# Theoretical Calculation

I'm totally lost as to where to start with this..

A spacecraft is launched from Earth. The spacecraft goes in an elliptical orbit, with minimum distance to the Sun of 1AU. At the maximum distance from the Sun along its orbit, the spacecraft arrives at Mars, at a distance of 1.5AU from the Sun. Assuming that Mars & Earth both have circular orbits, also ignore gravitational effects of the two planets on the spacecraft.

QESTION. The crew stays at Mars after the encounter. They relaunch into the same orbit, at such a time to be back at Earth at the earliest possibility. How long do they stay at Mars & how long is their total journey?

I've already worked out that the time to travel to Mars is approx 8.4months, however now i'm at a total loss as to where to go.

Any ideas?

## Answers and Replies

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marcus
Gold Member
Dearly Missed
for confirmation the CRC handbook gives around 243 days for the Hohmann ellipse trip to mars

it actually gives a range from 2.07E7 seconds to 2.40E7 seconds
I worked out 2.1E7 as a sample and it came to 243 days
it varies between 2.07 and 2.40 because the earth and mars orbits are not perfectly circular

that is 8.1 months or so, close enough to your 8.4
instead of 243 I will use 255 days, be closer to your number.

a handy idea is the "mean daily motion"
each day the earth pulls ahead of mars by 0.46 degree
the mdm of earth is around 0.99 degree per day
the mdm of mars is around 0.53 degree per day

you have reinvented the Hohmann transfer ellipse
a half-ellipse path for getting from one planet to another
and the CRC handbook of Chemistry and Physics has a table of all
the half-ellipse trips between all the planets, how long they take and what kind of rocket boost they need----what "delta vee".

but let us assume they are circular and make a rough calculation of how long the Hohmann voyagers would have to wait on mars before a good opportunity to return by the same half-ellipse path.

lets measure angles counterclockwise from 12 oclock
say earth starts out at 6 oclock position (180 degrees) and rocket arrives when mars is at 12 oclock position (zero degrees)

when they arrive mars is at zero degrees and earth is at 72 degrees

(it was at 180 and 255 days later it is at 180+252 = 432 same as 72)

now we have to wait until earth is catching up to mars and is still
252 - 180 = 72 degrees behind. then we launch from mars and during trip the earth will go 252 degrees while the rocket goes 180
and they meet

but 72 degrees behind is 288 degrees ahead and when they
land on mars the earth is already 72 degrees ahead so they have to wait while earth gains an additional 216 degrees on mars

now earth gains 0.46 degrees a day!

So they have to sit on mars 470 days!

that is dividing 216 degrees by 0.46 degrees per day.

mission planners often do not stick rigidly to Hohmann
ellipses which may economize on fuel but introduce time constraints.
what if 470 days seems like too long a time to sit?
better burn some extra fuel and try a different trajectory

but the Hohmann half-ellipse is still a very interesting benchmark.

Thanks very much, I just could not see where I was going with it, guess i'd spent too much time thinking hehe!

Thanks again.

marcus
Gold Member
Dearly Missed
Originally posted by MathematicalPhysics
Thanks very much, I just could not see where I was going with it, guess i'd spent too much time thinking hehe!

Thanks again.
my pleasure, really
this PF board can sometime work to get a fresh person
to think thru something, and it may actually help thinking
just by mixing minds like that
also someone else could see how to make my solution more elegant
in like, one line of algebra with fewer words, and post it
or you could improve on it yourself

are you trying the exercise of planning a mars mission
this is a very good exercise because one can motivate
learning all sorts of things about mars (as well as orbits
and deltavees for changing orbit and fuel/payload ratio and stuff)

if you are thinking thru a mars mission you could post your
ideas and it might interest someone here---never know

It actually origionated a couple of weeks ago with my Astrophysics lecturer (I'm in university) posing a question about a mission to Mars. Also after seeing quite alot of publicity about the recent missions to Mars, I was interested in developing my own knowledge.

All I have done so far is sketched the orbit of the spacecraft, shown the time to travel to Mars is approx 8.4 months and the of course with your help estimated the time the crew must stay on Mars, hence a total journey time.

I hope to deepen this into a more complex model and sometime hopefully weigh up the optimum time to stay on Mars, because as you say it may be better to use that little bit more fuel to get back sooner and choose a different return path.

For the moment though my next project takes over, which is based on trans-Neptunian objects.

I really like this course and in almost all my spare time im furthering my understanding.

When I get time to further this I will post my outcomes!