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 traveled 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)

How long it takes to escape from LEO to trans-Martian orbit

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

Fastest speed attained

Relative to what?

Total distance traveled 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)

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".