Gravity-compensating Martian brachistochrone

[sevenperforce]

The thought occurred to me that if you had an engine with sufficiently high energy to pull a brachistochrone (thrust prograde halfway to your destination, then retrograde until you arrive) to Mars, the ideal plan for a manned Martian mission would be to start from LEO at 1 gee, then gradually taper off thrust through the full transfer to 0.3 gees to Martian orbit. That way, your crew would be smoothly acclimatized to Martian gravity and have no adjustment period. The same could be done in reverse, starting at 0.3 gees and thrusting harder and harder (no innuendo intended) until you reached Earth at exactly 1 gee retrograde. 

Unfortunately I have absolutely no idea how much dV would be required for such a maneuver, nor how long the transfer would take. It would require like four nested integrals, and trying to set it up for iterative solution in Excel would be a nightmare. I don't even know if outgoing dV would equal incoming dV, due to the influence of old Oberth. 

Any ideas on how to calculate that?

Notably, such a thrust profile would be a prime candidate for beamed power... 

[fredinno]

8 minutes ago, sevenperforce said:

The thought occurred to me that if you had an engine with sufficiently high energy to pull a brachistochrone (thrust prograde halfway to your destination, then retrograde until you arrive) to Mars, the ideal plan for a manned Martian mission would be to start from LEO at 1 gee, then gradually taper off thrust through the full transfer to 0.3 gees to Martian orbit. That way, your crew would be smoothly acclimatized to Martian gravity and have no adjustment period. The same could be done in reverse, starting at 0.3 gees and thrusting harder and harder (no innuendo intended) until you reached Earth at exactly 1 gee retrograde. 

Unfortunately I have absolutely no idea how much dV would be required for such a maneuver, nor how long the transfer would take. It would require like four nested integrals, and trying to set it up for iterative solution in Excel would be a nightmare. I don't even know if outgoing dV would equal incoming dV, due to the influence of old Oberth. 

Any ideas on how to calculate that?

Notably, such a thrust profile would be a prime candidate for beamed power... 

Here: http://www.alternatewars.com/BBOW/Space/Direct_Traj_Calcs.htm

[lobe]

Would this be even possible? I mean, I think at even 0.3 g acceleration continuous the craft might accelerate beyond the ideal velocities along the route.

[sevenperforce]

4 minutes ago, lobe said:

Would this be even possible? I mean, I think at even 0.3 g acceleration continuous the craft might accelerate beyond the ideal velocities along the route.

The ideal velocity is what gets you there fastest. And you don't thrust prograde the entire way; you thrust prograde to the halfway point and then thrust retrograde to brake the rest of the way. At least in a constant-thrust brachistochrone. This is different; hence the mathematical conundrum. 

[lobe]

So it turns out that 0.3 g isn't that much, I thought the spacecraft might start reaching high relativistic velocity well before it reached Mars.

[mikegarrison]

Beamed power sounds great until you remember that the received power falls off with 1/r^2 due to spreading losses.

[Kerbart]

6 minutes ago, mikegarrison said:

Beamed power sounds great until you remember that the received power falls off with 1/r^2 due to spreading losses.

Not if the beam is narrow enough and, at a distance, you can still capture all of the beam. It's not like the beam loses energy, the 1/r² loss is in intensity per square unit. As long as you recover the entire area of the beam, you're compensating by capturing r² more area which will offset your reduced intensity.

[sevenperforce]

4 minutes ago, Kerbart said:

Not if the beam is narrow enough and, at a distance, you can still capture all of the beam. It's not like the beam loses energy, the 1/r² loss is in intensity per square unit. As long as you recover the entire area of the beam, you're compensating by capturing r² more area which will offset your reduced intensity.

Right. Beamed power would assumptively be coherent/in-phase (think laser). So power drops off due only to scattering, not as any consequence of geometry. That is why I was thinking that dropping power by 1/3 over the course of a Martian transfer would possibly correlate to the gradual attenuation of a beamed power arrangement.

[sevenperforce]

1 hour ago, fredinno said:

Here: http://www.alternatewars.com/BBOW/Space/Direct_Traj_Calcs.htm

The difficulty is that this is not a constant-acceleration trajectory.

[Bill Phil]

I don't think a few weeks of 0.3 (although it's closer to .38) would help much.

Maybe constantly accelerate at a low rate but have a centrifuge spin up to give .38 the whole time.

4 hours ago, lobe said:

So it turns out that 0.3 g isn't that much, I thought the spacecraft might start reaching high relativistic velocity well before it reached Mars.

It would take many months at 1g to get to relativistic velocities.

[sevenperforce]

18 hours ago, Bill Phil said:

I don't think a few weeks of 0.3 (although it's closer to .38) would help much.

Maybe constantly accelerate at a low rate but have a centrifuge spin up to give .38 the whole time.

A gradual decrease in gravity, even over the course of only a few weeks, should at least give the astronauts a better shot at getting used to moving around. Anyway this would be in the future, when we have high energy brachistochrone-capable engines. 

[Bill Phil]

Just now, sevenperforce said:

A gradual decrease in gravity, even over the course of only a few weeks, should at least give the astronauts a better shot at getting used to moving around. Anyway this would be in the future, when we have high energy brachistochrone-capable engines. 

But that won't help much. And you can use a centrifuge at Mars to acclimate, as well as one on a low acceleration torchship. 

Plus there's probably years of training in neutral buoyancy tanks that they have to go through.

[tater]

At anything like 0.3 to 1g acceleration, the trip times would be so short, there would be no reason to bother with acclimatizing people to variable g loads. You will always be fine going to a lower value, anyway.

[Bill Phil]

41 minutes ago, tater said:

At anything like 0.3 to 1g acceleration, the trip times would be so short, there would be no reason to bother with acclimatizing people to variable g loads. You will always be fine going to a lower value, anyway.

Exactly. That's why you use a smaller acceleration or just don't do fully brachisticrone, and use a centrifuge to vary the gravity.

[Bill Phil]

Let's use the average distance between the Earth and Mars ( best I can tell its 225 million km) and a constant .3g acceleration (lower end) along with some .5at^2 magic to find the time to Mars. 

About a week to Mars. Not enough time to acclimate at all. And that's only the lower end, you're starting at 1g and going down. Might take a few days max.

It's about 3.5 days at 1g constant.