# How long is the trip?

In earlier units you learned about the Hohmann Transfer Orbit (HTO) to Mars, where the spaceship is given just enough “delta-vee” to leave Earth on an eccentric orbit with perihelion at 1AU and aphelion at Mars’ orbit. This is the slow, cheap orbit – at least, it is cheaper than any other choice that will get us there.

You also considered the “launch window”: To ride the HTO we need to leave Earth at the right moment to arrive at Mars’ orbit when Mars is at the correct position.

1) What if we leave Earth a day later than planned? Is there a way to still get to Mars?

2) Could we leave Earth a day earlier than the ideal date?

The launch window includes the ideal time and a few days on either side of that ideal time. Here is what three orbits look like that all get to Mars when Mars is there:
The fat black orbit is the HTO, leaving Earth at time 2. The red (dashed) line
is the orbit followed by a spaceship that leaves Earth later and arrives later. The green (dotted) line is the orbit for one that leaves earlier, goes faster, and goes beyond Mars’ orbit before returning to meet Mars. (So you see! That is why leaving earlier also means going faster – it has to go farther!)

The fast mission:

If we are willing to use quite a bit more fuel, we can shorten the trip to Mars substantially. How? If we boost it with a larger “delta-vee” from HEO it will move on a larger ellipse, one that crosses the orbit of Mars. At all times the spacecraft on this larger orbit will be moving faster than one on a Hohmann orbit. Moving faster gets you there sooner. How much can we improve the time, and what will that cost?

To get there in half the time we should start out moving about twice as fast. The HTO requires us to start with about 33 km/s speed from near Earth. To get there twice as fast we’d need to start out around 60 km/s. That would mean delta-vee around 30 km/s instead of 3. This is ten times larger Dv, or ten times more fuel! We’d also have a big increase in the amount of fuel needed at Mars to slow down from such a high speed. Once at Mars, we’d be going more than 30 km/s too fast, so we’d again need 10 times as much fuel to slow down. Our one way trip would go from 2x2x2 to 2x10x10 – 7 units of fuel plus 1 unit of payload to 199 units of fuel plus one of payload! This would be a very expensive venture that really isn’t very practical unless there is something critical about a mission that calls for a fast transit.

What if we just double delta-vee at Earth? Then our trip would be about 10% faster, since we’d start at 36 instead of 33 km/s. We’d use double the fuel and only get there 10% faster – is that any use? You might think it isn’t, but it turns out there is a possible advantage to taking a faster trip out. If we travel to Mars and back using the HTO, our spacecraft has to stay on Mars for a long time – about a year and a half –before the orbits of the two planets are lined up again for the spacecraft to return back to Earth. If we use a quicker orbit to get there, then we have an opportunity to come home after only one month, making the whole round trip about 1 year shorter.

How long do we stay on Mars?

If your purpose is to settle on Mars, then this section isn’t important. If you want to return to Earth – such as with a sample return mission or short astronaut visit – then you need to figure out how long you can stay. Just as there are launch windows for leaving Earth to get to Mars, there are launch windows for leaving Mars to get to Earth. If we choose to travel by HTO to Mars, it turns out we have to wait about 1.5 years (as measured on Earth) before our first opportunity to come home.

3) If we go to Mars on the HTO, and miss the first opportunity to come home, when is our next opportunity to come home on a HTO?

The biggest advantage of traveling to Mars by the fast trip is that we have our first opportunity to return after only about 1 month on Mars. The biggest disadvantage is that we will be moving at the wrong speed in the wrong direction when we get to Mars, so the delta-vee to get into orbit around Mars will be bigger, again requiring more fuel. We will also need a bigger delta-vee to go from HEO to HTO at Earth. The fast trip NASA is considering takes about 20% less time spent traveling out, with about 3x as much fuel needed going into the transfer orbit and more than 5 times as much getting out again (because we have to change both speed and direction). The total cost, then, is about 2x3x5 = 30 = 29 units fuel, 1 unit payload vs. 2x2x2 = 8 = 7 units fuel, 1 unit payload for the HTO.

Fast or slow return trip?

On the way out, the fast trip was made faster by traveling in an orbit that was bigger than needed – one that would take us farther if we kept going. Suppose we just change our speed at Mars to return to Earth. In that case, do we get home sooner or later if we change the speed more? Answer: We’d have to slow down more than for the HTO, so the trip would take longer. In order to make a fast trip home, we’d have to change the direction of the spaceship’s motion at Mars. Changing the direction is very expensive in terms of the amount of fuel used, and the fuel to put us into the homebound orbit is very expensive because it had to be brought from Earth. Therefore, even if we take the fast trip out, it is likely that the trip home will go by a slower orbit. In fact, NASA is considering two mission plans for a mission to return from Mars:

A. Hohmann transfer both ways plus 1.5 years on Mars = 2.8 years.
B. Fast trip out plus 1 month on Mars plus slow trip home = 1.5 years.