4 ms·

How precisely do you have to get it right up front? Maybe it's OK to get your initial trajectory off by a 1/1000 part, and make an adjustment when you're 3/4 of

How precisely do you have to get it right up front? Maybe it's OK to get your initial trajectory off by a 1/1000 part, and make an adjustment when you're 3/4 of the way to Mars and your relative error is now more like 1/250.

CydeWeys 6y ago
There's no world in which it makes sense to save a few digits on pi and then risk running out of fuel while making your mid-course correction because your initial burn was too far off. The mid-course correction is because you can only be so precise when you're firing rocket engines, not because you're foolishly using imprecise constants for no good reason at all. So I'll reiterate, I want to see a source that NASA uses only 3.14 as the value for pi in making their trajectory calculations. And I want to point out the absurdity of even so much as having this debate in the very comment thread for an article on nasa.gov where NASA itself is saying that they use 3.141592653589793 for pi. We already have as good as a source as we're going to find, and there's a lot more sig figs in it than 3!
- dmurray 6y ago
  Sure. But the question asked was not about what value NASA use, but what precision you need for space travel. The rule of thumb I use when looking at noisy estimates is that you should shoot for twice the precision (one extra bit of information) that you have in your noisiest other value (or of your total noise - they're usually about the same thing). Any less precision than that and your estimate gets worse, but more precision doesn't materially improve your estimate. I don't know a rigorous proof for this but you can hand wave it from the Nyquist-Shannon sampling theorem [0]. So how much uncertainty comes from other sources when deciding on an initial burn for your trip to Mars? Contributing factors could be your fine control over the thrust from the rocket engine, but also solar radiation, gravity or - I suspect this is biggest - attitude control. I'd guess NASA can achieve a precision of better than 1/500 but probably not 1/50,000, so they'd need about 5 or 6 digits of pi. But I'm interested in hearing a more educated guess! [0] https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_samp...
  - CydeWeys 6y ago
    And so we're back to still needing a source to do better than guessing :/
kens 6y ago
Some factors are way more important than others for rocket navigation. For instance, the azimuth, the angle you're headed is very important, since a small error in angle gets you way off target. The Saturn V theodolite, used to calibrate this angle before launch had an accuracy of better than +/- 2 seconds of arc, i.e. 1 part in 648000. That suggests you'd need to use pi to at least 6 decimal places (probably more) for surveying and azimuth calibration. The book "Inventing Accuracy" is about missile guidance (which has mostly the same issues) and discusses the various sources of error in detail and the various contributions to the "error budget".

CydeWeys 6y ago

There's no world in which it makes sense to save a few digits on pi and then risk running out of fuel while making your mid-course correction because your initial burn was too far off. The mid-course correction is because you can only be so precise when you're firing rocket engines, not because you're foolishly using imprecise constants for no good reason at all. So I'll reiterate, I want to see a source that NASA uses only 3.14 as the value for pi in making their trajectory calculations. And I want to point out the absurdity of even so much as having this debate in the very comment thread for an article on nasa.gov where NASA itself is saying that they use 3.141592653589793 for pi. We already have as good as a source as we're going to find, and there's a lot more sig figs in it than 3!

dmurray 6y ago
Sure. But the question asked was not about what value NASA use, but what precision you need for space travel. The rule of thumb I use when looking at noisy estimates is that you should shoot for twice the precision (one extra bit of information) that you have in your noisiest other value (or of your total noise - they're usually about the same thing). Any less precision than that and your estimate gets worse, but more precision doesn't materially improve your estimate. I don't know a rigorous proof for this but you can hand wave it from the Nyquist-Shannon sampling theorem [0]. So how much uncertainty comes from other sources when deciding on an initial burn for your trip to Mars? Contributing factors could be your fine control over the thrust from the rocket engine, but also solar radiation, gravity or - I suspect this is biggest - attitude control. I'd guess NASA can achieve a precision of better than 1/500 but probably not 1/50,000, so they'd need about 5 or 6 digits of pi. But I'm interested in hearing a more educated guess! [0] https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_samp...
- CydeWeys 6y ago
  And so we're back to still needing a source to do better than guessing :/

kens 6y ago

Some factors are way more important than others for rocket navigation. For instance, the azimuth, the angle you're headed is very important, since a small error in angle gets you way off target. The Saturn V theodolite, used to calibrate this angle before launch had an accuracy of better than +/- 2 seconds of arc, i.e. 1 part in 648000. That suggests you'd need to use pi to at least 6 decimal places (probably more) for surveying and azimuth calibration. The book "Inventing Accuracy" is about missile guidance (which has mostly the same issues) and discusses the various sources of error in detail and the various contributions to the "error budget".