Delta-v budgets

Draft

Delta-v is the currency of mission design because it is the one quantity that converts directly into propellant mass through the rocket equation, and propellant mass is what a vehicle is mostly made of. A mission either closes on delta-v or it does not exist.

Why this quantity and not another

From the rocket equation, propellant fraction depends on delta-v exponentially:

\frac{m_p}{m_0} = 1 - e^{-\Delta v / (I_{sp}g_0)}

Two consequences make delta-v the natural accounting unit.

It is additive. Manoeuvres performed at different times and places add arithmetically, so a budget is a column of numbers. Velocities themselves are not additive, and neither is propellant mass across stages.

It is independent of the vehicle. A Hohmann transfer from LEO to GEO costs 3.9 km/s whether it is flown by a 100 kg smallsat or a 5 tonne communications platform. That lets mission design proceed before propulsion selection.

The cost is that delta-v is unforgiving. Adding 500 m/s to a budget late in a programme, on a stage with an $I_{sp}$ of 320 s, means roughly 15% more propellant, and the tanks were sized a year ago.

Launch to low Earth orbit

The orbital speed at 200 km is 7.78 km/s. A launch vehicle must deliver appreciably more, because three loss mechanisms consume energy that never becomes orbital velocity.

Term	Typical value	Cause
Orbital velocity at 200 km	7.78 km/s	The actual requirement
Gravity loss	1.2 to 1.5 km/s	Thrust spent holding the vehicle up during ascent
Drag loss	0.1 to 0.3 km/s	Atmospheric resistance, mostly below 50 km
Steering loss	0.1 to 0.5 km/s	Thrust vectored away from the velocity direction
Earth rotation credit	−0.1 to −0.465 km/s	Free eastward velocity from the launch site
Total ideal delta-v	9.3 to 9.5 km/s

Gravity loss is the largest and the least intuitive. While the vehicle accelerates vertically, part of the thrust is doing nothing but cancelling weight. Its magnitude is $\int g\sin\gamma\,\mathrm{d}t$ over the flight path angle $\gamma$ , so it is minimised by pitching over early and getting horizontal fast. That is why launch vehicles begin a gravity turn within the first minute rather than climbing straight up.

The rotation credit is only available for eastward launches and scales as the cosine of latitude. Kourou at 5.2° north recovers 0.463 km/s; Baikonur at 45.6° recovers 0.326 km/s. Polar and sun-synchronous launches recover essentially none of it, and retrograde sun-synchronous launches pay a penalty instead.

A delta-v map

All figures in km/s, from a 200 km circular parking orbit unless stated, and assuming impulsive burns with no aerobraking.

From	To	Delta-v	Note
Earth surface	LEO	9.3 to 9.5	Includes all losses
LEO	GTO	2.46	Perigee burn
GTO	GEO	1.48	Coplanar apogee burn only
LEO	GEO	3.9	Coplanar total
LEO	GEO from 28.5°	4.29	With the plane change combined at apogee
LEO	Escape	3.22	$C_3 = 0$
LEO	Trans-lunar injection	3.1
TLI	Low lunar orbit	0.8 to 0.9	Lunar orbit insertion
Low lunar orbit	Lunar surface	1.6 to 1.9	Powered descent
LEO	Trans-Mars injection	3.6	Varies with launch window
Mars arrival	Mars orbit	0.9 to 2.4	Propulsive; aerobraking cuts this dramatically
Mars arrival	Mars surface	0.5 to 0.9	With aeroshell doing most of the work
LEO	Sun-Earth $L_2$	3.2	Plus a few m/s per year to stay there

Two entries deserve emphasis. Lunar descent costs more than lunar orbit insertion, because the Moon has no atmosphere to help. And Mars orbit insertion varies by a factor of nearly three depending on whether aerobraking is used, which is why almost every Mars orbiter since Magellan has used it.

Worked example: GTO to GEO from Cape Canaveral

This is the most valuable single optimisation in routine mission design.

A satellite is delivered to GTO with apogee at GEO altitude and inclination 28.5°, the latitude of the launch site. At apogee its speed is 1.597 km/s; a geostationary orbit needs 3.075 km/s in the equatorial plane.

Doing the two jobs separately, circularise first, then rotate the plane:

\Delta v = (3.075 - 1.597) + 2(3.075)\sin\frac{28.5°}{2} = 1.477 + 1.514 = 2.991\ \text{km/s}

Doing them in one burn, applying the law of cosines to the vector triangle:

Combined manoeuvre

\Delta v = \sqrt{v_1^2 + v_2^2 - 2v_1v_2\cos\Delta i}

\Delta v = \sqrt{1.597^2 + 3.075^2 - 2(1.597)(3.075)\cos 28.5°} = 1.837\ \text{km/s}

The combined burn saves 1.15 km/s, close to 40% of the manoeuvre. On a satellite with a 320 s apogee motor that is roughly a third of the wet mass, and it is the difference between a viable commercial platform and an uncompetitive one.

Building a real budget

An operational budget has three distinct parts, and conflating them is a common error.

Deterministic: Manoeuvres known in advance: injection, transfers, orbit insertion, disposal. Computed from the reference trajectory.
Statistical: Corrections for errors that have not happened yet: injection dispersion, manoeuvre execution error, navigation uncertainty. Sized from Monte Carlo, usually at the 99th percentile.
Margin: Held against changes in the mission itself, not against known uncertainty. Typically 5 to 10% of the deterministic total.

The statistical term is not margin and must not be traded against it. It covers errors whose distribution is understood; margin covers the requirement moving. A programme that spends its statistical allocation on a new science objective has not gained capability, it has raised its probability of mission failure by an amount nobody computed.

Station keeping and disposal belong in the deterministic column and are regularly underestimated. A geostationary satellite spends about 50 m/s per year holding station, so a 15-year design life carries 750 m/s before any transfer is considered. That is comparable to the entire GTO-to-GEO circularisation.

References

Wertz, J. R., Everett, D. F. and Puschell, J. J. (eds.) Space Mission Engineering: The New SMAD, Microcosm Press, 2011, chapters 9 and 10.
Larson, W. J. and Wertz, J. R. (eds.) Space Mission Analysis and Design, 3rd ed., Microcosm Press, 1999.
Brown, C. D. Spacecraft Mission Design, 2nd ed., AIAA, 1998.
NASA/SP-2016-6105, NASA Systems Engineering Handbook, on margin policy.