Fun Times with Torchships

[Archgeek]

So, for reasons, I'm trying to get a probe to Eve as quickly as I reasonably can, for certain definitions of "reasonable" -- and decided the way to figure this out was to start with the basic kinematics equation of .5at²+vt+d and do a lot of algebra starting with the conceit that we want the acceleration portion to equal d.  This resulted in just deriving the brachistochrone equation like a doof: (total trip t = 2*sqrt(d/a), total dv = 2*sqrt(da))

Throw in the min distance from Kerbin to Eve being 3,668,900km, pick a comfortable acceleration, and there's the dv and burn time.  However, that's assuming crazy things like instant re-orientation, and for a = 1m/s, leads to a 22 tonne ion craft I already have around but also a pair of 16.8hr burns.  I could just make everything bigger to burn harder for less time, but that could make the lower stages get a bit out of hand.

So, we come to realm of having a coasting phase, which highlights the fact that the brachistochrone equation is a special case of a more general equation, and we've just set coast time to zero.

As such, instead of d being just a*t²/2, which was convenient, now it's a*t_a²/2 + v_maxt_coast where t_total = t_a+t_coast and v_max= roughly(a*t_a).  I think I can maybe constrain it with a chosen t_a -- say, 10hrs (I can handle a pair of 5 hour burns over a weekend), and a vaguely acceptable acceleration rate, say, 1m/s again, for ease.

So to solve for the times, we substitute a*t_a for v_max and t_total-t_a for t_coast

a*t_a²/2 + a*t_a * (t_total-t_a) = d => (d-a*t_a²/2)/a*t_a = t_total-t_a => d/a*t_a-t_a/2 = t_total - t_a => t_total =  d/a*t_a + t_a/2

This gives us 1m/s²*(36ks)²/2 + 36km/s*(t_total-36ks) = 648Mm + 36km/s*(t_total-36ks) = 3668.9Mm => 3020.9Mm/36km/s = t_total-36ks = 83913.89ks => t_total = not much more...that kinda gets overwhelmed by coast time...of over 23 thousand hours.  I might not've done that right.  I blame it being after 2:30.

Any thoughts/corrections?

[Freshmeat]

I am afraid you went off the rails completely. You are neglecting the gravity of Kerbol, as well as assuming that Eve does not move during the transfer.- Both are very much incorrect. If you want to crunch numbers, there really is no alternative but doing orbital math, OhioBob has an excellent website on the subject.

Otherwise, you can check the porkchop plot in AlexMoons Launch Window Planner and see what the dV cost is for a given travel time.

However, simplified physics with vague assumptions really get you nowhere in space.

[Chabadarl]

First, I am really not sure to understand you ... 

Second, as Freshmeat was faster than me, I'll just paraphrase, you're planning a transatlantic trip trying to use a Shanghai subway map. 

Third, you do the same thing twice hoping for a different result (did I tell you the definition of insanity?)
As the result was not good for you the first time (2*16h burn time is  too long) you have decided to set this result to the one of your choosing (2*5h burn time) 
but without other modification (same acceleration), you get what you ask for : ~9 Kerbal years of travel at low speed (but I will not bet on the result to be right) ...

And last, their are some errors in your calculation but you just get away with it because their is only one significant terme (d ~ v_maxt_coast~  a*t_a*t_coast)

[KerikBalm]

Freshmeat, a transfer window planner is absolutely not what he wants here. He's talking about sci-fi torchships with insane dV budgets. With such ships, you basically point at your target and thrust continuously until you get halfway, then you flip around 180 and start braking.

Sci fi torchships typically reach significant fractions of the speed of light, and thus the relative motions of the planets aren't so relevant... you only need to "lead" the target by a bit.

I don't have time at the moment to check the math though.

I also don't think any variation of this would work in stock KSP. There is no engine with high enough thrust and high enough Isp to make this sort of thing viable, you'd have to mod yourself an engine that gets you like 300,000 Isp (corresponding to a 30 km/sec exhaust velocity, which you could get with fusion) and gives your ship a 0.5:1 TWR

Edited October 11, 2016 by KerikBalm

[Chabadarl]

After @KerikBalm enlightment :

5 (hours of acceleration) *3600 (seconds per hour) * 1 (m/s² acceleration) =18km/s.

Not enough for just pointing at the target.

As a matter of fact, I have no idea how you get the 16.8hr burn ?

Edited October 11, 2016 by Chabadarl

[Chabadarl]

Ok. I get it now, 2 mistakes : you miss the distance during deceleration so (with td=ta) d = 2*(a*t_a²/2) + a*t_a * (t_total-t_a) 

but the biggest mistake :

t_total =2d/(a*t_a) + t_a/2 not  t_total =  d/a*t_a + t_a/2

so t_total = 416655 s or 116h

[Freshmeat]

@KerikBalm Actually, the launch window planner gives transfers down to a few days with corresponding dV budgets. And the OP is intending to coast some of the way, with an acceleration of 1 m/s^2 initially. However, given that c=3*10^8 m/s, no practical acceleration would reach some fraction of it. in your example:1:2 TWR ~ 5 m/s^2, still requiring 6*10^6 s for 0.1 c, or more than 1600 hours of continuous acceleration. Going the other way around, reaching 0.1 c in 10 hours (the original success parameter), would require acceleration of roughly 800g.

