# Delta-v budget

Delta-v in feet per second, and fuel requirements for a typical Apollo Lunar Landing mission.

In astrodynamics and aerospace, a delta-v budget is an estimate of the total delta-v required for a space mission. It is calculated as the sum of the delta-v required for the propulsive maneuvers during the mission, and as input to the Tsiolkovsky rocket equation, determines how much propellant is required for a vehicle of given mass and propulsion system.

Delta-v is a scalar quantity dependent only on the desired trajectory and not on the mass of the space vehicle. For example, although more thrust, fuel, etc. is needed to transfer a larger communication satellite from low Earth orbit to geosynchronous orbit, the delta-v required is the same. Also delta-v is additive, as contrasted to rocket burn time, the latter having greater effect later in the mission when more fuel has been used up.

Tables of the delta-v required to move between different space venues are useful in the conceptual planning of space missions. In the absence of an atmosphere, the delta-v is typically the same for changes in orbit in either direction; in particular, gaining and losing speed cost an equal effort. An atmosphere can be used to slow a spacecraft by aerobraking.

A typical delta-v budget might enumerate various classes of maneuvers, delta-v per maneuver, and number of each maneuver required over the life of the mission, and simply sum the total delta-v, much like a typical financial budget. Because the delta-v needed to achieve the mission usually varies with the relative position of the gravitating bodies, launch windows are often calculated from porkchop plots that show delta-v plotted against the launch time.

## General principles

The Tsiolkovsky rocket equation shows that the delta-v of a rocket (stage), is proportional to the logarithm of the fuelled-to-empty mass ratio of the vehicle, and to the specific impulse of the rocket engine. A key goal in designing space-mission trajectories is to minimize the required delta-v to reduce the size and expense of the rocket that would be needed to successfully deliver any particular payload to its destination.

The simplest delta-v budget can be calculated with Hohmann transfer, which moves from one circular orbit to another coplanar circular orbit via an elliptical transfer orbit. In some cases a bi-elliptic transfer can give a lower delta-v.

A more complex transfer occurs when the orbits are not coplanar. In that case there is an additional delta-v necessary to change the plane of the orbit. The velocity of the vehicle needs substantial burns at the intersection of the two orbital planes and the delta-v is usually extremely high. However, these plane changes can be almost free in some cases if the gravity and mass of a planetary body is used to perform the deflection. In other cases, boosting up to a relatively high altitude apoapsis gives low speed before performing the plane change and this can give lower total delta-v.

The slingshot effect can be used in some cases to give a boost of speed/energy; if a vehicle goes past a planetary or lunar body, it is possible to pick up (or lose) much of that body's orbital speed relative to the sun or a planet.

Another effect is the Oberth effect—this can be used to greatly decrease the delta-v needed, because using propellant at low potential energy/high speed multiplies the effect of a burn. Thus for example the delta-v for a Hohmann transfer from Earth's orbital radius to Mars's orbital radius (to overcome the sun's gravity) is many kilometres per second, but the incremental burn from LEO over and above the burn to overcome Earth's gravity is far less if the burn is done close to Earth than if the burn to reach a Mars transfer orbit is performed at Earth's orbit, but far away from Earth.

A less used effect is low energy transfers. These are highly nonlinear effects that work by orbital resonances and by choosing trajectories close to Lagrange points. They can be very slow, but use very little delta-v.

Because delta-v depends on the position and motion of celestial bodies, particularly when using the slingshot effect and Oberth effect, the delta-v budget changes with launch time. These can be plotted on a porkchop plot.

Course corrections usually also require some propellant budget. Propulsion systems never provide precisely the right propulsion in precisely the right direction at all times and navigation also introduces some uncertainty. Some propellant needs to be reserved to correct variations from the optimum trajectory.

## Budget

### Launch/landing

The delta-v requirements for sub-orbital spaceflight are much lower than for orbital spaceflight. For the Ansari X Prize altitude of 100 km, Space Ship One required a delta-v of roughly 1.4 km/s. To reach low Earth orbit of the space station of 300 km, the delta-v is over six times higher about 9.4 km/s. Because of the exponential nature of the rocket equation the orbital rocket needs to be considerably bigger.

• Launch to LEO—this not only requires an increase of velocity from 0 to 7.8 km/s, but also typically 1.5–2 km/s for atmospheric drag and gravity drag
• Re-entry from LEO—the delta-v required is the orbital maneuvering burn to lower perigee into the atmosphere, atmospheric drag takes care of the rest.

### Stationkeeping

Maneuver Average delta-v per year [m/s] Maximum per year [m/s]
Drag compensation in 400–500 km LEO < 25 < 100
Drag compensation in 500–600 km LEO < 5 < 25
Drag compensation in > 600 km LEO < 7.5
Station-keeping in geostationary orbit 50–55
Station-keeping in L1/L2 30–100
Station-keeping in lunar orbit 0–400 [1]
Attitude control (3-axis) 2–6
Spin-up or despin 5–10
Stage booster separation 5–10

### Earth–Moon space—high thrust

Delta-v needed to move inside the Earth–Moon system (speeds lower than escape velocity) are given in km/s. This table assumes that the Oberth effect is being used—this is possible with high thrust chemical propulsion but not with current (As of 2016) electrical propulsion.

