It is neat to see some of the old work done in the field, this looks like a pretty classic treatment of the topic. It looks like they were using a fourth-order Runge-Kutta integrator, which would likely limit long term integrations accuracy (though looks sufficient for their use case). Many algorithms I have seen typically use much higher order integration methods to beat down the accumulation of numerical error.
Source: Working on my PhD in orbital mechanics of asteroids/comets, here are my open source (python/rust) orbital integration tools:
https://github.com/dahlend/kete
My first thought was to upload the PDF to Qwen3 and ask it to reimplement in Python using NumPy, Astropy, etc. Have to work on the day job, but could be some educational fun learning and Jupyter plots in my near future. Anyway, the generated code looks promising and contains the requisite green tick and bar graph emojis, so what's not to like?
Idle question: in the days before TeX, when manuscripts like this were hammered out on Remington office typewriters, how did authors handle symbols?
In this manuscript for example you can see that power superscripts are really just regular numbers typed at an offset (perhaps rotating the paper around the platen one notch instead of the two that would be a whole line feed). But what about the vectors and the giant sigma? All hand drawn over the top of a typed manuscript?
Yes, I believe they're drawn or stenciled in. Some amount of care has been taken here to produce a more professional-looking result, but you can find plenty of old typed papers where math is obviously handwritten in. Like John Nash's thesis: https://library.princeton.edu/sites/g/files/toruqf6021/files...
These look like the fonts from the IBM Selectric with the changeable typeballs.
When you needed a greek letter you had to stop there, change typeballs, type the greek letter, then put the regular typeball back on.
On the equations the big stuff would be drawn in by hand from stencils.
On a diagram it could be a mechanical (assisted) drawing that was labeled by typing the same font size, like sketch (a).
When you get to sketch (b) though, this one is a reduced photocopy of the original page that was typed on when labeling the mechanical drawing to begin with.
You can see the way that all of the equations and illustrations could very well be place-held in the text draft until a perfect equation or diagram could then be added later by cutting the proper size horizontal strip of paper containing the original drawing, and "pasting" it over the blank spot in the text where the figure goes. Before photocopying to arrive at the priceless original like this where you couldn't always tell where it was cut-and-pasted.
The Fortran printouts from the IBM line printer look like they could be pasted in both at full-size and photoreduced, on one page at page 20.
But a good typist could avoid that for most equations, they could blaze through the text but when an equation came up it took more time to get one equation right than to type many more pages of text.
Mostly they didn't .. it was handballed to the secretaries of the math and physics typing pool who used stencils, high end typewriters, and other template mechanisms.
A good many such secretaries were reasonably talented math and physics graduates themselves who had limited opportunity to be hired to do "a man's work".
When I was a grad student at the Institute for Theoretical Physics in Heidelberg in 2014, they still had a few of these secretaries who would type out handwritten manuscripts by professors who couldn't or wouldn't write LaTeX. Maybe they still do.
Non-secretary women were not in short supply at NASA at this time though. The work of Virginia E. Morrell and Bonnie J. McBride a few years earlier, at this same research center springs to mind.
There were some to be sure .. I wouldn't imply otherwise, none the less many existed that were not hired and most STEM heavy companies and institutions were very male biased.
(With the few and obvious exceptions such as Dame Vera Stephanie "Steve" Shirley's ventures)
FWiW I learnt a wee bit from Cheryl Praeger, Cathleen Morawetz, Robyn Owens, et al.
Love seeing stuff like this. The corrections for an oblate spheroid threw me for a loop at first, until I realized "yeah, of course". I've only ever played around with ideal bodies when simulating the n-body problem (sounds a bit raunchy...) so never even considered the fact that a rotating planet isn't perfectly spherical.
From NASA, 1963, by William C. Strack, Wilbur F. Dobson, and Vearl N. Huff
As described herein, this code is designed to operate on an IBM 704 computer that has an 8000 word (8 K) memory and at least 1 K of drum.
Even so constrained it includes means of changing coordinate base when approaching asymptotes inducing loss in numerical accuracy, variable step size control, etc.
Takes me back to when I lived and breathed such code for early geophysical and remote sensing work.
It looks like they chose to use the "universal gravitational constant" "k" instead of Newton^s constant, "G": p.23, "k^2 = universal gravitational constant, 1.32452139x10^20, m^3/(sec^2)(sun mass units)"
But the value and unit of "k" given in the Wikipedia page is different. Do you know what NASA document means by "universal gravitational constant" in modern sense?
