How would constant acceleration affect a Hohmann Transfer Orbit?

[Sojourner01]

OK, I'm doing an assignment on spaceflight and I'm looking for a more complicated answer to a Hohmann Transfer. I know the conventional one, but that assumes the spacecraft has a constant velocity through the majority of the orbit.

What if it doesn't? How would one calculate the time needed to reach aphelion if the craft has a small constant acceleration due to a thruster burn (say, an ion engine)?

 

[enigma]

I think you've confused yourself a little bit.

Hohmann Transfers do not have constant velocity during the transfer. As the object goes outward from the gravitating body, it loses velocity, just the same as you lose upwards velocity when you jump.

What remains constant is the total Energy. The spacecraft loses velocity (which can also be counted as Kinetic Energy), but it gains Potential Gravitational Energy. The two different types of energy flip back and forth as the satellite proceeds on its orbit, but the sum of the two remains constant.

Doing a thruster burn changes the energy of the system, because it converts chemical or electrical potential energy into Kinetic. Unless the orbit is circular, there is no time that the velocity needs to stay constant.

For ion thrusters and other similar types of propulsion, the equations become much more complicated. To give you an idea, in my space propulsions class which is a 4th year class in an aerospace engineering major they only said _how_ it's done... we didn't have to actually do it.

Basically, you know the starting position and velocity of the spacecraft , and you consider the thrust, mass, direction of burn, etc. If you take a small enough time step, you then can know where the spacecraft is at the initital time + the time step.

The basic procedure is nothing more than Euler's Method for numerical solutions.

When you couple this with a multivariable controller, you can get a grip on the maximum error you can expect to see over long durations. If that error is less than the mission envelope says it can be (for example you want the spacecraft to arrive at a planet at a certain time + or - 2 minutes) the spacecraft can get where it's going.

There are several software packages which are used to do calculations like this. One of them is http://www.stk.com/ or Satellite Toolkit.

Hope that helps. Welcome to PF!

 

[Sojourner01]

Heh, thanks.

I knew that about ordinary Hohmann orbits, I just got the wording wrong. I was thinking constant linear acceleration i.e. zero.

The thing about ion engines is that they can make a continuous burn all the way from here to the Kuiper belt (where this is going). Obviously this will be complicated; not in terms of physical principles but definitely difficult to solve. I may either abandon the 'boosted Hohmann' idea entirely, or take your advice and use said software to calculate a possible value.