The return to LEO figures assume that a heat shield and aerobraking/aerocapture is used to reduce the speed by up to 3.2 km/s. The heat shield increases the mass, possibly by 15%. Where a heat shield is not used the higher from LEO Delta-v figure applies, the extra propellant is likely to be heavier than a heat shield. LEO-Ken refers to a low Earth orbit with an inclination to the equator of 28 degrees, corresponding to a launch from Kennedy Space Center. LEO-Eq is an equatorial orbit.

The reference for most of the data[2] no longer works, and some things are not clear, such as why there is such a big difference between going from L2 to LEO versus going from L1 to LEO. The figure for LEO to L2 comes from a paper by Robert W. Farquhar.[3] (One could probably use a similar tactic to get to L1 for about the same delta-v.) Note that getting to one of the Lagrange points means not just getting to the right place but also adjusting the final velocity in order to stay there. Another source gives values from LEO to GEO, L1, and lunar surface.[4]

∆V km/s from/to LEO-Ken LEO-Eq GEO EML-1 EML-2 EML-4/5 LLO Moon C3=0
Earth 9.3–10
Low Earth orbit (LEO-Ken) 4.24 4.33 3.77 3.43 3.97 4.04 5.93 3.22
Low Earth orbit (LEO-Eq) 4.24 3.90 3.77 3.43 3.99 4.04 5.93 3.22
Geostationary orbit (GEO) 2.06 1.63 1.38 1.47 1.71 2.05 3.92 1.30
Lagrangian point 1 (EML-1) 0.77 0.77 1.38 0.14 0.33 0.64 2.52 0.14
Lagrangian point 2 (EML-2) 0.33 0.33 1.47 0.14 0.34 0.64 2.52 0.14
Lagrangian point 4/5 (EML-4/5) 0.84 0.98 1.71 0.33 0.34 0.98 2.58 0.43
Low lunar orbit (LLO) 1.31 1.31 2.05 0.64 0.65 0.98 1.87 1.40
Moon 2.74 2.74 3.92 2.52 2.53 2.58 1.87 2.80
Earth escape velocity (C3=0) 0 0 1.30 0.14 0.14 0.43 1.40 2.80

### Earth–Moon space—low thrust

Current electric ion thrusters produce a very low thrust (milli-newtons, yielding a small fraction of a g), so the Oberth effect cannot normally be used. This results in the journey requiring a higher delta-v and frequently a large increase in time compared to a high thrust chemical rocket. Nonetheless, the high specific impulse of electrical thrusters may significantly reduce the cost of the flight. For missions in the Earth–Moon system, an increase in journey time from days to months could be unacceptable for human space flight, but differences in flight time for interplanetary flights are less significant and could be favorable.

The table below presents delta-v's in km/s, normally accurate to 2 significant figures and will be the same in both directions, unless aerobraking is used as described in the high thrust section above.[5]

From To delta-v (km/s)
Low Earth orbit (LEO) Earth–Moon Lagrangian 1 (EML-1) 7.0
Low Earth orbit (LEO) Geostationary Earth orbit (GEO) 6.0
Low Earth orbit (LEO) Low Lunar orbit (LLO) 8.0
Low Earth orbit (LEO) Sun–Earth Lagrangian 1 (SEL-1) 7.4
Low Earth orbit (LEO) Sun–Earth Lagrangian 2 (SEL-2) 7.4
Earth–Moon Lagrangian 1 (EML-1) Low Lunar orbit (LLO) 0.60–0.80
Earth–Moon Lagrangian 1 (EML-1) Geostationary Earth orbit (GEO) 1.4–1.75
Earth–Moon Lagrangian 1 (EML-1) Sun-Earth Lagrangian 2 (SEL-2) 0.30–0.40

[5]

### Interplanetary

The spacecraft is assumed to be using chemical propulsion and the Oberth effect.

From To Delta-v (km/s)
LEO Mars transfer orbit 4.3[6] ("typical", not minimal)
Earth escape velocity (C3=0) Mars transfer orbit 0.6[7]
Mars transfer orbit Mars capture orbit 0.9[7]
Mars capture orbit Deimos transfer orbit 0.2[7]
Deimos transfer orbit Deimos surface 0.7[7]
Deimos transfer orbit Phobos transfer orbit 0.3[7]
Phobos transfer orbit Phobos surface 0.5[7]
Mars capture orbit Low Mars orbit 1.4[7]
Low Mars orbit Mars surface 4.1[7]
Earth–Moon Lagrange point 2 Mars transfer orbit <1.0[6]
Mars transfer orbit Low Mars orbit 2.7[6] (not minimal)
Earth escape velocity (C3=0) Closest NEO[8] 0.8–2.0

