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| Forum Home > Physics > Introduction to Astrophysics - Yale University, Prof. Charles Bailyn, PhD | ||
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Site Owner Posts: 133 |
Planetary Orbits a^3 = p^2GM / 4pi^2 G is the gravitational constant. If using solar masses, years and AU, then G = 4pi^2, so: a^3 = p^2M a is distance of the semi-major axis of the elliptical orbit of the planet from the star in Astronomical Units (AU). One AU is the distance of the semi-major axis of the Earth to the sun, ie. @ 93 million miles. p is the orbital period in years. M is the mass of the objects in solar masses, the planet being negligable compared to the star. | |
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Site Owner Posts: 133 |
Exoplanets Measuring the distance between extrasolar planets from their parent stars: A (Earth)__________________D1_______________________B (star) D2 (exoplanet) * D3 C A right triangle is formed between the observer, point A, the star, point B, and the exoplanet, point C. The sides of the triangle are D1, D2 and D3. The angle at point B is a right angle. Trigonometry tells us that: the sin of angle a = D2 / D3. For small angles like this, if the angle is in radians, sin a = a. Also, D1 is virtually the same as D3. So, a = D2 / D1. D2 in AU can be found by use of the formula for planetary orbits: a^3 = p^2M. M can be assumed to be 1 solar mass, since most stars are about that and we will find the cube root anyway, so the difference is negligable. D1 is in parsecs. D2 is in AU. Angle a is in arc seconds. Small angle formula: a = D2 / D1 | |
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Site Owner Posts: 133 |
Detecting Exoplanets by Star's Motion Newton's 3rd Law: for every action, there is an equal and opposite reaction (conservation of momentum) MpVp = M*V* Mp = mass of the planet M* = mass of the star Vp = velocity of the planet V* = velocity of the star (planet) o________________(center of mass) .__* (star) The planet and star both orbit around the center of mass in opposite directions. If the system is viewed edge on the Doppler shift in the stars spectrum reveals it's velocity. When the shift is red, the star is moving away from us and the velocity is positive. When the shift is blue, the star is moving toward us and the velocity is negative. change in wavelength / rest wavelength = Vr / c (Vr = radial velocity, c = speed of light) The velocity can be plotted over time. If the orbit is nearly circular, this produces a sin wave. The time between crests reveals the planet's orbital period. For nearly circular orbits only: V = 2pi(a / p), this is derived from the equation v = d/t (velocity = distance/time) V = velocity a = distance from center of mass (radius of orbit) 2pi(a) = circumference of orbit (total distance travelled) A circle's circumference is 2pi(r) where r is the radius. p = orbital period (time) The star and planet are always on opposite sides of the center of mass, therefore their orbital periods must be the same. | |
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Site Owner Posts: 133 |
The belief that star wobble was due to planets was strongly confirmed by transits of planets with edge-on orbits to our line of sight which periodically dimmed the light of the star exactly in accordance with their orbital period as shown by the wobble of the star. | |
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