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Ok, I think I've now completed the mapping of the centre manifold orbits of the L1 and L2 points of the Earth-Moon system. For the mathematics aficionados in the Orbiter Land, this 'mapping' consists of calculating the coefficients of a Lindstedt-Poincaré expansion of the centre-manifold to...

Over the last couple of months I've been busy working out the parametric representing the closed 'centre manifold' orbits of the L1 and L2 Lagrange points of the Earth-Moon and Sun-Earth systems. And I've just about finished. Essentially, this is a mapping exercise. Just as Keplerian orbits...

Very nice. As with BrianJ, it looks as if I'm going to have download GMAT.

Ajaja Congrats! I wondered if anyone was making headway with this. Although deceptively simple, this isn't an easy challenge: one needs to recognise that this orbital transfer problem can't be solved using simple analytical techniques; and the standard Orbiter suite of trajectory planning...

I always have to do these by reverting to the basic Euler angle transformations between ecliptic and equatorial coordinates and then grinding methodically through the algebra. But having just done that I find that the equinox corresponding to the 0 degree axis is indeed (as per 'BrianJ') just...

This note is going to be about building a high-fidelity integration engine for Orbiter that takes into account the gravitational influence of other bodies (tidal forces) and the 'non-spherical gravity sources' that Orbiter attributes to each body. In a subsequent post, I will show how this...

Target practice with Delta Gliders OK, so what's interesting about this? It's just a screen shot of two Delta Gliders rather unrealistically passing through each other. Yes, true. However.... Each Delta Glider is orbiting at around 7550 m/s in orbital planes oriented at right angles to...

This note sets out the transformations that Orbiter uses to convert points in its 'body-centred inertial' (BCI)reference frame to its 'body-centred, body-fixed' reference frame. (The BCI reference frame is a generalisation of the Earth-centred inertial (ECI) reference frame applying to any...

Yes, it does seem miraculous that the offsetting forces do appear in the equations of motion. I’m sure there is a clear explanation for this offset and I will rummage through the internet in search of one. If I am successful, I will post.

Yes, the speed of gravity is finite. And so you might think that the direction of the gravitational force should not act towards the instantaneous position of the gravitating body. But this isn’t the case. In analogy with a relativistic treatment of electromagnetism, there are counter-balancing...

Just as a quick comment: if you are integrating in an inertial frame of reference centred on SSB then you don’t need the tidal terms. Using them would be ‘double counting’ and will lead to an error. It’s only if you are integrating in a non-inertial ECI coordinate system that you do need to add...

Are you saying that checking a simulation with another simulation is better than checking a simulation against reality? No, I'm not saying that. All that I'm saying is that Orbiter can provide a controlled test of model formulation. As I understand it, you have two possibilities: either you...

OK, I guess that this means that we can rule out DOPRI as a significant source of error. Let's focus on the TLEs.. My understanding is that you are using the TLEs from one epoch to inform the initial state vectors of the DOPRI integrator and then, after computing the final state vectors using...

I also don't know much about the DOPRI integrator either. Most integrators are subject to errors of one kind or another. It's always a good idea to run an integrator against a known analytical solution (e.g., pure Keplerian elliptical motion, to see how accurate the integrator is under...

Here's my experiment: 1) calculate the state of the GPS PRN 07 satellite with the SDP4 propagator for the TLE 18001.44476542 2) propagate the state with the DOPRI853 integrator 3) for all the subsequent TLEs, calculate the difference between the DOPRI position and the TLE position for the TLE...

I've had a chance to run a test of the tidal contributions in the equations of motion in the ECI reference frame. A brief description of the test Before presenting the results, I should describe the specific test performed. * I placed a Delta-Glider in a circular orbit around the Earth with an...

Indeed, your expressions are vectors. In vector form, the ECI equations of motion are: \mathbf{Q}''(t) = -\frac{\mu _e}{\Delta r_e^3}\mathbf{Q}(t) + \sum_i\left(\frac{\mu _i}{\Delta r_i^3}\left(\mathbf{Q}_i(t)-\mathbf{Q}(t)\right) - \frac{\mu_i}{\Delta r_{e,i}^3}\mathbf{Q}_i(t)\right) where...

The signs are correct, I think. But note that r_xx is always greater than zero because one is taking the dot product of the vector and then the positive square root. So, the sign shouldn't matter. I'll double check via numerical experiment over the weekend that I haven't slipped up on a sign...

Starting from a clean sheet, I re-derived the ECI equations of motion in Mathematica using a slightly different approach (Lagrangian mechanics) and got the same result. So, unless I'm having a brain-freeze, I think the equations of motion are OK. Also note that: temp = y[sat] - y[earth]...

Yes, basically, the sum of the three accelerations is just the right-hand side of the following equations of motion in the ECI reference frame: X''(t) = -\frac{\mu _e}{\Delta r_e^3}X(t) + \sum_i\left(\frac{\mu _i}{\Delta r_i^3}\left(X_i(t)-X(t)\right) - \frac{\mu_i}{\Delta...