Speed at perihelion and aphelion?

[Master J]

SO the equations for the speed at the aphelion and perihelion are, respectively:

v=SQRT[GM( (2/r_p) -(1/a) )]

v=SQRT[GM( (2/r_a) - (1/a) )]

where M is mass of sun, r_a & r_p are distances from sun at aphelion and perihelion , and a is length of semi major axis.

How do you derive them? I am having trouble seeing where they come from, and a quick Google turns up NOTHING on them unfortunately.

ANy help?

 

[D H]

Those equations result simply from inserting the perifocal and apofocal distances into the vis-viva equation,

$$v^2 = GM\left(\frac 2 r - \frac 1 a\right)$$

This equation follows directly from conservation of energy. The total energy (kinetic plus potential) of a point mass m separated by a distance r from some other point mass M and moving with a velocity v relative to that other point is

$$E = \frac 1 2 m v^2 - \frac {G M m}{r}$$

The total energy of a point mass in an elliptical orbit is also given (see any intermediate-level classical mechanics text) by

$$E = - \frac {G M m}{2a}$$

Equating the two expressions leads directly to the vis-viva equation.

 

[astrokreng]

The equations for the speed at aphelion and perihelion can be derived using the principles of orbital mechanics. Specifically, they are derived from the law of universal gravitation, which states that the force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them.

In the case of a planet orbiting the sun, the force of gravity acting on the planet is equal to the centripetal force keeping it in orbit. At aphelion, the planet is furthest from the sun and therefore experiences a weaker gravitational force, while at perihelion, the planet is closest to the sun and experiences a stronger gravitational force.

Using the law of universal gravitation and the equation for centripetal force, we can derive the equations for the speed at aphelion and perihelion. The first step is to equate the force of gravity to the centripetal force:

F = GMm/r^2 = mv^2/r

Where F is the force of gravity, G is the gravitational constant, M is the mass of the sun, m is the mass of the planet, r is the distance between them, and v is the velocity of the planet.

Solving for v, we get:

v = SQRT[GM/r]

Next, we need to consider the distances from the sun at aphelion and perihelion. At aphelion, the distance is equal to the semi-major axis plus the distance between the sun and the planet, or r_a = a + r. At perihelion, the distance is equal to the semi-major axis minus the distance between the sun and the planet, or r_p = a - r.

Substituting these values into the equation for v, we get:

v_a = SQRT[GM( (2/r_p) - (1/a) )]

v_p = SQRT[GM( (2/r_a) - (1/a) )]

Therefore, the equations for the speed at aphelion and perihelion are derived from the law of universal gravitation and the equation for centripetal force. These equations show the relationship between the distance from the sun and the speed of a planet in its orbit, and are important in understanding the dynamics of our solar system.