According to Marsden and Ross, "The energy levels of the Sun–Earth L1 and L2 points differ from those of the Earth–Moon system by only 50 m/s (as measured by maneuver velocity)."[9]

We may apply the formula

(where μ = GM is the standard gravitational parameter of the sun, see Hohmann transfer orbit) to calculate the Δv in km/s needed to arrive at various destinations from Earth (assuming circular orbits for the planets). In this table, the column labeled "Δv to enter Hohmann orbit from Earth's orbit" gives the change from Earth's velocity to the velocity needed to get on a Hohmann ellipse whose other end will be at the desired distance from the sun. The column labeled "v exiting LEO" gives the velocity needed (in a non-rotating frame of reference centred on Earth) when 300 km above Earth's surface. This is obtained by adding to the specific kinetic energy the square of the speed (7.73 km/s) of this low Earth orbit (that is, the depth of Earth's gravity well at this LEO). The column "Δv from LEO" is simply the previous speed minus 7.73 km/s.

Destination Orbital radius (AU) Δv to enter Hohmann orbit
from Earth's orbit
v exiting LEO Δv from LEO
Sun 0 29.8 31.7 24.0
Mercury 0.39 7.5 13.3 5.5
Venus 0.72 2.5 11.2 3.5
Mars 1.52 2.9 11.3 3.6
Jupiter 5.2 8.8 14.0 6.3
Saturn 9.54 10.3 15.0 7.3
Uranus 19.19 11.3 15.7 8.0
Neptune 30.07 11.7 16.0 8.2
Pluto 39.48 11.8 16.1 8.4
Infinity 12.3 16.5 8.8

To get to the sun, it is actually not necessary to use a Δv of 24 km/s. One can use 8.8 km/s to go very far away from the sun, then use a negligible Δv to bring the angular momentum to zero, and then fall into the sun. This can be considered a sequence of two Hohmann transfers, one up and one down. Also, the table does not give the values that would apply when using the moon for a gravity assist. There are also possibilities of using one planet, like Venus which is the easiest to get to, to assist getting to other planets or the sun. The Galileo spacecraft used Venus once and Earth twice in order to reach Jupiter.

### Delta-vs between Earth, Moon and Mars

Delta-v needed for various orbital manoeuvers using conventional rockets.[7][10] Red arrows show where optional aerobraking can be performed in that particular direction, black numbers give delta-v in km/s that apply in either direction. Lower-delta-v transfers than shown can often be achieved, but involve rare transfer windows or take significantly longer, see: fuzzy orbital transfers. Electric propulsion vehicles going from Mars C3=0 to Earth C3=0 without using the Oberth effect need a larger deltaV of between 2.6 km/s and 3.15 km/s.[11] Not all possible links are shown.
Abbreviations key: Escape orbit (C3), Geostationary orbit (GEO), Geostationary transfer orbit (GTO), Earth–Moon L5 Lagrangian point (L5), low Earth orbit (LEO).
The figure 2.5 for LEO to GTO is higher than necessary[lower-alpha 1] and the figure of 30 for LEO to the sun is also too high.[lower-alpha 2]

### Near-Earth objects

Near-Earth objects are asteroids that are within the orbit of Mars. The delta-v to return from them are usually quite small, sometimes as low as 60 m/s, using aerobraking in Earth's atmosphere.[12] However, heat shields are required for this, which add mass and constrain spacecraft geometry. The orbital phasing can be problematic; once rendezvous has been achieved, low delta-v return windows can be fairly far apart (more than a year, often many years), depending on the body.

However, the delta-v to reach near-Earth objects is usually over 3.8 km/s,[12] which is still less than the delta-v to reach the Moon's surface. In general bodies that are much further away or closer to the sun than Earth have more frequent windows for travel, but usually require larger delta-vs.

## Notes

1. The sum of LEO to GTO and GTO to GEO should equal LEO to GEO. The precise figures depend on what low Earth orbit is used. According to Geostationary transfer orbit, the speed of a GTO at perigee can be just 9.8 km/s. This corresponds to an LEO at about 700 km altitude, where its speed would be 7.5 km/s, giving a delta-v of 2.3 km/s. Starting from a lower LEO would require more delta-v to get to GTO, but then the total for LEO to GEO would have to be higher.
2. Earth's speed in its orbit around the sun is, on average, 29.78 km/s, equivalent to a specific kinetic energy of 443 km2/s2. One must add to this the potential energy depth of LEO, about 61 km2/s2, to give a kinetic energy close to Earth of 504 km2/s2, corresponding to a speed of 31.8 km/s. Since the LEO speed is 7.8 km/s, the delta-v is only 24 km/s. It would be possible to reach the sun with less delta-v using gravity assists. See Solar Probe Plus. It is also possible to take the long route of going far away from the sun (Δv 8.8 km/s) and then using a very small Δv to cancel the angular momentum and fall into the sun.

## References

1. Frozen lunar orbits