[Pyramid] Issues with the "Halfway to Anywhere" article in Pyramid 3/79

[Michael Thayne]

Reading the article "Halfway to Anywhere" in Pyramid #3/79, I've noticed a couple apparent mistakes and one thing that could at least be clearer. Can anyone check my reasoning? I'd also appreciate it if others have noticed other issues.

First apparent mistake: for "reaching orbit", it has you flat add the atmospheric pressure in Earth atmospheres. This would be a remarkable coincidence if it were true. This Stack Exchange thread makes me think the actual cost is negligible for game purposes.

Second apparent mistake: the section on brachistochrone transfers gives the time to make one (in days) as 58.11 * delta-V/A. However, this seems at odds with the formula in Spaceships for time required to burn a given amount of delta-V. Based on Spaceships (and generating the unit conversions from scratch myself), the formula for time in days ought to be 0.0019 * delta-V / A.

Finally, it mentions that low-thrust ships cannot use Hohmann transfers. How low-thrust is low-thrust? Is there any way to adjust for low-thrust situations?

[Anthony]

Originally Posted by Michael Thayne

Finally, it mentions that low-thrust ships cannot use Hohmann transfers. How low-thrust is low-thrust? Is there any way to adjust for low-thrust situations?

The math for hohmann transfers assumes instantaneous acceleration, so at any thrust level you're not going to quite get a hohmann. The error won't be very large unless your acceleration time is a large fraction of your orbital period, however.

[Michael Thayne]

Originally Posted by Anthony

The math for hohmann transfers assumes instantaneous acceleration, so at any thrust level you're not going to quite get a hohmann. The error won't be very large unless your acceleration time is a large fraction of your orbital period, however.

That makes sense. What's a "large fraction", though? 1%? 5%? 20%?

Also: I'm surprised that one of the constants in the brachistochrone equation is exactly 1500. Would a more accurate number (for "D") be 1482, like the equation for "Continuous Acceleration with Reaction Drives" in Spaceships?

[Humabout]

1) The flat 1 mps per atm is something of a fudge factor because atmospheric drag is anything but neglibile. Unfortunately, the amount of drag is highly dependent on many factors that are very far below GURPS resolution (e.g., surface roughness, coefficient of drag, crossectional area), and some that vary greatly as you ascend through the atmosohere, specifically air density. None of these are addressed in Space or Spaceships, and thus fell outside the scope of the article.

2) The formula for brachistocrone transfers is actually the more accurate constant acceleration approach because nothing js a straight line in space, and the formula in Spaceships assumes. Instead, a brachisto hrone transfer is a curve, because the peimary's gravity always forces a ship's trajectory into a curve.

3) All of the slower transfers (hohmann and bielliptic) assume an instant thrust (impulsove thrust). This never happens in real life, but anything that is over 1 g is definitely close enough. Anything below 0.1 g is decidedly low thrust. In between might be viable but inefficient. That calculations for how inefficient was beyond the scope of the article.

4) GURPS generally uses two significant figures. The actual formulas for use with exact SI units are included in the article, if you want additional resolution; these don't require conversion factors that may be rounded.

Hope that helps!

[Michael Thayne]

Originally Posted by Humabout

1) The flat 1 mps per atm is something of a fudge factor because atmospheric drag is anything but neglibile. Unfortunately, the amount of drag is highly dependent on many factors that are very far below GURPS resolution (e.g., surface roughness, coefficient of drag, crossectional area), and some that vary greatly as you ascend through the atmosohere, specifically air density. None of these are addressed in Space or Spaceships, and thus fell outside the scope of the article.

2) The formula for brachistocrone transfers is actually the more accurate constant acceleration approach because nothing js a straight line in space, and the formula in Spaceships assumes. Instead, a brachisto hrone transfer is a curve, because the peimary's gravity always forces a ship's trajectory into a curve.

3) All of the slower transfers (hohmann and bielliptic) assume an instant thrust (impulsove thrust). This never happens in real life, but anything that is over 1 g is definitely close enough. Anything below 0.1 g is decidedly low thrust. In between might be viable but inefficient. That calculations for how inefficient was beyond the scope of the article.

4) GURPS generally uses two significant figures. The actual formulas for use with exact SI units are included in the article, if you want additional resolution; these don't require conversion factors that may be rounded.

Hope that helps!

Thanks for this. Question about (2): I'd expect the fact that a brachistochrone trajectory is curved to affect amount of delta-V expended in the transfer. I wouldn't expect it to affect the rate at which delta-V is burned, since I thought a brachistochrone transfer was a constant-acceleration maneuver? What am I missing?

[Humabout]

Originally Posted by Michael Thayne

Thanks for this. Question about (2): I'd expect the fact that a brachistochrone trajectory is curved to affect amount of delta-V expended in the transfer. I wouldn't expect it to affect the rate at which delta-V is burned, since I thought a brachistochrone transfer was a constant-acceleration maneuver? What am I missing?

I'm not completely sure I understand what you're asking, here, so apologies if this doesn't answer your question.

Delta-V literally just means the change in velocity. A change in velocity over a period of time is defined as acceleration. For the purposes of this particular orbital transfer, acceleration is fixed, so transfer time is given by [delta-v] / [acceleration].

Remember that the arclength of any curve between two points is longer than a straight line between them. This means it's going to take more delta-v to traverse the curve. And based on the equation above, delta-v is in the numerator, so as it increases, time increases.

Does this help?

[AlexanderHowl]

Atomic Rockets has probably more than you ever want to know on space travel topics (http://www.projectrho.com/public_html/rocket/).

[Humabout]

A better site for orbital mechanics would be Rocket and Space Technology (http://www.braeunig.us/space/orbmech.htm).

If covers a much broader range of orbits with worked examples, and has information on rocket engines, as well (these I haven't really looked into, so I can't say if they're good, but his orbital stuff is).

[EDIT]
There is also a good example of how to get to Mars and other planets here (http://www.braeunig.us/space/interpl.htm)