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The Mathematics of Star Trek. Lecture 11: Extra-Solar Planets. Outline. Finding Extra-Solar Planets The Two-Body Model A Celestial Cubic Example-51-Pegasi. Finding Extra-Solar Planets. Recent discoveries of planets orbiting stars rely on a type of problem known as an inverse problem.

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The Mathematics of Star Trek

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The mathematics of star trek l.jpg
The Mathematics of Star Trek

Lecture 11: Extra-Solar Planets


Outline l.jpg
Outline

  • Finding Extra-Solar Planets

  • The Two-Body Model

  • A Celestial Cubic

  • Example-51-Pegasi


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Finding Extra-Solar Planets

  • Recent discoveries of planets orbiting stars rely on a type of problem known as an inverse problem.

  • In the paper “A Celestial Cubic”, Charles Groetsch shows how the orbital radius and mass of an unseen planet circling a star can be obtained from the star’s spectral shift data, via the solution of a cubic equation!


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The Two-Body Model

  • Assume a far-off star of mass M is orbited by a single planet of mass m<M, with a circular orbit of radius R.

  • The star and planet orbit a common center of mass (c.o.m.).

  • To an observer on Earth, the star will appear to wobble.

    • Think of a hammer thrower spinning around—the thrower is the star and the hammer is the planet!


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The Two-Body Model (cont.)

  • On earth, we see this wobble as a Doppler shift in the wavelength of the light from the star.

    • As the star moves towards us, the light shifts towards the blue end of the spectrum.

    • As the star moves away from us, the light shifts towards the red end of the spectrum.

  • The magnitude of these shifts determine the radial velocity of the star relative to Earth.

  • The time between successive peaks in the wavelength shifts gives the orbital period T of the star and planet about their center of mass.


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The Two-Body Model (cont.)

  • For our model, we assume the following:

    • The star orbits the center of mass in a circle of radius r with uniform linear speedv.

    • The Earth lies in the orbital plane of the star-planet system.

    • The distance D from the Earth to the center of mass of the star-planet system is much greater than r (D >> r).


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The Two-Body Model (cont.)

  • Recall from trigonometry that v =  r, where  is the angular speed.

  • Also recall that  =/t, where  is the angle in radians traced out in t seconds by the star as it orbits around the center of mass.

D

c.o.m.

Earth

r


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The Two-Body Model (cont.)

  • Since v is constant, it follows that  is also constant, so when t = T,  = 2, and thus  = 2/T.

  • Using this fact, we can write the radial velocity, given by V(t) = d’(t), as follows:

  • Hence, V is sinusoidal, with amplitude equal to star’s linear speed v, and period equal to the star’s period T about the center of mass!


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The Two-Body Model (cont.)

  • Measuring wavelength shifts in the star’s light over time, a graph for V(t) can be found, from which we can get values for v and T.

  • Then, knowing v and T, we can find the orbital radius r of the star about the center of mass:

  • Finally, the mass M of the star can be found by direct observation of the star’s luminosity.


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The Celestial Cubic

  • At this point, we know M, v, T, and r.

  • We still want to find the radius R of the planet’s orbit about its star and the mass m of the planet.

  • From physics, the centripetal force on the star rotating around the c.o.m. is equal to the gravitational force between the planet and star.


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The Celestial Cubic (cont.)

  • The centripetal force is given by

  • Parameterizing the star’s orbit about the center of mass, we find the planet’s position vector to be:


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The Celestial Cubic (cont.)

  • Differentiating twice, we see that the acceleration of the star is given by:

    so the magnitude of the centripetal force on the star is


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The Celestial Cubic (cont.)

  • The magnitude of the gravitational force is

    where G is the universal gravitation constant

  • Equating forces, we get


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The Celestial Cubic (cont.)

  • We now have one equation that relates the unknown m and R.

  • To get another equation, we’ll use the idea of finding the balance point (center of mass) for a teeter-totter.

  • Archimedes discovered that the balance point (center of mass) for a board with masses m1 and m2 at each end satisfiesm1r=m2r2 (Law of the Lever).

Balance Point

m2

m1

r1

r2


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The Celestial Cubic (cont.)

c.o.m.

r

R-r

  • Thinking of the planet and star as masses on a teeter-totter, the Law of the Lever implies,

  • Solving (2) for R and substituting into (1), we find



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The Celestial Cubic (cont.)

  • Dividing (3) by M2, and setting

    and

    we find that x and  satisfy the following cubic equation:


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Measured wavelength shifts of light from the star 51-Pegasi show that

v = 53 m/s,

T = 4.15 days, and

M = 1.99 x 1030 kg.

Use Mathematica to find r, , x, and m by finding the roots of (4) directly.

Example-51-Pegasi


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Example-51-Pegasi (cont.) show that

  • Repeat, using a fixed-point method to solve the following equation which is equivalent to (4):

  • Groetsch argues that equation (4) can be solved by iteration of (5), via

  • Try this with Mathematica and compare to the solution above.


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References show that

  • C.W. Groetsch, “A Celestial Cubic”, Mathematics Magazine, Vol. 74, No. 2, April 2001, pp. 145 - 152.

  • C.P. McKeague, Trigonometry (2cd ed), Harcourt Brace, 1988.

  • J. Stewart, Calculus: Early Transcendentals (5th ed), Brooks - Cole, 2003.

  • http://zebu.uoregon.edu/51peg.html


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