Planet Nine' hypothesis", by C. de la Fuente Marcos and R. de la Fuente Marcos, Universidad Complutense de Madrid, Ciudad Universitaria, E-28040 Madrid, Spain, Monthly Notices of the Royal Astronomical Society: Letters, July 2017. DOI: 10.1093/mnrasl/slx106.


10. "Transiting Exoplanet Survey Satellite,", authors George R. Ricker, et al. in Journal of Astronomical Telescopes, Instruments, and Systems, Vol 1, No.1, Article No. 014003 (24 October 2014). https://doi.org/10.1117/1.JATIS.1.1.014003

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