The code appears to use in some places GK2M, which is the Newtonian constant of gravity, and in other places SQRDK, which is inappropriately described as a "gravitational constant", but it actually is the mass of the Sun expressed in some special units.
Newton's constant is known only with a very high uncertainty, i.e. a very low precision.
For the great bodies of the Solar System, e.g. the Sun and the planets, one knows with a high accuracy the product between their mass and Newton's constant, because that can be measured by the force with which they attract a body of known mass, e.g. an artificial satellite or an interplanetary probe.
Computing their mass in kilograms would be pointless in most cases, because that would introduce great uncertainties in the computations. So for the Sun and the planets one expresses their masses by the products between their mass and Newton's constant, whenever that is possible, i.e. whenever one needs to compute their attraction force exerted upon a small object.
Wikipedia names the product mass-Newtonian constant as "the standard gravitational parameter of a body", but I believe that this is a misleading name, because this product is just the mass of the body expressed in different (non-SI) units. Expressing a mass by its product with the Newtonian constant is not different from expressing the mass in pounds instead of kilograms. Using the Newtonian constant instead of some random unit conversion factor just has the advantage of removing the uncertainties from some expressions computing forces of gravity.
Gauss's constant k is defined as sqrt(G), but for a while the international standard was to define k and then compute G as k^2, which is why NASA refers to it that way.
I think it's just units. From wikipedia: "and its value in radians per day follows by setting Earth's semi-major axis (the astronomical unit, au) to unity, k:(rad/d) = (GM)0.5·au−1.5."
the value given in the paper assumes the distance in meters I think.
It is neat to see some of the old work done in the field, this looks like a pretty classic treatment of the topic. It looks like they were using a fourth-order Runge-Kutta integrator, which would likely limit long term integrations accuracy (though looks sufficient for their use case). Many algorithms I have seen typically use much higher order integration methods to beat down the accumulation of numerical error.
Source: Working on my PhD in orbital mechanics of asteroids/comets, here are my open source (python/rust) orbital integration tools: https://github.com/dahlend/kete
I'm interested. Could you give examples ?
That is cool! I encourage you to post as a Show HN with a comment explaining more if you want
My first thought was to upload the PDF to Qwen3 and ask it to reimplement in Python using NumPy, Astropy, etc. Have to work on the day job, but could be some educational fun learning and Jupyter plots in my near future. Anyway, the generated code looks promising and contains the requisite green tick and bar graph emojis, so what's not to like?
Scipy implements similar algorithms, but delegates the heavy lifting to - you guessed it - Fortran. Example: https://github.com/scipy/scipy/blob/v1.16.1/scipy/integrate/...
The combination of Fortran and AI here starts to be close to 2001. ;)
You mean 2001 of 1969?
The X-files AI episodes of the '90's are about as close to 1969 as to today.
All anybody could do then was to use their imagination, but is it all that much different today?
Idle question: in the days before TeX, when manuscripts like this were hammered out on Remington office typewriters, how did authors handle symbols?
In this manuscript for example you can see that power superscripts are really just regular numbers typed at an offset (perhaps rotating the paper around the platen one notch instead of the two that would be a whole line feed). But what about the vectors and the giant sigma? All hand drawn over the top of a typed manuscript?
Yes, I believe they're drawn or stenciled in. Some amount of care has been taken here to produce a more professional-looking result, but you can find plenty of old typed papers where math is obviously handwritten in. Like John Nash's thesis: https://library.princeton.edu/sites/g/files/toruqf6021/files...
After the Remington era, I once used this :-) IBM Selectric typewriter math fonts, I found this description: https://www.duxburysystems.org/downloads/library/texas/apple...
Godement in his analysis exposition briefly talks about this, it was all hand drawn.
These look like the fonts from the IBM Selectric with the changeable typeballs.
When you needed a greek letter you had to stop there, change typeballs, type the greek letter, then put the regular typeball back on.
On the equations the big stuff would be drawn in by hand from stencils.
On a diagram it could be a mechanical (assisted) drawing that was labeled by typing the same font size, like sketch (a).
When you get to sketch (b) though, this one is a reduced photocopy of the original page that was typed on when labeling the mechanical drawing to begin with.
