Calculation of Earth-Mars trip for book

andrewbee

Hi,

I am researching a book, and would love to have the help of a space scientist (or similar) who can calculate orbits etc.

Specifically I need accurate figures for an Earth to Mars trip, in a Hohmann transfer orbit, not using conventional propulsion but using a VASIMR engine.

Givens for the plot of the book:

- Total transfer time from low Earth orbit to the orbit of Phobos is about 6 months
- Thrust is continuous, accelerating for the first half of the trip and then decelerating for the rest

I need to know, given

- Level of thrust, assuming constant thrust, to achieve transfer from LEO to the orbit of Phobos (around 3200 miles above Martian surface) in 6 months (given in m/s2)

Given the above level of thrust:
- How long it takes to escape from LEO to trans-Martian orbit
- How long in Hohmann orbit (i.e. trans-Mars)
- Fastest speed attained
- Total distance travelled from Earth to Mars (I believe it to be on the order of 500M kilometers in Hohmann orbit, but more exact figure would be nice)

I will take care of all the breakthroughs in propulsion and nuclear reactors required to achieve this :)

Thanks

mfb

This might be interesting

For some rough estimates:

delta_v for a Hohmann orbit (8-9 months) is ~5km/s, but that requires a high thrust (to accelerate in a low earth orbit), VASIMR will need more.

earth->mars with a constant acceleration of 0.01g (0.1m/s^2) needs a delta_v of ~300km/s. Based on those numbers, it requires ~3*10⁶s or ~40 days.

Let's consider the proposed "1-2" trajectory: ~50km/s delta_v and probably something like 6 months travel time with constant acceleration. This requires an acceleration of 0.003m/s^2 or 3N/ton of mass.
With the medium thrust design values proposed there (80N, 150km/s exhaust velocity), we can accelerate ~30 tons, including about 10 tons of reaction mass (and the way back needs some cheaper trajectory)

In high thrust mode, maybe some days to a week (rough estimate). In low thrust mode: You don't want that.

Relative to what?

Relative to what, and where is the significance?

andrewbee

Thanks, that's some useful info. The link is interesting too - a sci-fi writer's dream!

As far as distance goes, I had to think about what I really meant, and I guess it really comes down to how far the ship moves itself under its own power. So, for 50km/s delta_v, you spend 3 months accelerating to 25km/s and another 3 months decelerating to 0, the distance is about 192M km. It's just bragging rights for the astronauts, lol.

I guess that also answers what the fastest speed is - I really meant the delta_v at the point before it begins to decelerate, which is 25 km/s.

mfb

I think there is an implicit misconception hanging around here: There is no absolute velocity in physics. On earth, you can ask for the speed of a car - and what you mean, is the speed of the car relative to the ground. In space, there is no ground - no obvious frame to measure a velocity.
What is the velocity of the ISS? Relative to earth, about 8km/s. Relative to the sun, it varies between ~22 and ~38km/s every 90 minutes. Relative to the center of our galaxy, it is about 200km/s, with +- 30km/s seasonal variation.

delta_v of 25km/s, in the best case, means that a spacecraft leaves earth with 25km/s. It can be more or less, depending on details of the acceleration process. But this relative velocity changes within months, as spacecraft and earth have different orbits around the sun. For a spacecraft somewhere in the solar system, the velocity relative to earth is quite meaningless (unless you care about Doppler shifts in communication, but that is a different topic).

If you have some really futuristic propulsion system - probably fusion-related, with delta_v capabilities of 1000km/s or more, you can treat the planets as nearly motionless, and your velocity relative to everything in the solar system is similar. This would give a natural way to talk about velocities again.

andrewbee

Yeah, I knew what you were asking. That's why I reframed it as a max. delta_v attained, rather than relative to a celestial body.

mfb

That is 50km/s, accumulated over the whole trip ;).
"Braking" is not the opposite direction of "accelerating".