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?
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".
(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.
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 :)
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?
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 value given in the paper assumes the distance in meters I think.
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.
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.
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
Here is a heavily used method in astronomy, this involves a higher order polynomial expansion than RK4.
This method has been extended a few times, my code uses a variation of it, and I know of several other projects which are also descended from it.
defrost•5mo ago
Takes me back to when I lived and breathed such code for early geophysical and remote sensing work.