Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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150
votes
3answers
23k views

How did the Apollo computers evaluate transcendental functions like sine, arctangent, log?

Navigation with a sextant or maneuvers using gimbal angles might be two examples of cases where an Apollo computer might need to do trigonometry. Trigonometric functions like sine, arctangent, etc. ...
33
votes
3answers
45k views

How to programmatically calculate orbital elements using position/velocity vectors?

I would like to build some orbital mechanical software from scratch. I feel that this would be a great way to learn the steps required to calculate different Kepler orbital elements of an object, plot ...
28
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3answers
9k views

Time at 1 g acceleration to travel 100 000 light years

How long would it take to go 100,000 light years at a constant 1 g acceleration?
26
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1answer
2k views

What are some specific examples of the calculations human “computers” did for the Mercury space program?

The movie Hidden Figures tells the story of three black women who were "computers" for the Mercury space program. They performed calculations related to orbits, launches, aerodynamics, etc. What are ...
25
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2answers
9k views

How complex was the math and physics necessary to place Apollo 11 on the moon?

Specifically, I'm interested in how closely the models used to calculate the various burns and course corrections represented reality. Was standard Newtonian mechanics sufficient or were relativistic ...
24
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3answers
21k views

Delta-V chart mathematics

Could someone explain me the mathematics behind the Delta-V charts one can find on internet such as:                 &...
22
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1answer
4k views

Who developed the mathematics used to correct the trajectory of Apollo 13?

The 1967 NASA technical note D-4246, Midcourse Guidance Procedure with Single Position Fix Obtained From Onboard Optical Measurements describes correcting the trajectory of a spacecraft traveling ...
18
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4answers
5k views

What is the “pendulum rocket fallacy” as it relates to analogizing a pencil balanced on a finger to maintaining attitude of a hovering rocket?

I've received comments that explain to me that the analogy between rocket attitude control during a hovering maneuver and the act of balancing pencil on the end of a finger is a helpful and good one; ...
17
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1answer
3k views

What branches of mathematics are widely used in astrodynamics?

We knew that Voyager 1 and 2 exploited the gravitational slingshot from Saturn and Jupiter back in 70s, and the trajectories were predicted by Newton's laws of motion, and this law we used a lot in ...
17
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1answer
394 views

What is the optimal inclination change strategy?

Consider a transfer between two circular orbits of similar radius, the only difference being the inclination difference, $\alpha$. What's the minimal $\Delta v$ required to perform this transfer? ...
15
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1answer
5k views

Can I derive a combined equation for velocity of solar sail?

In this article on The Planetary Society website, I read this about solar sails: At an acceleration rate of 1 millimeter per second per second (20 times greater than the expected acceleration for ...
14
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2answers
2k views

What is the “specific impulse”?

One of the most important values when talking about rocket engines is the so-called specific impulse. I have read the Wikipedia article but still have trouble understanding what it really is and what ...
14
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3answers
1k views

Is there an intuitive reason for why the shape of the orbit at perigee is the mirror image of that at the apogee?

Is there an intuitive reason for why the shape of the orbit at perigee is the mirror image of that at the apogee? It seems strange to me, because the situation is so different at the perigee compared ...
12
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2answers
2k views

Solar sail force calculation: Angle vector

In this Wikipedia article about solar sails, I read the following: The force on a sail and the actual acceleration of the craft vary by the inverse square of distance from the sun (unless close ...
11
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3answers
1k views

Physics and math behind flight through solar system [closed]

I wrote a program that simulates a solar system. I was able to calculate the locations for every planet on its elliptical route for any given time. In a second project, I managed to simulate newtonian ...
11
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2answers
6k views

Solar sail thrust calculation

In Space Mission Engineering: The new SMAD, page 555, section 18.7.2, the following thrust formula is given for a solar sail: $$F=\frac{2RSA}{c}\sin^2\theta=9.113\times10^{-6}\frac{RA}{D^2}\sin^2\...
11
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3answers
5k views

How to best think of the State Transition Matrix, and how to use it to find periodic Halo orbits?

I'll state my mathematical question about the state propagation and state transition matrices first, then show you a simple problem for which I would like to use these concepts to generate a densely ...
10
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2answers
2k views

How are gravity assists conceived?

