Cristiano

You're right (but I need 2*pi because my original n is in revolution/day). What about da/dt= -2 * 10096.676 / n3 * dn/dt / (3 * a2) ?

I know a, n and dn/dt. a3= 10096.676 / n2. I need to calculate da/dt. 398600.8 has unit km3/s2, n is rad/s, dn/dt is rad/s2. EDIT: Probably I'm wrong in the unit of measurement of da/dt; I thought it was km/s2, but it should be km/s, right?

The original formula is a3 = 398600.8 / (2 * pi * n)2 With dn/dt= 6.07340239 * 10-12 rad/s2 I get da/dt= -1.51970883 * 10-5 km/s2 which is -113445.6 km/day2, while I expect about -1 to -10 km/day2.

If I put the real numbers, I get a wrong result. Could you please confirm that the formula is correct?

I know the value of dn/dt (e.g. dn/dt = 123). a3 = 345 / n2 Is there any way to calculate da/dt?

I meant a recursive formula like: a = A/U * (a + b) (I wrote just an example for clarification).

Supposing that I correctly calculate the integral to obtain the circumference and the area (I'm not sure right now), is there any numerical method that I can try? EDIT: I used the formula C= pi * (a + b) [ 1 + sum... to calculate the circumference and my above result (145485.418131498) is good, while the area is wrong.

I know that the area of an ellipse is 1403761773.43497 and its circumference is 145485.418131498. Is there any way to calculate from those numbers that the semi-major axis is 26534.9039306654?

Thank you all. I've never heard of absement, but unfortunately it doesn't seem useful for my problem, which is related to astronomy. I need to calculate the mean radius vector of an Earth's artificial satellite. But probably this is not the right forum. I first need to ask some basic questions about the ellipse... I switch to the math subforum.

If I integrate the speed I obtain the space, but if I integrate the space, what do I get?

I still have a little doubt. We have: v0 = v v' = f(t, v), v= RK(v') b' = f(v, b) When I wrote RK for b', should I first calculate the new v and use this new v as the input for b' or should I use v0, the v before the update v= RK(v')?

Ah! Ok. The same that I do for the n-body simulation. Now it's clear. Thank you very much.

I have 3 variables: v, b and h. I know how to calculate v’, b’ and h’. Starting the simulation from an initial condition, I need to calculate v, b and h after a given time using a numerical integrator (say RK4). If I use the simple Euler method, I write; v= v + dt * v’ b= b + dt * b’ h= h + dt * h’ but when I switch to RK4, I don’t know what to write. The simulation is explained here: https://mintoc.de/index.php?title=Gravity_Turn_Maneuver For example, h’= v * cos(b) doesn’t seem a differential equation; does it mean that h= h + dt * h’ is correct and that there is no reason to use RK4? Also, the second RK4 step is: k2 = f(x + dx / 2, y + k1 / 2) and we have that m’ = constant, v’= f(v, m). When I write the above k2 f(x, y) function, I suppose that I need to consider also m at time x + dx / 2, not only v, in other words f(x, y) is: m = m_initial – m’ * x v’ = f(m, v) and not only v’ = f(v).

Probably the best answer I can get comes from an orbital mechanics book, where it is said that for an (unperturbed) elliptical orbit, the conservation of energy may be written: V2 / 2 = mu / r - mu / (2 * a)

While I'm very thankful for your explanations and for your patience, I still don't see a clear cause-and-effect rule. Probably I should find something that links the kinetic energy with the potential energy for the orbital motion…