Transfer Energy

[PB666]

Last week there was a thread created that discussed the basic requirements of deltaV required to get into various positions of the moon.
Other than the launch variables the statement was made or asked if deltaV tables was the best way to handle this.

I looked at the from an energy perspective, first off I need to add that the classic formula for calculating delta-V between two circular orbits is

  - SQRT(u/r₀) for the first burn  (r is r₀ in this case in the wiki image, ignore the v = ) r can either be an apoapse or periapsis

and

SQRT(u/r₁)  -   (r is r₁ in this case in the wiki image) for the second burn.  r can either be an periapsis or apoapse

The perfect energy requirement equal to the is close to this at in the case of the lowest and highest eccentricities (e = 1)

but in the middle ranges it is considerably different. The basic problem is that elevation of a circular orbit neccesarily requires two burns. During small burns the change of velocity is small and as a consequence little momentum is lost. In changing to very eccentric orbit much momentum is lost, but the dV required to establish the second orbit is small fractional to the energy required to create the transfer orbit. At minimum escape velocity its zero. In eccentricities (e) of transfer orbits around 0.7 (e.g a geosynchronoous from LEO transfer) have substantial inefficiency because considerable momentum is lost as the satellite slows to its apoapse at which it needs to burn. So for example a station keeping burn is perfectly efficient, and also a escape orbit (minimal) is perfectly efficient (but because of N-body problems more or less a theoretical exercise)

The energy requirement works within tolerances if the correction factor for eccentricity is provided dV (total)/((1-e)+LN(1+e^1.9)), up to about e=0.75 but becomes inaccurate after this. Its not perfect. I tested this with a number of orbits, a is irrelevant the error is a function of e. This means without using a table one has a minimum requirement for a single step energy plot of knowing e as well as initial radius and final radius. Its not hard to calculate e but in creates also a two step operation. Ergo the OP is correct, the two step dV plots are as simple as any other means of plotting the dV requirements of an orbital change.

Edited April 4, 2016 by PB666

[PB666]

16 hours ago, PB666 said:

Last week there was a thread created that discussed the basic requirements of deltaV required to get into various positions of the moon.
Other than the launch variables the statement was made or asked if deltaV tables was the best way to handle this.

I looked at the from an energy perspective, first off I need to add that the classic formula for calculating delta-V between two circular orbits is

  - SQRT(u/r₀) for the first burn  (r is r₀ in this case in the wiki image, ignore the v = ) r can either be an apoapse or periapsis

and

SQRT(u/r₁)  -   (r is r₁ in this case in the wiki image) for the second burn.  r can either be an periapsis or apoapse

The perfect energy requirement equal to the is close to this at in the case of the lowest and highest eccentricities (e = 1)

but in the middle ranges it is considerably different. The basic problem is that elevation of a circular orbit neccesarily requires two burns. During small burns the change of velocity is small and as a consequence little momentum is lost. In changing to very eccentric orbit much momentum is lost, but the dV required to establish the second orbit is small fractional to the energy required to create the transfer orbit. At minimum escape velocity its zero. In eccentricities (e) of transfer orbits around 0.7 (e.g a geosynchronoous from LEO transfer) have substantial inefficiency because considerable momentum is lost as the satellite slows to its apoapse at which it needs to burn. So for example a station keeping burn is perfectly efficient, and also a escape orbit (minimal) is perfectly efficient (but because of N-body problems more or less a theoretical exercise)

The energy requirement works within tolerances if the correction factor for eccentricity is provided dV (total)/((1-e)+LN(1+e^1.9)), up to about e=0.75 but becomes inaccurate after this. Its not perfect. I tested this with a number of orbits, a is irrelevant the error is a function of e. This means without using a table one has a minimum requirement for a single step energy plot of knowing e as well as initial radius and final radius. Its not hard to calculate e but in creates also a two step operation. Ergo the OP is correct, the two step dV plots are as simple as any other means of plotting the dV requirements of an orbital change.

I have updated this with formula for the alterantive, It assumes the user will manage the dV. There it is only for space craft design. Its convinient because it might shorten the list of post-launch-minimal orbit dV needed

u of course is the celestials gravitational constant. Earth is 3.986E14 (in standard metric units). Eccentricity (e) is (r_apo - r_pe) / (r_{apo +} r_pe)

Happy square root day http://www.illinoisscience.org/2016/04/happy-square-root-day/

e	Correction factor (k)		e	Correction factor (k)		e	Correction factor (k)
0.001	1.001		0.26	1.2176		0.65	1.4019
0.002	1.002		0.27	1.2245		0.66	1.4038
0.004	1.004		0.28	1.2313		0.67	1.4056
0.006	1.006		0.29	1.2380		0.68	1.4071
0.008	1.008		0.30	1.2445		0.69	1.4084
0.010	1.010		0.31	1.2510		0.70	1.4094
0.015	1.015		0.32	1.2574		0.71	1.4103
0.020	1.020		0.33	1.2636		0.72	1.4108
0.025	1.025		0.34	1.2698		0.73	1.4112
0.030	1.029		0.35	1.2758		0.74	1.4112
0.035	1.034		0.36	1.2818		0.75	1.4110
0.040	1.039		0.37	1.2876		0.76	1.4104
0.045	1.044		0.38	1.2934		0.77	1.4096
0.050	1.048		0.39	1.2990		0.78	1.4083
0.055	1.053		0.40	1.3045		0.79	1.4068
0.060	1.057		0.41	1.3099		0.80	1.4048
0.065	1.062		0.42	1.3152		0.81	1.4024
0.070	1.067		0.43	1.3204		0.82	1.3995
0.075	1.071		0.44	1.3255		0.83	1.3961
0.080	1.076		0.45	1.3304		0.84	1.3922
0.085	1.080		0.46	1.3353		0.85	1.3877
0.090	1.084		0.47	1.3400		0.86	1.3825
0.095	1.089		0.48	1.3446		0.87	1.3766
0.100	1.093		0.49	1.3491		0.88	1.3699
0.110	1.102		0.50	1.3535		0.89	1.3622
0.120	1.110		0.51	1.3577		0.90	1.3535
0.130	1.119		0.52	1.3618		0.91	1.3435
0.140	1.127		0.53	1.3658		0.92	1.3321
0.150	1.135		0.54	1.3696		0.93	1.3189
0.160	1.143		0.55	1.3733		0.94	1.3035
0.170	1.151		0.56	1.3769		0.95	1.2854
0.180	1.159		0.57	1.3803		0.96	1.2637
0.190	1.167		0.58	1.3836		0.97	1.2367
0.200	1.174		0.59	1.3867		0.98	1.2016
0.210	1.182		0.60	1.3896		0.99	1.1504
0.220	1.189		0.61	1.3924		0.995	1.1104
0.230	1.196		0.62	1.3951		0.9975	1.0802
0.240	1.204		0.63	1.3975		0.99875	1.0577
0.250	1.211		0.64	1.3998		0.9999	1.0169

Edited April 4, 2016 by PB666
Added Easter Egg

[fredinno]

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