So there are some very complex gravity assists out there. The Cassini mission, for instance, used a VVEJ (Venus Venus Earth Jupiter) gravity assist to make it to Saturn. Similar flybys have been done ...
10
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1answer
600 views

Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch?

For fun I'd like to use Hubble Astrometric data for example the observations listed at the bottom here to numerically estimate the orbit of 2014 MU69 as well as its uncertainties. My plan is to use ...
10
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4answers
18k views

Converting Orbital Elements to Cartesian State Vectors

Updated 2016-12-12 I'm writing an orbital mechanics program in Unity. I convert an orbiting object's position and velocity into orbital elements, then converting the orbital elements back into ...
10
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0answers
365 views

Mathematics used for F9R flyback and landing [duplicate]

I was wondering if anyone may have useful links to the mathematics used by the F9R that allowed it to fly back to LZ-1 and land successfully. I'm hoping to find some papers that will give the broadest ...
9
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4answers
1k views

Calculate flight path angle given semi-major axis, eccentricity and distance from the focal point

How can I, given the semi-major axis (α), the eccentricity (ε), and the distance from the focal point (r), calculate the flight path angle (φ)? (As shown above)
9
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1answer
213 views

Launching east from a mountain on the equator at midnight during a new moon; ranking of each contribution?

This answer is intriguing and I'm curious if it's correct, and especially which effects are most important. If you want to avoid gravity assists, the most fuel-efficient way out of the Solar System ...
9
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2answers
367 views

Explanation of weird values in New Horizons data?

In this answer I computed (methods given later in post) the estimated time, distance, range, and resolution for the science capture for New Horizon's closest approach to Ultima Thule (2014 MU69). ...
9
votes
1answer
264 views

Star-shaped artifacts in SAR images of the “Suez Canal traffic jam seen from space”

Wikipedia's 2021 Suez Canal obstruction links to the Sentinel-1 synthetic aperture radar (SAR) image of the Traffic jam in the Gulf of Suez caused by the obstruction as seen by the Sentinel-1 ...
8
votes
2answers
955 views

What's the eccentricity of an orbit (trajectory) falling straight down towards the center?

Discussion below this answer has got me thinking. A circular orbit has eccentricity of $\epsilon = 0$, ellipses $ 0 < \epsilon < 1$ (between but not including 0 and 1), and parabolic and ...
8
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2answers
3k views

In “spacecraft talk” is nadir just a fancy word for “down”?

Assume the approximation that the Earth is spherically symmetric — without higher order gravitational multipoles, and other effects that might be related to gravity from other bodies or the ...
8
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2answers
3k views

How can I predict what an object's orbital state vectors will be in the future?

I have a position and velocity vector for an orbiting object and can calculate the orbital elements. From a simple two-body situation, how can I predict where in the orbit the object will be at some ...
8
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1answer
1k views

Gravity drag Delta-v calculation for launches to orbit

It would appear that gravity drag has a exact formulation for an instantaneous moment in a rocket launch. I'm tempted to think that some calculus could get you an exact number for Delta-v of an ...
8
votes
2answers
550 views

How can I calculate the delta-v correctly, this way does not seem to be correct?

I'm learning about Astrodynamics, and I would like to ask how can I calculate the $\Delta V$ for an interplanetary mission. I used a website for the porkchop plots (http://sdg.aero.upm.es/index.php/...
8
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1answer
505 views

Laplace Method for Orbit Estimation

I have a question about Laplace angle-only method for orbit determination where the line of sight vectors are being interpolated. I read somewhere that the method fails (due to a matrix being singular)...
7
votes
2answers
991 views

From the General Thrust Equation towards Tsiolkovsky, how to explain dropping these terms along the way?

The NASA Glen Research Center tutorial page Rocket Thrust Equation links to the General Thrust Equation page which starts with: Which transcribes into MathJax as: $ \ \ \ F = \dot{m}_\mathrm e V_\...
7
votes
1answer
503 views

How to numerically solve an optimal thrust-vectoring problem?

This question is a follow-up to this question and is linked to this question, where if needed more details can be found on the parameters used in the following. For an in-plane non-impulsive orbital ...
7
votes
1answer
262 views

Negative photon pressure? What is the physics behind this expression?

