What is the most efficient flight path into orbit?

[Wicked Pilot]

Get out your calculus book out.

[Darth Wong]

Get out your calculus book out.

The integration is easy. Force F=-Gm1m2/r^2 (if we treat "upwards" as positive in this case), so if you integrate force over distance r, you get energy Fr=U=-Gm1m2/r. No need to crack open a textbook.

[[BL]Phalanx]

In order to figure out minimum acceleration to achieve an orbit of given altitude in a given timeframe, you must figure out the orbital velocity (that's the velocity you need for centripetal acceleration to match gravitational acceleration). Acceleration is final velocity divided by time. That's the angular component of acceleration.

Okay, I think I did figure that out somewhere already. At an altitude of 1921 kilometers, you'd be 5,318,000 meters from the center of Mars.

Plugging that into:

g = (m * G) / r^2
m = 6.4219E23 kg
G = 6.67E-11

I get 1.514 meters/sec^2

This should be equal to:

a = v^2 / r

So

v = (a * r)^(1/2)

v = 2838 meters/sec

The initial velocity is what the craft starts with by launching eastward from the equator, which is 247 meters/sec, so the change in velocity is:

delta_v = 2838 - 247 = 2591 meters/sec

Divide this by 18 seconds and get an acceleration of:

143 meters/sec/sec

Look okay?

Then, use the x=0.5at^2 formula to determine minimum radial acceleration (upwards, to gain altitude). Put them together, and you've got minimum acceleration.

Do I also have to account for the ship slowing down to reach that altitude? Because:

a = (2 * x) / t^2
x = 1921000 meters
t = 18 seconds

a = 11858 meters/sec/sec

But at that acceleration and time, the final speed upwards, gaining altitude, is gonna be 213 km/sec. It's gonna shoot past that altitude.

The use of gravitational PE affects the energy requirement (energy required to reach orbit is GPE+KE), but not the acceleration requirement, which is a simple matter of velocity and time.

Oh, okay. Thanks.

[Zoink]

I gave this some thought,

Perhaps you should break the problem into segments.

So for your total time period t = t1+t2+t3

First time segment (t1), vertical acceleration. Accelerate the rocket vertically to a fixed speed.

Second time segment (t2), rotate the rocket to maintain constant vertical speed, and begin accelerating in the "orbital" direction, up to your orbital speed.

Third time segment (t3), vertical deceleration into orbital path. Use gravity to decelerate the rocket vertically, while maintaining orbital acceleration.

For a total vertical travel,
(1) h = 1/2*a*t1^2 + (a*t1)*t2 + (a*t1) - 1/2*g*t3^2

(2) t=t1+t2+t3

orbital speed
(3) v = (t2+t3)*(a)

vertical velocity = 0 at time t,
(4) a*t1 = g*t3

4 equations, 4 unkowns. Those are some basic equations, to make it more complicated (and accurate), g isn't constant, mass of the rocket isn't constants, but then you're into differential equations (mx'') to which you could add a drag term (which isn't constant). So it depends on how accurate you want the number.

[victorhadin]

I've got plenty of data for air density to 100km (above that assume zero). I could always write out an Excel program to simulate it at varying angles of ascent if I had time, and note varying drag coefficients and suchlike.

I dunno what this is, mind. Presumably the spacecraft will vary thrust and angle of ascent constantly. I dunno if my mind can handle that on a Friday evening.

[[BL]Phalanx]

vict0r! I didn't know you posted here as well....

[victorhadin]

*Takes a bow.*

Sporadically. I drift in and out.