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):r=\frac{-L^2}{km[1+e\cos(\theta{}-\theta_{o})]} where k=Gm₁m₂, m=m₁m₂/(m₁+m₂), e=\sqrt{1+\frac{2EL^2}{k^{2}m}}, E is the total energy of the system, L is its total angular momentum, and \theta_{o} is the initial angle.

By definition of the center of mass frame (in which this problem is usually analysed), m₁r₁=m₂r₂ and \theta_{1}=\theta_{2}+\pi{}\Rightarrow{}\dot{\theta_1}=\dot{\theta_2}\Rightarrow{}m_{1}v_{1}=m_{2}v_{2} 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=mr^{2}\dot{\theta}=2.6612E40, E=\frac{1}{2}[m\dot{r}^{2}+mr^{2}\dot{\theta}^{2}]-\frac{Gm_{1}m_{2}}{r}=-2.6516E33 J, k=7.9304E44 kg*m³/s², m=5.9736E24 kg, e=1.7103E-2. Therefore, r=\frac{1.4950\mathsf{x}10^{11}}{1+0.017103\cos(\theta)} 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, \dot{\theta}=\frac{L}{mr^2}\Rightarrow{}\ddot{\theta}=\frac{-2L\dot{r}}{mr^3} and from radial equation of motion, \ddot{r}=\frac{L^2}{m^{2}r^3}-\frac{Gm_{1}m_{2}}{mr^2} 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.