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Enjo I'll expand on this a little later, but there are two controllable inputs and two outputs that need to be set to zero. The controllable inputs in the routine are: 1. 'r', the orbital radius at which the orbit insertion burn commences; and 2. 'tm', the total burn time. These two...

And for anyone interested, here (below) is a simple, stand-alone C program that performs the integration of the equations of motion for the example given above. This implementation uses a Taylor series method to carryout the integration. Given initial conditions for the dynamical variables...

As a specific example of the above, consider an unloaded - but fully-fuelled - Shuttle A approaching the Moon. The semi-major axis and eccentricity of its approach orbit are: a = 84324191.4 m e = 0.9599018 This is a high eccentricity elliptic orbit with periapsis altitude of 1643 km...

As promised, this note outlines an algorithm for finding the time when a retrograde burn must be started (and stopped) so as to insert into spacecraft into a circular orbit around a central gravitating body such as a planet or moon assuming constant engine thrust during the manoeuvre and while...

Survived the typhoon. Now girding loins to write up a detailed note on the original topic.

Enjo I haven't forgotten about this topic. I've given the matter some thought over the last couple of days and have mentally worked out the structure and content of what I want to say on the algorithm. But a significant-ish typhoon is now bearing down on us here in Hong Kong and I have a...

Enjo I'm going to assume that you want an extended range of accuracy - and not necessarily a 100% accurate solution spanning the entire input parameter space. So, basically, I'm going to move forward with a perturbations scheme, since, for me, a 100% accurate solution everywhere simply isn't...

Thorsten Again, it isn't going to be quite that simple, I'm afraid. But, give it a go, and let's see what you come up with.

Enjo, Yes, I agree: we have to assume a retrograde orientation for the burn. It might be helpful if you clarify the regime over which you want improved accuracy. Is it just for relatively short burns, but where BTC's circular orbit injection solution is noticeably, but marginally, in...

Thorsten The problem is not in integrating the equations of motion forward (or backwards) in time. I agree, that's quite a trivial task. The task here is, I think, to work out what manoeuvre is needed to bring a vessel into a circular orbit (or other prescribed orbit) from its initial...

Enjo In a way, Thorsten is right in his comments, at least in as far as he suggests that the equations of motion derived in the opening post are not particularly profound. All that they do, really, is make clear the mathematical problem to be solved to properly account for (Keplerian) gravity...

Yes, I guess one could include other forces if one so desires. ---------- Post added at 02:51 PM ---------- Previous post was at 02:48 PM ---------- Quite so. And yes it does work. I checked this evening on a low thrust high delta v orbit entry burn. More on this later. ---------- Post...

You are quite correct, it isn't a minimum fuel solution - although if the maximum fuel flow rate is 2 kg/s for the Delta Glider then I can imagine that it might well be. At the end of the day, this simply serves as as demonstration of a generalised form of the ideal rocket equation - i.e., to...

The subject of this post is a generalisation of the well-known ideal rocket equation \Delta V = v_e\,\log\frac{m_i}{m_f} to a Keplerian gravitational field. The resulting equations of motion are relevant to the calculation of optimal launch trajectories (for bodies without atmospheres) and...

Yes, I quite agree. For me, the issue has been knowing 'how' Orbiter forms the product 'GM' and not so much the 'what' the values of 'G' and 'M' actually are. It is probably worthwhile just reflecting on the magnitude of this non-problem. Although, Orbiter's value of 'G' differs markedly...

To the extent that Orbiter is 'a world unto itself', there is no need to update values of G and celestial body masses. For most users, most of the time, the differences between current and possible updated values would be too small to observe differences in the resulting trajectories. And to...

Because of the sensitivity of dynamics near collinear Lagrange points, precise estimation of gravitational parameters - GM - for Orbiter's gravitating bodies is important for precision trajectory planning in the vicinity of those points (particularly when approach speeds to a Lagrange points ia...

A series of four videos are now up on Youtube showing how to use the spreadsheet panning tool to get out to the the Earth-Moon L1 Lagrange point - and rendezvous with a vessel parked there. Videos can be found here: Spreadsheet trajectory planning tool A screenshot of the final rendezvous...

ADSWNJ, what do you need to begin? I have a good understanding of the maths/physics and workflows needed. And I have working versions of pieces of the tool suite needed to target Lagrange points in terms of Lua scripts, VB code and spreadsheets. And, if necessary, I convert these to a series...

So far, a spreadsheet-based trajectory planning tool has been developed that incorporates an accurate fourth-order symplectic integrator along with some rudimentary visualisation and targeting information for getting to the Moon. However, the original purpose of this too development was to...