all right so here is a research paper
published by nasa and it credits t h
skopinski and katherine johnson it was
written at the langley research center
and was written in september
1960. that's a year before alan shepard
would become the first american in space
and before john glenn would become the
first american to orbit the earth a
legacy of people like katherine johnson
has helped to bring a lot of attention
to the work done at nasa around this
time calculating how to put people up in
space and how to bring them back down
safely the movie hidden figures
especially highlighted the role of human
computers in that work but until now i
never really knew what that work would
have looked like i knew that katherine
johnson's work as shown in the movie had
something to do with orbital mechanics
or analytic geometry but i was curious
to know what the actual calculations and
equations were so that's why i went
looking through the archives to find the
actual documents this is the first
report that officially includes
catherine's name on the cover page but
it's not the first report she worked on
it's called the termination of azimuth
angle at burnout for placing a satellite
over a selected earth position let's go
through some of the terms the azimuth
angle is measured from north and is the
angle that you're appointing when you
launch a rocket often and is limited by
safety we don't really want to launch
rockets over populated areas or over
houses and so if you're launching you
often want to have your azimuth angle
such that you're going to launch out
over the ocean
and burnout refers to the shut off of
the rocket engine so after the burnout
point the orbital parameters are
essentially locked in and we can then
just work with newton's laws and other
equations to find out where it's going
to be because the azimuth angle at
launch will affect where the satellite
is going to end up after a certain
number of orbits
this paper is trying to find out what
launch azimuth angle you would need in
order to end up at a selected earth
position after a set number of orbits so
say you want your astronaut to splash
down in a certain region on earth so
that you can go and retrieve them well
the math in this paper is working out
what you would need to do at launch and
able to ensure that they start by
listing out a bunch of symbols we have
things like the semi-major axis and the
eccentricity relating to the shape of
the elliptical orbit
and we have the inclination angle of the
orbital plane and some of the parameters
make reference to the perigee location
which is the point when the satellite is
closest to the center of the earth
it says here that if we know the initial
conditions of speed radial distance and
elevation angle at burnout those will
all determine the orbital
characteristics of a satellite so going
on to work out a way to relate all of
these parameters they start with a bit
of elementary orbital mechanics for
which they reference an introduction to
celestial mechanics published in 1914
these are all derivations working
towards that ultimate relationship
between the azimuth angle and the
longitude and latitude that we wish to
land at
these calculations were done by hand
likely by katherine johnson herself but
then it says here that the results for
both cases are then compared with the
results obtained by a method of a three
degree of freedom calculation
performed on an ibm 704 electronic data
processing machine
this would be around the time that nasa
got its very first computers its ibm
machines which would have been huge and
taken up an entire room and very
difficult to use they go on to present
two example orbit calculations
one for a satellite that is launched
westward and one for satellite launched
eastward just to show that this method
is not limited by the direction of
launch this page here is a summary of
all of the equations that were derived
and that will be useful in the
calculation and it actually involves a
bit of an iterative procedure because we
don't initially know anything about the
orbital angle or the time it takes to
get to a certain position we just have
this first approximation for the time
it's taken to get to a selected point we
can feed that approximation into another
equation and that value into another
until we start to see some convergence
in the values we're getting for the
angle after we're happy with this much
we can then finally feed the parameters
into an equation to find the azimuth
angle and the inclination angle telling
us what we need to launch the satellite
there's also a mention of oblateness
effects because
the earth is not perfectly spherical and
the sublightness is taken into account
with small modifications to the latitude
and longitude values after the first
iteration
and these are the exact equations here
that katherine johnson would have been
calculating in her role as a computer
but for this paper she also had a lot to
do with deriving them in the first place
in 1960 no woman at nassau had ever had
her name included on a report catherine
johnson worked on this report with ted
skurpinski and their supervisor
encouraged skopinski to finish the
report and just put his name on it
before going away on a trip to houston
but ted said catherine should be the one
to finish it she's done most of the work
anyway
and so it became the first of 26 reports
to include catherine's name if you take
a look at these photos you might notice
these calculator looking devices on
people's desks and these were mechanical
calculating aids here is a video from
youtube of a restored one and showing
the kind of process that it uses there's
no electronics in one of these but they
can do arithmetic like addition and
subtraction quite easily and through a
bit more of a convoluted process can
also do multiplication and division
and even find things like square root
that is all done through a series of
amazing and intricate mechanical
movements
catherine likely had access to one of
these to help her step through the
calculations and something else that she
might have had would have been a log
table
working out big tables of logarithms and
also trigonometry values was something
that motivated us inventing computers in
the first place to make calculating
these tables easier and ensure that
there are less mistakes
engineers around this time would have
also used the slide rule but i'm
thinking that here these calculations
need to be done to
really a high degree of accuracy maybe
four or five decimal places at least and
so a slide rule might not be able to
offer that kind of precision but a log
table especially one made for engineers
can go right up to
five six or seven significant figures
for the first example the eastwood
launch we have a given launch position
and also a selected reentry position
and we're going to do three orbits these
are the values that catherine worked out
using those equations on the previous
page and using her first approximation
we can do a little recreation of this
using the log table taking the first
equation and the values we've been given
which fills out to this we can do the
subtraction by hand although the
division and the multiplication is a bit
harder in the grand scheme of things
though this equation is pretty short and
simple so maybe she wouldn't have needed
to use log tables for this but we'll use
them for the sake of illustration
a quick reminder about logarithms if you
have a thousand that could be rewritten
as 10 to the power of 3
so the log in base 10 of a thousand is
three
the log of a hundred is two and the log
of ten is one for a log of base ten it
is rewriting your number as ten to the
power of something and that something in
the power is your logarithm converting
our numbers into logarithms is useful
because instead of dividing two numbers
you can just subtract their logarithms
and instead of multiplying two numbers
you can just add their logarithms
first we want to find the log of 91.585
there are two significant figures before
the decimal point so we do two minus one
is one
so our number is going to start with one
point something to find the rest we go
to the table
this is a five place log table and we're
looking for nine one five eight five
the first three digits are here we come
down to nine one five
then across to 8.