You can see the way that all of the equations and illustrations could very well be place-held in the text draft until a perfect equation or diagram could then be added later by cutting the proper size horizontal strip of paper containing the original drawing, and "pasting" it over the blank spot in the text where the figure goes. Before photocopying to arrive at the priceless original like this where you couldn't always tell where it was cut-and-pasted.
The Fortran printouts from the IBM line printer look like they could be pasted in both at full-size and photoreduced, on one page at page 20.
But a good typist could avoid that for most equations, they could blaze through the text but when an equation came up it took more time to get one equation right than to type many more pages of text.
As can be seen, it was obviously worth it :)
> how did authors handle symbols?
Mostly they didn't .. it was handballed to the secretaries of the math and physics typing pool who used stencils, high end typewriters, and other template mechanisms.
A good many such secretaries were reasonably talented math and physics graduates themselves who had limited opportunity to be hired to do "a man's work".
When I was a grad student at the Institute for Theoretical Physics in Heidelberg in 2014, they still had a few of these secretaries who would type out handwritten manuscripts by professors who couldn't or wouldn't write LaTeX. Maybe they still do.
Non-secretary women were not in short supply at NASA at this time though. The work of Virginia E. Morrell and Bonnie J. McBride a few years earlier, at this same research center springs to mind.
There were some to be sure .. I wouldn't imply otherwise, none the less many existed that were not hired and most STEM heavy companies and institutions were very male biased.
(With the few and obvious exceptions such as Dame Vera Stephanie "Steve" Shirley's ventures)
FWiW I learnt a wee bit from Cheryl Praeger, Cathleen Morawetz, Robyn Owens, et al.
Love seeing stuff like this. The corrections for an oblate spheroid threw me for a loop at first, until I realized "yeah, of course". I've only ever played around with ideal bodies when simulating the n-body problem (sounds a bit raunchy...) so never even considered the fact that a rotating planet isn't perfectly spherical.
It’s amazing how many times I search for some space-related technical thing and I end up with a paper from NASA Glenn (used to be Lewis)
From NASA, 1963, by William C. Strack, Wilbur F. Dobson, and Vearl N. Huff
Even so constrained it includes means of changing coordinate base when approaching asymptotes inducing loss in numerical accuracy, variable step size control, etc.Takes me back to when I lived and breathed such code for early geophysical and remote sensing work.
It looks like they chose to use the "universal gravitational constant" "k" instead of Newton^s constant, "G": p.23, "k^2 = universal gravitational constant, 1.32452139x10^20, m^3/(sec^2)(sun mass units)"
I think "k" was also known as "Gaussian gravitational constant" https://en.wikipedia.org/wiki/Gaussian_gravitational_constan...
But the value and unit of "k" given in the Wikipedia page is different. Do you know what NASA document means by "universal gravitational constant" in modern sense?
The code appears to use in some places GK2M, which is the Newtonian constant of gravity, and in other places SQRDK, which is inappropriately described as a "gravitational constant", but it actually is the mass of the Sun expressed in some special units.
Newton's constant is known only with a very high uncertainty, i.e. a very low precision.
For the great bodies of the Solar System, e.g. the Sun and the planets, one knows with a high accuracy the product between their mass and Newton's constant, because that can be measured by the force with which they attract a body of known mass, e.g. an artificial satellite or an interplanetary probe.
Computing their mass in kilograms would be pointless in most cases, because that would introduce great uncertainties in the computations. So for the Sun and the planets one expresses their masses by the products between their mass and Newton's constant, whenever that is possible, i.e. whenever one needs to compute their attraction force exerted upon a small object.
Wikipedia names the product mass-Newtonian constant as "the standard gravitational parameter of a body", but I believe that this is a misleading name, because this product is just the mass of the body expressed in different (non-SI) units. Expressing a mass by its product with the Newtonian constant is not different from expressing the mass in pounds instead of kilograms. Using the Newtonian constant instead of some random unit conversion factor just has the advantage of removing the uncertainties from some expressions computing forces of gravity.
It's just regular old G, defined in mass-of-sun units: https://en.m.wikipedia.org/wiki/Gravitational_constant (fourth item in the first table: NASA also uses meters whereas Wiki uses km)
Gauss's constant k is defined as sqrt(G), but for a while the international standard was to define k and then compute G as k^2, which is why NASA refers to it that way.
I think it's just units. From wikipedia: "and its value in radians per day follows by setting Earth's semi-major axis (the astronomical unit, au) to unity, k:(rad/d) = (GM)0.5·au−1.5."
the value given in the paper assumes the distance in meters I think.