Leaving now (December), at a constant acceration of 1g, how long to turnover for deceleration at 1 g?

Question

If a spaceship leaves Earth now, with no lunar or Venusian boost, how long would it take to reach the halfway point for turnover and the beginning of deceleration to arrive at Mars at a relative velocity (relative to Mars) of zero.

Answer 1

With a 1G acceleration you can forget about orbits and just treat the problem as one of crossing the distance between Earth and Mars. How far is Mars from the Earth right now? For this exercise lets say 100 Million Miles. Then you just need to figure out how long it would take to go 50 Million Miles with a 1G acceleration. The second 50 Million Miles will be used for deacceleration and would take the same amount of time as the original acceleration.

If I did the numbers correctly for this problem it would take about 37 hours to go 50 Million Miles and then another 37 hours to deaccelerate. So with a 1G propulsion we could get to Mars in about 3 days. Lets Go!

Answer 2

Not enough info! You need to know the distance X. This is a function of mars's position now, extrapolated to where it will be at intercept. Since we don't know the trip time yet, you have to do successive approximations knowing mars's current position relative to earth and its relative velocity in orbit. So you will iterate equations that look something like this:
do
t=sqrt(2*.5*X/g) (turnover time, derived from s=.5*a*t^2))
t2=2*t (trip time)
ThetaMars = ThetaMars(initial) + t2 * orbital angular rate
compute location XMars from ThetaMars
X = mag(vector distance from Earth(initial) to XMars)
loop until converged
EDIT: The 1st answerer makes a good point. NASA wouldn't do it that way, but a good approximation can be made when the trip time is so short that Mars barely moves from its initial position.

Answer 3

Yikes! This is NOT a simple question.

In order to answer this, you must know the distance between Earth and Mars during the voyage. Since both planets are in orbit around the sun, the distance is constantly changing. You must find the optimum velocity and direction to set the spaceship on its way.

The positions of Earth and Mars can be found using the JPL Ephemeris. See:
http://ssd.jpl.nasa.gov/horizons.cgi

Good luck! Sounds tough!