## Wednesday, June 27, 2012

### Inflated Delta Vs

"What's delta V from Earth orbit to Mars orbit?" -- a common question in science fiction or space exploration forums. The usual answer given is around 6 km/s, the delta V needed to go from a low, circular Earth orbit to a low, circular Mars orbit. A misleading answer, in my opinion.

There are a multitude of possible orbits and low circular orbits take more delta V to enter and exit. A science fiction writer using 6 km/s for Earth orbit to Mars orbit has a needlessly high delta V budget.

There are capture orbits that take much less delta V to enter and exit. By capture orbit I mean a periapsis as low as possible and apoapsis as high as possible. A capture orbit's apoapsis should be within a planet's Sphere Of Influence (SOI).

On page 124 of Prussing and Conway's Orbital Mechanics, radius of Sphere Of Influence is given by:

rsoi = ( mp / ms ) 2/5 rsp

where

rsoi is radius of Sphere Of Influence
mp is mass of planet
ms is mass of sun
rsp is distance between sun and planet.

The table below is modeled after a mission table at Atomic Rockets, a popular resource for science fiction writers and space enthusiasts.

• Departure and destination planets are along the left side and across the top of the table.
• Numbers are kilometers/second
• Numbers below the diagonal in blue are delta V's needed to go from departure planet's low circular orbit to destination planet's low circular orbit. These are about the same as the blue quantities listed at Atomic rockets.
• Numbers above the diagonal in red are delta V's needed to go from departure planet's capture orbit to desitnation planet's capture orbit.

 Venus Earth Mars Jupiter Saturn Uranus Neptune Venus .7 3.6 5.6 6.7 7.5 7.5 Earth 6.8 1.1 3.5 4.6 5.3 5.4 Mars 7.9 5.7 3.0 4.5 5.6 5.8 Jupiter 25.8 24.0 21.8 .1 .3 .3 Saturn 20.0 18.1 16.2 27.8 .1 .2 Uranus 16.6 14.7 13.2 23.8 16.6 .03 Neptune 17.3 15.4 14.1 24.5 17.3 13.1

It's easy to see the red numbers are a lot less than the blue numbers. I used this spreadsheet to get these numbers. The spreadsheet assumes circular, coplanar orbits.

A  graphic comparing delta Vs from earth to various destination planets:

If a low circular orbit at the destination is needed, it's common to do a burn to capture orbit with the capture orbit's periapsis passing through the upper atmosphere. Each periapsis pass through the upper atmosphere sheds velocity, lowering the apoapsis. Thus over time the orbit is circularized without the need for reaction mass. The planets in the table above have atmospheres, so the drag pass technique can be used for all of them.

A delta V budget is from propellant source to destination. If propellant depots are in high orbit, the needed delta V is closer to departing from a capture orbit than departing from a low circular orbit.

Thus it would save a lot of delta V to depart from Earth-Moon-Lagrange 1 or 2 (EML1 or EML2) regions. The poles of Luna have cold traps that may have rich volatile deposits. This potential propellant is only 2.5 km/s from EML1 and EML2. Entities like Planetary Resources have talked about parking a water rich asteroid at EML1 or 2. Whether EML propellant depots are supplied by lunar or asteroidal volatiles, they would greatly reduce the delta V for interplanetary trips.

Mars' two moons, Phobos and Deimos, have low densities. Whether that is from volatile ices or voids in a rubble pile is still unknown. If they do have volatile ices, these moons could be a propellant source. It would take much less delta V departing from Deimos than low Mars orbit.

All the gas giants have icey bodies high on the slopes of their gravity wells. However the axis of Uranus and her moons are tilted 97 degrees from the ecliptic. The plane change would be very expensive in terms of delta V. So the moons of Uranus wouldn't be helpful as propellant sources.

Venus has no moon. So of all the planets listed above, only Uranus and Venus lack potential high orbit propellant sources.

Anyway you look at it, the blue numbers from conventional wisdom are inflated.