The Wikipedia article on Radiation Pressure gives an equation for the pressure on a perfect reflector produced by a beam of photons with an energy flux $E_f$ (in units of power/area) as $$P=\frac{...
7
votes
1answer
976 views

Using of the coordinate systems in orbit propagation

For the orbit propagation in ECI coordinate system, considering the effects of zonal harmonics and Sun/Moon perturbations the following procedure was formed: Convert the state vector to ECEF ...
7
votes
2answers
201 views

Implementing control system to track position and velocity references simultaneously

I've been trying to implement a PI (proportional-integral) controller to allow a launch vehicle to track 2 reference states simultaneous, namely position and velocity, so that the launch vehicle will ...
7
votes
0answers
1k views

For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other?

The term $J_2$ seen in discussions of nodal precession of orbits (e.g. sun-synchronous) appears to come in two flavors. For example, in the Wikipedia article Geopotential model it has a value of $1....
7
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0answers
431 views

Understanding formula for geocentric position of Moon

I am trying to understand a basic formula in a Celestial Mechanics reference. The formula is for the position of the Moon in a geocentric frame. The reference claims in line (981) that this should be ...
6
votes
1answer
338 views

Analytical expression for the ground track of the International Space Station

For a maths task in school, I am investigating the orbit of the International Space Station around the Earth. I understand that when the 3D movement in space is represented on a 2D surface, the ...
6
votes
1answer
1k views

How were Frenet frames and Gram–Schmidt orthonormalization used in spacecraft orbit calculations?

I haven't seen the film Hidden Figures yet, but I have read and listened to some dialogue clips in this NPR Review. In it they mention a Frenet frame and the Gram–Schmidt process for ...
6
votes
1answer
480 views

I've almost learned to spell Chebyshev, why has JPL switched to Hermite interpolation for DE438?

According to @ChrisR's thorough answer One caveat however is that the interpolation in DE438 is a Hermite interpolation, and no longer a Chebychev interpolation, so you may need to update your code....
6
votes
2answers
617 views

Generating plot of LV's performance as a function of the C3 parameter

Following my previous question about Ariane 5 performance data for escape missions, I would like to know how the plot of launch vehicle capabilities like the next one are obtained: More specifically, ...
6
votes
1answer
208 views

How (the heck) was coherent synthetic aperture radar (SAR) implemented using photographic emulsion aboard Apollo 17?

This answer to Is there really microfilm on the Moon? includes a block quote which includes the following subquote: The equipment was installed in the service module and consisted of a coherent ...
6
votes
1answer
106 views

Displaying spacecraft attitude from rotation matrix or quaternion?

Navigation and guidance systems generally use rotation matrices or quaternions to uniquely specify 3D rotations. But these are not the least bit intuitive, so we like to convert them to Euler angle ...
6
votes
0answers
165 views

What are the contributions of ancient Indian astronomers in mapping of the solar system and galaxies?

In the history of astronomy we usually come up with the name of European astronomers like Kepler and Galileo. Earlier Indian astronomers, for example Aryabhata and Bhaskaracharya, are not usually ...
5
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1answer
1k views

How does the poliastro python package “Going to Mars with Python” example work? What's it really doing?

@astrojuanlu's answer mentions the existence of the Python package poliastro, and in the documentation there I found an example titled Going to Mars with Python using poliastro among others. Input ...
5
votes
1answer
3k views

Why does the eccentricity vector equation always equal -1?

This is the eccentricity vector equation, $$e=\frac{1}{\mu}[(v^2-{\mu \over r})r-(r \cdot v)v]$$ $$e=|e|$$ Now this equation is written differently from many different sources but they essentially ...
5
votes
2answers
1k views

Calculating the planets and moons based on Newtons's gravitational force

I am trying to understand how to calculate the orbits of solar system bodies in an n-body framework, based on the pair-wise gravitational interaction between the objects. At present, I am considering ...
5
votes
2answers
174 views

Do all rigid bodies of radius r have at least one stable orbit with perihelion p such that 2r > p > r?

In reading about the TESS mission, I was ultimately led to this NASA commentary from 2006, discussing how lumpy Earth's moon is. One of the points made is that there exist only 4 orbital inclinations ...
5
votes
1answer
626 views

Spiraling out from circular orbit to escape via low thrust, what is γ (gamma)?

@MarkAdler's comment led me to ask Why would a slow spiral from a C3 of zero take about 2.4 times as much ΔV as an impulsive maneuver? which resulted in this tidy and efficient @MarkAdler answer which ...