303
votes
Accepted
How did the Apollo computers evaluate transcendental functions like sine, arctangent, log?
Since the Apollo 11 code is on GitHub, I was able to find the code that looks like an implementation of sine and cosine functions: see here for the command module and here for the lunar lander (it ...
- 2,223
83
votes
Time at 1 g acceleration to travel 100 000 light years
Nonrelativistic solution
The variables used will be
$x$ for the distance travelled
$v$ for velocity
$a$ for acceleration ($1~\mathrm{g}$)
$t$ for the time
$c$ for the speed of light.
Non braking
...
- 3,566
41
votes
How did the Apollo computers evaluate transcendental functions like sine, arctangent, log?
You also asked for the logarithm, so let's do this as well. As opposed to the sine and cosine functions, this one is not implemented with a Taylor series-like approach only. The algorithm is based on ...
- 713
38
votes
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Person falling from space
Your question is under-specified (you don't give the size or posture of your subject), so I'm assuming an average-sized woman falling in the classic face-down skydiver posture. I'm also modeling this ...
- 14.6k
37
votes
What is the "pendulum rocket fallacy" as it relates to analogizing a pencil balanced on a finger to maintaining attitude of a hovering rocket?
The pendulum fallacy is the belief that rockets would be passively stable with engines at the top, with the rocket "hanging" from them. The error lies in expecting gravity to pull the body ...
- 10.7k
35
votes
Accepted
Who developed the mathematics used to correct the trajectory of Apollo 13?
The authors of the paper are Harold A. Hamer, Katherine G. Johnson, and W. Thomas Blackshear. Of these, the name Katherine Johnson might ring a bell with people, as she was one of the protagonists in ...
- 14.1k
29
votes
Accepted
How complex was the math and physics necessary to place Apollo 11 on the moon?
Was standard Newtonian mechanics sufficient or were relativistic effects included?
Relativistic effects didn't have to be modeled; other sources of error
would have swamped the effects of relativity,...
- 167k
24
votes
What is the "pendulum rocket fallacy" as it relates to analogizing a pencil balanced on a finger to maintaining attitude of a hovering rocket?
In the inverted pendulum problem:
gravity exerts a vertical force on the pendulum, at the center of gravity
the support of the pendulum (like the finger under the pencil) exerts a vertical force on ...
- 368
16
votes
What is the "specific impulse"?
If you want to understand how the 'seconds' value fits the greater image, there's this rather contrived definition (which nobody uses because it's contrived and mostly useless but evocative enough.)
...
- 54.9k
15
votes
How does the poliastro python package "Going to Mars with Python" example work? What's it really doing?
Full disclaimer: I'm the author and main developer of poliastro.
The most important step before doing anything is somehow retrieving the positions and velocities of the planets of the Solar System. ...
- 1,052
14
votes
Accepted
Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch?
This is not a complete answer. It is instead an extended comment to the following:
I understand I will have to retard the gravity from each source by its particular light-time, as well as correct ...
- 73.2k
14
votes
What's the eccentricity of an orbit (trajectory) falling straight down towards the center?
The eccentricity is 1.0.
The eccentricity $e$ of an orbit can be found from the radius of apoapse and periapse as:
$$e=\frac{r_a-r_p}{r_a+r_p}$$
and the semimajor axis $a$ can as well, from:
$$a=\...
- 651
14
votes
Accepted
What is the velocity of the ISS relative to the Earth's surface?
JPL Horizons has trajectory data for the International Space Station, SPKID = -125544
Revised: Nov 23, 2022
Trajectory is TLE-based. Predicts run for 4 weeks into future, but are of low accuracy for ...
- 2,564
13
votes
Accepted
Why would a slow spiral from a C3 of zero take about 2.4 times as much ΔV as an impulsive maneuver?
If you're in a circular orbit, your velocity is $\sqrt{\mu\over r}$. Escape velocity at that distance is $\sqrt{2\mu\over r}$. So the impulsive $\Delta V$ to reach escape velocity starting from that ...
