Gravity assists for Mars mission?

[Robert Clark]

This is a question involving the orbital mechanics of spaceflight. There are a few different components to the question. First, if our Mars rocket departed from the Moon or a Lagrange point propellant depot fully fueled towards Earth at, say, 11 km/s or more, so it's moving at speeds beyond Earth's escape velocity, then in just passing by the Earth it should pick up additional speed equal to Earth's escape velocity about 11 km/s. So at least temporarily it should have a speed of 22 km/s. But the problem is that it still will be slowed down by the Earth as it proceeds to Mars, so it will lose some of this speed. How much speed will it lose?

 

What I want to do is leave Earth's vicinity at such high speed so that you don't have the long travel times of the Hohmann orbit, and in fact so that the trajectory approximates a straight-line path and if you do it at closest approach of Mars then the travel time could be say 60,000,000 km/22 km/s = 2,700,000 s, about 31 days. (You would have the problem of aerocapture at Mars at such highly elevated speeds but I'll leave that to another discussion.) So another question I have is at what high speed would you need so that the path is approximately straight-line?

 

This is just using Earth flyby. Could we in addition also use a Venus flyby? You would need an orbital arrangement where both Venus and Mars are near the Earth at the same time. Say you are now traveling at 22 km/s towards Venus, minus the amount you're slowed by leaving the Earth. You can likewise pick up about 11 km/s additional speed by just passing by Venus on the way to Mars, perhaps arranging it so that the path is bent by Venus to aim the craft towards Mars. So you could conceivably be traveling now at 33 km/s, again though I need to know how much speed you would lose in leaving Venus. You would also have to factor in the additional time it takes to get to Venus and the longer straight-line distance to Mars from Venus. Also, in being within Venus's orbit around the Sun, the greater gravitational effects of the Sun will have a greater effect to curve the trajectory.

 

Finally, could we use repeatedly the gravitational boosts of Earth and Venus? Suppose we are now at 33 km/s, more or less, after leaving Venus but we arrange it so our path is bent completely around to head back towards Earth. Could we once more get an additional 11 km/s to bring our velocity to 44 km/s after the Earth boost? Could we do this repeatedly to get arbitrarily high speeds?

 

 

Bob Clark

 

 

Edited May 31, 2012 by Robert Clark

[Enthalpy]

Hi Bob and everybody!

 

Depending on whether I got properly your intent of gravity assist at Earth, it might be flawed. The simplest representation of a gravity assist is to think it first as seen from the celestial body: the craft arrives at some far velocity and later moves away with that same velocity but in a different direction (meanwhile it has been faster as it was closer to the body). Then, seeing the flyby from a different location, say from our Sun, what was "same speed, different direction" can become "more speed" (or less speed, very useful to capture a craft using Saturn's moons) if the directions subtract the celestial body's speed first and add it later.

 

Now, a craft starting from a Lagrangian point or the Moon moves essentially with the same speed as Earth does before nearing it, and will regain this nearly zero speed vs Earth as it moves away. No gain possible, because no initial speed difference is available. The craft will depart with the same 30km/s vs our Sun as Earth has.

 

One trick of fuel depots at Lagrangian point is not a slingshot effect but an Oberth effect (Wikipedia): the craft gets free 11km/s from Earth, it accelerates there, and loses speed as it leaves Earth's vicinity - but this is more efficient because it's all a matter of energy vs speed. That is, 3km/s near Earth bring (11+3km/s)^2 of which 11^2 are lost, instead of 3^2 if accelerating far from Earth.

 

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Aerocapture at high speed must be possible at Mars but needs a craft with some lift. Please refer to our other forum, I described it there - maybe in the X37 thread.

 

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A Venus flyby would make the trip longer because of distance, and Venus doesn't permit a slingshot at high speed because it's only as massive and big as Earth is. Repeated use won't help. Worse, reaching Venus takes a higher delta-V than Mars. Venus slingshots have been used to obtain decent speeds from very low initial speed, both to reach Jupiter (or Saturn?) and Mercury.

 

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I couldn't find a straight-path travel to Mars, far less so with chemical propulsion. My best results (other forum) were a short trip OR a short stay (near opposition) with chemical propulsion and high-speed aerobraking both at Mars and Earth plus descent-ascent and return modules preset at Mars, and a short trip AND a short staywith Solar thermal propulsion and preset modules. Both use a trip passing nearer to our Sun than Earth is; I put a downloadable spreadsheet there.

[Janus]

Enthalpy is correct. You can't use the Earth for a gravity assist for an object that starts out with the same orbital velocity as the Earth.

 

A gravity assist is more or less like an elastic collision, with gravity as the mediator.

 

Here's an idealized example:

 

Assume that you launch a probe from an inner planet to an outer planet. It's orbit is such that its aphelion is at the planet's orbit. When it gets there it will be moving much slower than the Outer planet. We assume that we time things so that the probe arrives so that it is just a bit ahead of the planet. For arguments sake we'll say the the probe is moving at pro-grade 5 km/sec and the planet at 11 km/sec. This means that the probe is moving at 6 km/sec retrograde with respect to the planet. As the probe falls in toward the planet it picks up speed. We will also assume that the probe takes a parabolic orbit around the planet, meaning that it will leave the planet at the same speed that it came in at but in the opposite direction.

 

When the probe reaches the distance from the planet where it was originally moving at 6 km/sec retrograde wrt the planet, it is now moving 6 km/sec pro-grade. It's orbital velocity wrt to the Sun is now 11+6 = 17 km/sec, whereas before its was 5 km/sec. It has picked up 12 km/sec. Notice that 12 km/sec is twice the difference in velocity.

 

Now imagine that the probe started at the Lagrange Point for the planet. It starts off with zero velocity difference wrt to the planet, so it is moving at 11 km/sec. It falls in toward the planet, does the parabolic orbit and climbs back out. When it reaches Lagrange point distance again, it will be moving at 0 velocity with respect to the planet, and once again is moving at 11 km/sec. It has gained no velocity with respect to the Sun.