now the first two digits are missing off
of this value which says 180 and the
first two digits are over here they are
9 6. so our logarithm is
9 6
one eight zero
but then because we had
nine one five
eight
five we also need to add a two point
five giving us nine six
one eight
two five
doing the same thing for our other
numbers we end up with the three
logarithms that we want we want to add
the logs of these two values since
they're multiplied and subtract this one
that gives us 0.88610
but we're not at an answer we're in a
strange realm of logarithms and to get
back to our final answer we need to take
the anti-log of it there's a table for
that too we are looking for 0.88610
0.88
6
1
that will be 7 6 9 1 plus 2
7 6 9 3
our decimal point will be placed between
the 7 and the 6
to give us exactly the same answer that
catherine johnson has recorded down here
take
7.693 and feed it into here with our
other known values and you'll get 32.162
take the cosine of that multiply it by
the cosines of the latitudes plus the
sines of the latitudes
take the inverse cosine of this whole
thing
and then add on the angle at burnout and
you should get 51.89
and taking the sines cosines or inverse
cosines can be done by looking it up in
the table for example the sine of phi 2
our latitude 34 degrees would be
34 degrees .0 the sine is 0.559
from these four expressions you'll get
your first approximations of these four
values
we are given a reference value of t of
theta one so you can use that and the
new value you've worked out for t
theta 2 e to go back and calculate a new
value of this parameter
feed that into these equations again and
you'll get your second iteration but
also for that use new values for your
latitude and your longitude that are
corrected for the ablateness of the
earth doing that whole process four
times gives you this table of iterations
because we've reached a point where
we've got some kind of convergence to a
value here we can take these values
and use them with our last two equations
to work out the azimuth angle and the
inclination angle going back to compare
our results with that of the new ibm
computer
it says that using the azimuth angle
calculated by the method of this paper
an orbit was calculated by the computer
for an ablate rotating earth and it's
the trace of that resulting orbit that
is presented in figure 5a so this is
actually what the computer has given us
for what the orbit will look like doing
it by hand like in this paper indicates
that the selected position would be
reached in
281.316 minutes
but the computer data indicates that at
the same time the satellite would be
18.6 nautical miles west of the desired
position and wouldn't reach the desired
position until
281.398 minutes which is just a little
bit later so this agreement is
considered good
now we can do these things pretty much
instantly here's some code on
mathematica which generates a graphic
where you can just move the slider and
see how moving the azimuth angle will
change the location of the satellite
after however many orbits you like
when the big ibm computer was first used
to calculate the trajectory of a flight
astronaut john glenn actually asked if
catherine johnson could verify the
calculations so that he could feel
confident going up she reproduced some
of the computers calculations by hand to
make sure it was doing the right thing
in the hidden figures movie there's also
a scene where the character who plays
catherine johnson
is talking about euler's method
she says that they can use euler's
method to solve a problem about going
from an elliptical orbit to a parabolic
one she runs off to the side and finds a
book that she opens up to learn about
the topic and from that tiny screen grab
i did manage to find out which book that
was it's this one the mathematics of
physics and chemistry by marginal and
murphy
and here we can find the same page it's
about the modified euler method and this
is about taking small iterative steps to
solve a differential equation we didn't
have any differential equations in the
paper that we just looked at so this
would have been used for different work
entirely likely having to do with
actually trying to plot the path of the
rocket perhaps using the rocket equation
which is a differential equation
but i think that's enough for this video
thank you for watching and a special
shout out to today's patreon cat of the
day june
you