Acceleration in elliptical orbits ?

[Bjarne]

How is acceleration between aphelion and perihelion calculated?
Which equation can be used?

 

[IsometricPion]

That depends on what information is already known. For instance, if the velocity is known at a point then the acceleration will be perpendicular to it (in the direction of the body being orbited) with a magnitude given by the (magnitude of the) force due to gravity at the distance of the point from the body being orbited.

 

[Bjarne]

That depends on what information is already known.

For instance, if the velocity is known at a point then the acceleration will be perpendicular to it (in the direction of the body being orbited) with a magnitude given by the (magnitude of the) force due to gravity at the distance of the point from the body being orbited.

Let us say the Earth
Aphelion; 152,052,232 km
Perihelion 147,098,290
Speed let us just say speed at Aphelion = 30 km/s ( I don’t know where to find anything else as a average.)
Acceleration Due to Gravity (Aphelion)
GM/r^2 = 6.67e-11*2e30/152098232000^2 = 0.005766436287721685 m/s^2

Can you based on these data explain / show the equation how to calculate the acceleration towards perihelion?

 

[IsometricPion]

(in the direction of the body being orbited) with a magnitude given by the (magnitude of the) force due to gravity at the distance of the point from the body being orbited.

Sorry, this part of my previous post is inaccurate.

The wikipedia http://en.wikipedia.org/wiki/Kepler_problem" on the Kepler problem gives the distance between the two bodies in terms of the angle of line connecting them (wrt the x-axis):[tex]r=\frac{-L^2}{km[1+e\cos(\theta{}-\theta_{o})]}[/tex] where k=Gm₁m₂, m=m₁m₂/(m₁+m₂), [tex]e=\sqrt{1+\frac{2EL^2}{k^{2}m}}[/tex], E is the total energy of the system, L is its total angular momentum, and [tex]\theta_{o}[/tex] is the initial angle.

By definition of the center of mass frame (in which this problem is usually analysed), m₁r₁=m₂r₂ and [tex]\theta_{1}=\theta_{2}+\pi{}\Rightarrow{}\dot{\theta_1}=\dot{\theta_2}\Rightarrow{}m_{1}v_{1}=m_{2}v_{2}[/tex] so if the subscript 2 denotes the properties of Earth in this coordinate system, (using the data from this http://nssdc.gsfc.nasa.gov/planetary/planetfact.html" for G) one needs to know the mass of each object, the velocity of one of the objects and their separation at given time. Using the values for the Earth and Sun found on the linked site, and the distance and velocity of the Earth at aphelion (rounding to 5 places): m₁=1.9891e30 kg, m₂=5.9736E24 kg, aphelion= 1.5210E11 m, min. orbital velocity (aphelion velocity)=29290 m/s. L=[tex]mr^{2}\dot{\theta}[/tex]=2.6612E40, [tex]E=\frac{1}{2}[m\dot{r}^{2}+mr^{2}\dot{\theta}^{2}]-\frac{Gm_{1}m_{2}}{r}[/tex]=-2.6516E33 J, k=7.9304E44 kg*m³/s², m=5.9736E24 kg, e=1.7103E-2. Therefore, [tex]r=\frac{1.4950\mathsf{x}10^{11}}{1+0.017103\cos(\theta)}[/tex] in meters, having taken θ_o to be zero (i.e., x-axis aligned with Sun-Earth line at aphelion). Since total angular momentum is conserved, [tex]\dot{\theta}=\frac{L}{mr^2}\Rightarrow{}\ddot{\theta}=\frac{-2L\dot{r}}{mr^3}[/tex] and from radial equation of motion, [tex]\ddot{r}=\frac{L^2}{m^{2}r^3}-\frac{Gm_{1}m_{2}}{mr^2}[/tex] which are the magnitudes of the acceleration in the θ and r-directions, respectively. So, the magnitude of the acceleration in each direction can be written solely as a function of r or θ. A little more work will similarly give the magnitude and direction of the acceleration vector in whatever coordinate system one chooses.

 

[pmacias]

Acceleration in an elliptical orbit is the rate at which the velocity of an object changes as it moves along its orbital path. This acceleration is caused by the gravitational force of the central body, which varies in strength depending on the distance between the object and the central body.

To calculate the acceleration between aphelion and perihelion in an elliptical orbit, the equation used is the Newton's Second Law of Motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass. In the case of an elliptical orbit, the net force is the gravitational force of the central body, and the mass is the mass of the object in orbit.

The equation for acceleration in an elliptical orbit is:

a = G * M / r^2

Where:
a = acceleration
G = gravitational constant
M = mass of the central body
r = distance between the object and the central body

This equation can be used to calculate the acceleration at any point along the elliptical orbit, including between aphelion (the point farthest from the central body) and perihelion (the point closest to the central body). The distance (r) used in the equation will vary depending on the position of the object in its orbit.

In summary, the acceleration between aphelion and perihelion in an elliptical orbit can be calculated using the Newton's Second Law of Motion equation, taking into account the mass of the central body and the distance between the object and the central body.