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In astrodynamics, the vis-viva equation is one of the equations that model the motion of orbiting bodies.It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight which is the gravitational force determined by the product of the mass of the object and the strength of the surrounding gravitational field.
Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation ¨ = ¨ ¨ = = (+) where we have again used Newton's third law F 12 = −F 21 and where r is the displacement vector from mass 2 to mass 1, as defined above. The force between the two objects, which ...
The orbits of a test particle of infinitesimal mass about the central mass is given by the equation of motion = (+). where is the specific relative angular momentum, = = and is the reduced mass. This can be converted into an equation for the orbit = (+), where, for brevity, two length-scales, = and =, have been introduced. They are constants of ...
For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit , it is equal to the excess energy compared to that of a parabolic orbit.
The choice of solar mass, M ☉, as the basic unit for planetary mass comes directly from the calculations used to determine planetary mass.In the most precise case, that of the Earth itself, the mass is known in terms of solar masses to twelve significant figures: the same mass, in terms of kilograms or other Earth-based units, is only known to five significant figures, which is less than a ...
If Jupiter had Mercury's orbit (57,900,000 km, 0.387 AU), the Sun–Jupiter barycenter would be approximately 55,000 km from the center of the Sun ( r 1 / R 1 ≈ 0.08). But even if the Earth had Eris's orbit (1.02 × 10 10 km, 68 AU), the Sun–Earth barycenter would still be within the Sun (just over 30,000 km from the center).
Two bodies orbiting a common center of mass, indicated by the red plus. The larger body has a higher mass, and therefore a smaller orbit and a lower orbital velocity than its lower-mass companion. The binary mass function follows from Kepler's third law when the radial velocity of one binary component is known. [1]
Depiction of asteroid Pallas' 18:7 near resonance with Jupiter in a rotating frame (click for animation). Jupiter (pink loop at upper left) is held nearly stationary. The shift in Pallas' orbital alignment relative to Jupiter increases steadily over time; it never reverses course (i.e., there is no libration).