We could set some other constraints. Assuming we could go with point and shoot, the planets should not move a lot.The orbital velocity of Eve is roughly 11*10^3 m/s, give or take a bit, and the SOI is 85*10^6 m, which ought to represent the target we must hit That means we must reach Eve in 7700 s, or about 2 hours. If we are talking a torchship, then we need to work a bit on the equation of motion, although it is easy to see why we can ignore the influence of Kerbol. The easy way of doing the math is doing the halfway trip twice, each time starting at v=0 and moving with constant acceleration, so d/2=1/2*a*(1/2*t)^2 <=>d=a*(1/2*t)^2 <=> a=d/(1/2*t)^2. Given our numbers, a = 22g. Actually not that bad, but still far away from the original post. And having the itsy bitsy dV requirement of 1900000 m/s

Edited October 11, 2016 by Freshmeat
forgot dV

[KerikBalm]

yea, but the transfer window calculator assumes instantaneous dV, whereas a brachistichrone trajectory/a torchship assumes you spend a lot of time accelerating.

Its true that he did mention coasting. Also most SF torchships involve interstellar travel, where one considers the relative motions of stars in the galaxy, not planets. Obviously that also means much longer acceleration times.

At the very least it seems like something you'd reserve for journeys to the outer planets with a mod like OPM.

I suppose in this case, simply picking a fast transfer window does conform to what the OP was talking about, despite his use of the words "brachistochrone" and "torchship"

[Archgeek]

Ooof, it seems I made a mess of this one, starting with my stretching the definition of "torchship" a bit with my plan of a nuke burn and ion brake, as well as by omission of some math I'd done earlier still:Eve's got an orbital period of 261 d 5 h 39 m 21s, or 5,657,961s. This period can be expressed in degrees, minutes, and seconds of arc.  This breaks down to 15,717s per degree, 262s per minute, and 4.366s per second of arc.

...and looking at these numbers again, with Eve pulling a degree for every 4hrs 22min, I can see that my initial tack of a 1m/s average acceleration is a bit low for a trip time of 33.65 hours. Eve would scuttle off by nearly 7 degrees, which is a bit much. Relatedly, I just found I'd dropped a '6' from the 3,668,900km distance figure, messing that up by an order of magnitude, hence my oddly low trip time earlier.  Still, a few degrees is hard even aim an ejection, so 3 to 4 degrees wouldn't be too hard to correct for as the burn progressed... just aim a little ahead of where Eve is, preferably so that you're intercepting it at its closest approach to Kerbin, and even a low-ish TWR craft can get the job done.  With 5m/s, the trip's barely over 15 hours, but the dv gets right out of hand.  If I just double it, it's only reduced to 23.8hrs, but 5.45 degrees isn't two hard to roughly lead the target by, and the dv comes to 171.3km/s.  I can easily get 85.6km/s out of an ion probe, but it won't be near as light as the 60.7km/s version.

---

Still, long story short, I was off by an order of magnitude (which is why I considered coasting at all, I thought sacrificing some portion of the max possible speed could save me some real-life time), my biggest constraint is Eve's orbital movement, and our good friends the brachistochrone equations still rule the day: 

total trip t = 2*sqrt(d/a), total dv = 2*sqrt(da)

Though, I get the sense that I may have to correct for Kerbin's orbital movement, as that'll induce some drift that could well reduce the synodic motion from the probe's perspective.  Should I go with Eve's synodic period of 14,686,947s (11.33 hours per degree) instead?

[Freshmeat]

Your assumptions are off. Calculation of dv would be a half way acceleration and a half way deceleration. The derivation would end up v_max=sqrt(2*d*a), but we need twice that, over half the distance. Plugging in 5 m/s and (3.67*10^6 km)/2, we get v_max = 135 km/s, thus dv = 271km/s. And that means a vessel with a fuel mass of 700 times the dry mass, using ions. This again means that the acceleration of the ship will change _significantly_, so we cannot use the equations of motion for constant acceleration. I think this could be solved by a bit of numerical integration, but frankly, I am to lazy to write it up unless I get really bored.

[Archgeek]

6 hours ago, Freshmeat said:

Your assumptions are off. Calculation of dv would be a half way acceleration and a half way deceleration. The derivation would end up v_max=sqrt(2*d*a), but we need twice that, over half the distance. Plugging in 5 m/s and (3.67*10^6 km)/2, we get v_max = 135 km/s, thus dv = 271km/s. And that means a vessel with a fuel mass of 700 times the dry mass, using ions. This again means that the acceleration of the ship will change _significantly_, so we cannot use the equations of motion for constant acceleration. I think this could be solved by a bit of numerical integration, but frankly, I am to lazy to write it up unless I get really bored.

Yeah, if you read the post above, you'll note I'm abandoning the coasting idea, as the omission of a '6' had me off by an order of magnitude.  5m/s allows one to squeeze in too much delta-v for a stock craft, but 2 should be manageable (note that only the braking phase, post turn-around will be ions -- the accelerating phase is planed to be a nuke monstrosity, so if I need 120km/s, that's just a 23 tonne 60km/s ion ship, and a nuclear monstrosity to propel it 60km/s), and there may be further shenanigans involving Kerbin's orbital motion reducing the amount of apparent motion Eve can pull off while underway.  Ship accelleration will be kept in check by shedding engines and droptanks as needed, aiming for a target average.

Also of note, it seems my intial question was already answered elsewhere: http://www.projectrho.com/public_html/rocket/torchships.php#brachistochrone (scroll down to "The Equations")

"Timothy Charters worked out the following equation. It is the above transit time equation for weaker spacecraft that have to coast during the midpoint

T = ((D - (A * t^2)) / (A * t)) + (2*t)"