Rocket in nonuniform gravitational field

[mistergrinch]

Hello, I'm trying to solve the rocket equation in a nonuniform gravitational field. The standard rocket equation at a distance x from the center of a planet of mass M gives:

m*d^2x/dt^2 + v_e*dm/dt = -GmM/x^2

where v_e is the relative exhaust velocity.

Multiplying by dx/dt and assuming dm/dt=a (const.), this can be written as:

d/dt( 1/2m(dx/dt)^2 - GMm/x + v_e*a*x ) = 0 =>

1/2m(dx/dt)^2 - GMm/x + v_e*a*x = K (const.)

I can then separate variables and integrate, which gives me a very ugly expression for t in terms of x.

Another approach is to write the original equation as:

dv/dt + 1/m*dm/t*v_e = -GM/x^2 which can be integrated to give
v = v_e * ln(m0/m) - GM * integral_0_t{1/x^2 * dt}

I.e. the rocket eqn. modified by an integral of the gravitational force. But I have no idea what to do with this integral so I don't see how this helps me!

I was wondering if anyone has a better idea how to solve this problem. How do they solve problems like this at NASA, or are the rocket burns short enough that they just assume F_gravity is constant? This seems like a useful problem to be able to solve -- I saw a reference at google saying Tsiolkovsky was the first to solve it, but I can't find any references to any solutions. Does anyone have any ideas?

 

[rcgldr]

NASA uses numerical integration to solve the issue of launching through atmoshpere and diminishing strength of gravity with distance from the earth, moon, planet, sun, ... . I'm not sure if the math is directly solvable once beyond the effects of the atmosphere.

The math for the time it takes for two objects released at some distance apart to collide due to gravity can be solved directly. Credit goes to Arildno for figuring how to solve one of the integrals:

point masses (post #8 in this thread):
https://www.physicsforums.com/showthread.php?t=360987

masses of fixed size:
https://www.physicsforums.com/showthread.php?t=306442

Once in space, navigation can done relative to distant stars to use as a frame of reference to determine a position and direction in space. Tracking objects from the Earth as the Earth rotates and orbits is also done, to first locate the Earth relative to the stars, then locate the object relative to the earth. It's not perfect, so there are mid-course corrections made using on board rocket engines.

 

[D H]

NASA uses numerical integration to solve the issue of launching through atmoshpere and diminishing strength of gravity with distance from the earth, moon, planet, sun, ... . I'm not sure if the math is directly solvable once beyond the effects of the atmosphere.

Factor in a rocket following a curved trajectory through a complex atmosphere with a variable thrust and non-constant ballistic coefficient around an object with a non-spherical mass distribution, and no, there is no closed form solution.

Once in space, navigation can done relative to distant stars to use as a frame of reference to determine a position and direction in space.

You can get attitude from the distant stars, but not position.

 

[rcgldr]

You can get attitude from the distant stars, but not position.

I'm not sure about Earth tracking external objects, but in the Apollo missions, one of the means of navigation was to use star charts, a scanning telescope, and the equivalent of a sextant. Only 2 stars or 1 star and the Earth's horizon (moon's horizon could have also been used) was needed according to the articles below. I assume that multiple readings taken over time are needed to get precise position and velocity, just like naval navigation.

http://www.ion.org/museum/item_view.cfm?cid=4&scid=19&iid=293

http://en.wikipedia.org/wiki/Apollo_PGNCS#Optical_unit

For Apollo 13, they used the terminator line dividing day and night on the Earth to help navigate the spacecraft back to the proper course for re-entry into earth. I think they could also measure the subtended angle of the diameters of the Earth and moon to help locate relative distance from Earth and moon.

http://www.universetoday.com/62763/...lo-13-part-6-navigating-by-Earth's-terminator

 

[D H]

I'm not sure about Earth tracking external objects, but in the Apollo missions, one of the means of navigation was to use star charts, a scanning telescope, and the equivalent of a sextant. Only 2 stars or 1 star and the Earth's horizon was needed according to these articles. I assume that multiple readings taken over time are needed to get precise position and velocity, just like naval navigation.

Suppose you know absolutely nothing about your state (position, velocity, attitude, attitude rate) and you take a set of star sightings accurate to within one arc minute (which is rather precise for star trackers). This most certainly will tell you your attitude. If your star sightings include a nearby star, say one that is 4.35 light years away, it will tell you something about your position, but only to within 80 AU or so. Moreover, star trackers typically exclude nearby stars such as Alpha Centauri precisely because of its ~750 mas annual parallax. So not much help there if you are worried about your position along a trajectory between the Earth and the Moon.

If you add in a sighting to one much closer object such as the Moon (or the Earth), you now do know something about your position. In particular, you know you are somewhere along a line segment (fuzzy cone due to the inexactness of the measurements) extending from the Moon (or the Earth) in a certain direction. Where along that fuzzy cone? You don't know.

Suppose instead that you are not lost in space, that you have an inertial navigation system which by means of dead reckoning propagates your state over time. Due to orbital mechanics, the errors in the dead-reckoned state are much worse along track than across track. The covariance matrix is ellipsoidal, with the long axis of the ellipsoid being aligned more or less with your velocity vector. Now the sighting of the Moon (or Earth) plus a star tells a lot about where you are. It gives the navigation system information needed to update the estimated state.

 

[sophiecentaur]

And just think how smart the old astronomers must have been to work out all those orbits and make such accurate predictions. No computers. Hats off to them, I say.

 

[D H]

Those old astronomers had lots of computers. Here's a photo of some of those computers at work in 1949:

 

[mistergrinch]

^^ Haha I have to admit I find the old computers a lot more appealing than the new ones!

My calculation would be useful for launches from the Moon or any other large body with no atmosphere. I was hoping to find out what Tsiolkovsky's solution was to see if he had a simpler form for the equation. I find it strange that a google search brings up absolutely nothing on this topic.

rcgldr thanks for those links. The 2 mass problems were fun -- I couldn't solve the integral even numerically without that substitution. Nice trick!

 

[rcgldr]

point masses (post #8 in this thread):
https://www.physicsforums.com/showthread.php?t=360987

Some terms were left out in 3 of the intermediate steps in post #8, the corrected steps are shown in post #15. The initial steps and the integral were OK.

 

[D H]

My calculation would be useful for launches from the Moon or any other large body with no atmosphere.

The ideal rocket equation, even if it is modified to account for a uniform gravity field, is useless as is. Rockets don't go straight up.