From: alain245@videotron.ca   
      
   On Mar/11/2021 at 15:00, Alain Fournier wrote :   
   > On Mar/11/2021 at 14:11, JF Mezei wrote :   
      
   >> But that brings up ECLSS and food. If you have to slingshot and be   
   >> autonomous for another year to get back to Earth, that means that when   
   >> you leave earth, you leave with a hell of a lot of supplies and water.   
   >> And it also means you need some pretty fancy ECLSS to reclaim as much as   
   >> you can for water and oxygen.  I have to wonder if SpaceX would take ISS   
   >> system designs or make their own from scratch.  In later case, they are   
   >> a long way from having a Mars crewed mission.   
   >   
   > A mission profile with a free return (nearly free return) trajectory   
   > follows a path similar to a mission with a short stay on Mars. It   
   > wouldn't take a year. More like 7 months. But it is more energy   
   > expensive than a mission following a Hohmann transfer trajectory.   
      
   I thought I could entertain you a little more about orbital mechanics.   
      
   First, let's look at the base case. You want to go to Mars with the   
   least energy expensive trajectory. That is a Hohmann transfer orbit.   
    From Earth, you fire your rockets to increase your speed in the   
   direction of Earth's orbit around the sun. After that burn you are in an   
   orbit with its perihelion at Earth's orbital altitude and aphelion at   
   Mars' orbit.  You just need to time your departure so you arrive at   
   aphelion when Mars happens to be at that same spot. That would be a   
   trajectory you would use for a probe, not for humans because the trip   
   from Earth to Mars takes 259 days (8.5 months). Your trip from Earth to   
   Mars is half of an elliptical orbit. When you want to go back to Earth,   
   you basically do the second half of that orbit, but once again you have   
   to time your departure so Earth is at the right spot when you arrive at   
   perihelion. So you have to stay about 9 months on Mars (or 9 months +   
   n*26 months).   
      
   If all you want to do on Mars is to plant the Canadian flag and come   
   back, the trip with the Hohmann transfer orbits takes you 26 months.   
   That is a little long for the five minutes it takes to actually plant   
   the Mexican flag. So you might want to spend a little more fuel and do   
   it faster.   
      
   To see how you could do the trip faster it helps to imagine what would   
   happen if an inhabitant of Mercury decided to go to Mars with a Hohmann   
   transfer orbit. The elliptical orbit with perihelion at Mercury's   
   altitude and aphelion at Mars' altitude has a semi-major axis of about 1   
   astronomical unit (more like 0.95, but to keep things simple let's   
   assume it is 1). That means that the orbital period of that orbit is   
   about 1 year. So if you are on such an elliptical orbit and you time it   
   so it passes near Earth, then you will pass near Earth each year. Of   
   course, if what you want to do is go from Earth to Mars, you don't need   
   to time it so it passes near Mercury, you just depart from Earth on this   
   trajectory at a time which is suitable so you arrive near Mars at   
   aphelion. Now we know that on an elliptical orbit you have higher speed   
   near periapsis and slower speed near apoapsis. So between Earth and   
   Mars, your angular rate will be less than the average rate over a   
   complete orbit of one revolution per year. So if you go to Mars and then   
   try to come immediately back with such a trajectory, when you will reach   
   Earth's orbit, Earth will be farther away and you won't like that.   
      
   What you want to do is the opposite of an Hohmann transfer orbit from   
   Mercury. Imagine this time that you leave Earth on a Hohmann transfer   
   orbit for Neptune. In the lower part of the orbit (e.g. between Earth   
   and Mars) your angular rate will be higher than that of Earth. So if you   
   go to Mars with such a trajectory, plant the Kazakh flag (for Make   
   Benefit Glorious Nation of Kazakhstan) and immediately come back to   
   Earth with a trajectory again with periapsis at Earth orbit and apoapsis   
   at Neptune orbit, then you will get to Earth orbit in front of Earth,   
   since your angular rate is higher than that of Earth. You can easily   
   solve that problem by staying on Mars for a while after having planted   
   the flag of China. Since Mars' orbital angular rate is lower than that   
   of Earth, Earth will catch up during your stay. The problem with that is   
   that you spent enough fuel to reach Neptune just to get to Mars. So if   
   you want to stay only five minutes on Mars to plant the flag of   
   Zimbabwe, you choose a transfer orbit with an apoapsis at X, where X is   
   a distance from the sun somewhere between Mars (which would return you   
   behind Earth) and Neptune (which would return you in front of Earth)   
   such that you return at Earth (X is in fact somewhere in the asteroid belt).   
      
   If you want to go even faster than that, you can choose the trajectory   
   of an orbit with periapsis lower than Earth and apoapsis higher than X   
   mentioned above. But that will cost you in delta-V.   
      
   Now, about the free return trajectory if you decide that you don't   
   really want to plant the flag of [choose your country here] on Mars.   
   When you reach Mars, you have the right speed to return to Earth, your   
   only problem is that you are not going in the right direction. But by   
   passing by Mars, gravity will change your direction, you just need to   
   properly choose at what altitude you do your Mars flyby.   
      
      
   Alain Fournier   
      
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