Its always Hohman transfer time somewhere…

This post is about freedom.  Freedom to travel to and from Mars any time you want not just once every 26 months when the Hohman transfer window is open.

To explain how to do this I am going to make a simplification and describe the Earth and Mars as round orbits.  This doesn’t change the result by much as you will see later.  So imagine both orbits are round and we are looking down on the Sun so that the Earth and Mars rotate clockwise.

 

hohman

 

We start out at E1 which is the Earth initial point.  This is outside the Hohman window because the correct departure point is H1 because at the end of the Hohman transit Mars will be at M1 which corresponds with leaving at H1 and the earth isn’t at H1.  But we want to go to to Mars without waiting the for Earth to line up with the H departure point.  So we chart a course towards the sun so that we can rotate faster than the Earth and reach point H2 while the Earth is still at point E2.  Point H2 is lined up with new Mars position M2 so that leaving at H2 will arrive at M2 upon completion.

This produces 3 phases of the journey.  A up to 17 km/s delta V moving towards the sun which will cause the space craft to increase in orbital velocity and increase in angular velocity in terms of rotation speed around the sun.  Then there will be a delay which depends on how far out of phase we were at the start and at the end of the delay the spacecraft will execute an a up to 17 km/s delta V move away from the sun back to the Earth orbit which will put it at point H2 which is lined up to take a Hohman transfer and arrive at M2 just as Mars is in that position.  The final delta V maneuver is to execute a simple Hohman transfer from point H2 which will be ~6100 m/s.

Because the Earth and Mars orbits are not perfectly circular the Hohman transfer from Earth to Mars will vary slightly from 6100 m/s and the two <17 km/s burns may not be perfectly equal to each other.  But at you can see below increasing the 6100 m/s doesn’t change the result much.

 

OK now you are wondering why did I design this set of maneuvers that appear to be in the wrong direction???

 

The answer is that the phase adjust was moved close to the sun so that solar sail spacecraft can maneuver faster in this region and to minimize the velocity difference relative to Earth.

Consider a one way trip to mars in 100 m/s segments.  There will be up to 401 segments each way to create 40.1 km/s.  Using 25.5 seconds Isp the mass fraction for a 100 m/s delta V is 1.5 ship dry mass.  This means that for each of the 802 segments that make up a round trip we will need 0.5 times the dry mass of the spacecraft of reaction mass provided that we can reload reaction mass that is synchronized perfectly every 100 m/s.  This is difficult to do tactically but is physically possible provided someone put all the reaction mass in exactly the correct position and velocity and we navigated the trajectory exactly as planned.  In practice additional reaction mass will be loaded at each stop to provide manuring margin for the rendezvous with the refueling points.

 

So the mass fraction is 401 times ship dry mass of fuel and 1 times ship dry mass of ship for a mass fraction of 402 to travel a distance of up to 40.1 km/s each way for a total of 80.2 km/s which would give us an Effective Isp of 1364 seconds provided someone else put all our reaction mass out for us.

Now imagine a fleet of solar sail space craft put the reaction mass exactly where it needed to be to be at the correct position and velocity when we arrived.  Again this is difficult tactically but physically possible.  Now we are ready for the final twist.  Instead of putting reaction mass in our spaceship’s path imagine the many solar sail powered spacecraft are the reaction mass.

The advantage of using the solar sail powered spacecraft is that after our transit that uses all 401 times dry mass worth of reaction mass we can just wait some period of time for the solar sail powered spacecraft to navigate and thus rearrange the reaction mass into position for the next round trip transit.  This is done by entering the Earth orbit at a single point and then returning to the future desired trajectory.  So in-between transits the solar sail spacecraft are up to -17lm/s (closer to the sun) and +6.1 km/s (closer to Mars) relative to Earth orbit.  From this position the solar sail spacecraft must travel 2 times the deltaV because it has to come to Earth and then return to its holding pattern.

So the solar sail spacecraft need to travel 34 km/s and 12.2 km/s respectively.  The solar sail powered spacecraft between Earth and Mars will travel at an effective speed of 288 m/s per month while the solar sail spacecraft closer to the sun will travel faster than 460 m/s per month.  This will give us a “reset” time in-between transits of 42.36 months (mars side) and 73.91 months (sun side).  Lets imagine for a moment that the reliability of the solar sail powered spacecraft is such that the entire fleet will last 100 years before any of the spacecraft lose functionality.  If that were the case 100*12/74=16.22 so there will be 16 round trips to Mars ignoring the Hohman transfer for a initial mass fraction of 402.

This effectively gives a total delta V of 40.1 km/s one way *2 ways *16 transits for a total of 1,283,200 m/s delta v from a reaction mass ratio of 402.  If this were performed using rocket fuel or ion propulsion that would be 21,836 seconds Isp.  Which doesn’t sound that impressive until you realize that the actual physical Isp is 25.5 so this is high acceleration capable thrust that can keep up with 453 seconds Isp rocket fuel.

To put his in perspective if you used 380 second Isp rocket fuel to perform round trips to Mars (12200 m/s each roundtrip) and only refueled at Earth you would require more than 401 times ship dry mass of rocket fuel!  This non hohman transfer is able to outperform a 380 second hohman transfer provided its a round trip.

And the final kicker is provided the entire fleet of solar sail spacecraft and the carrier ship all last 100 years they all end up in LEO at the end of the 100 years and could be melted down and reformed to start the process again with an effective new mass sourced from inside Earth’s gravity well far lower than 401 times the dry mass of the carrier ship!  In addition, no mass has been vented into space and permanently lost to mankind except the photons from the sun which would have bene lost anyway.

Its possible to imagine that one day the derivative mass fraction will be effectively unity with no new mass from Earth being needed to fuel the next 16 journeys.

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