- 58.5k
12
votes
Accepted
Star-shaped artifacts in SAR images of the "Suez Canal traffic jam seen from space"
Discrete Fourier techniques introduce errors in their terms that track with the sinc function. Any target in the image will produce side lobes like those in the following graph.
Strong reflections ...
- 473
11
votes
Accepted
What are some specific examples of the calculations human "computers" did for the Mercury space program?
(This is adapted from my question/answer at Day-to-day tasks of human computers, ala Hidden Figures movie - History of Science and Mathematics Stack Exchange)
I was also fascinated by the film Hidden ...
- 671
11
votes
Why is specific impulse equivalent to effective exhaust velocity?
Rockets produce thrust by ejecting reaction mass at some velocity. The fundamental quantities involved are mass flow rate and exhaust velocity, thrust is the consequence of these.
It's no coincidence ...
- 10.7k
11
votes
Accepted
Is there a neat formula for hyperbolic Kentucky windage?
Using Perifocal polar coordinates, where the x-axis points from the central body to the periapsis, and the polar equation for conic sections:
$$r=\frac{a(1-e^2)}{1+e \cos(f)}$$
Provided parameters
$\...
- 5,410
11
votes
Object slowest at periapsis - despite correct position calculation
If I understand correctly, your $p, q$ are essentially cartesian coordinates, and you're trying to get the velocity components in those two orthogonal directions.
However, you're taking the derivative ...
- 149k
10
votes
Accepted
In "spacecraft talk" is nadir just a fancy word for "down"?
There are some subtleties here. The fields where the concept of nadir are most important are nadir-pointing Earth observation satellites, satellites formation flying, and rendezvous and proximity ...
- 73.2k
10
votes
Accepted
What is the highest number of impulses required for an optimal orbital transfer?
Whether you realize it or not, this is a very fundamental and challenging question in astrodynamics. It's personally one of my favorite topics in the field, and has been very rigorously studied by ...
- 116
10
votes
Is there an intuitive reason for why the shape of the orbit at perigee is the mirror image of that at the apogee?
If we assume Keplerian/Newtonian mechanics, then we can see a way to rendering the same local curvature of the path at perigee and at apogee (terms for orbiting Earth, of course).
At both points the ...
- 8,445
9
votes
Accepted
Deriving the changes in Keplerian Elements induced by small impulses
If we assume a perfect two-body problem, absent perturbations from external bodies or non-spherical gravity sources (i.e., perfect conic orbits with no precession or variation), your constraints ...
- 17.2k
9
votes
Accepted
What is the "specific impulse"?
In simplest terms it is just the thrust produced divided by the propellant flow rate. "How much thrust am I getting for the propellant I am expending?"
So bigger is better - you are getting more ...
- 178k
9
votes
Accepted
How to calculate the flight path angle, γ, from a state vector?
This is a problem that has plagued groups of people very knowledgeable about orbital dynamics but who learned using different textbooks: there are two different definitions of "flight path angle"!!
...
- 18.2k
9
votes
Accepted
Launching east from a mountain on the equator at midnight during a new moon; ranking of each contribution?
Breaking that down:
launch due East
site in the Ecuadorean Andes
sometime before local midnight
on a July 4
when there's a new moon
Launch due east.
With the exception of launch ...
- 73.2k
9
votes
Accepted
What is the formula for Legendre Polynomials from EGM96 calculated in F447.f program?
This is not a recurrence for a Legendre polynomial $P_{n}(\mu)$
but an Associated Legendre Function (ALF).
This1 states much more than is needed here.
ALFs are often denoted $P_{n,m}(\mu)$
in ...
- 1,124
9
votes
Accepted
Why does the eccentricity vector equation always equal -1?
The expression on the right is meant to give the eccentricity vector but the vector notation has been lost.
Here it is in this answer:
$$ e = {v^2 r \over {\mu}} - {(r \cdot v ) v \over{\mu}} - {r\...
- 149k
9
votes
Accepted
Object slowest at periapsis - despite correct position calculation
As uhoh said, the velocity is the derivative of the position w.r.t. to time, not true anomaly. I.e., if we denote time by $t$, the components of the velocity vector are $\left(\frac{dp}{dt}, \frac{dq}{...
- 2,040
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