# Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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### Usage 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 ...
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### variables affect the trajectory in a stochastic way

What are the different variables affecting the trajectory of a rocket in a stochastic way for getting potential three-sigma trajectory?
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### Low-thrust spiraling to escape, is the flight path angle (gamma) at C3=0 always 39 degrees?

@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 ...
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### How to calculate the flight path angle, γ, from a state vector?

@Julio's excellent answer describes a flight path angle, and explains that it is the angle between the tangential direction (perpendicular to the radial vector to the central body) and the current ...
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### 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 ...
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### Why would a slow spiral from a C3 of zero take about 2.4 times as much ΔV as an impulsive maneuver?

I just read a fascinating comment that I don't understand. In part it says ...the Oberth effect does not depend on the mass of the object. A slow spiral in to a low orbit from a C3 of zero will ...
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### 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 ...
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### What does this paper say is wrong (quantitatively and procedurally) with WISE & NeoWISE asteroid data?

I don't have daily access to the journal Icarus so when I read the New York Times article Asteroids and Adversaries: Challenging What NASA Knows About Space Rocks; Two years ago, NASA dismissed and ...
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### 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....
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### How can I verify my reconstructed gravity field of Ceres from spherical harmonics?

Based on @DavidHammen's very helpful answer I've made progress reconstructing the gravity field of Ceres from the Dawn radiometric data. The question there contains further information, but suffice it ...
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### 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/...
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### Roughly how many kinds of closed or periodic orbits are there in the circular restricted three-body problem?

Question: Roughly how many kinds of closed or periodic orbits are there in the circular restricted three-body problem? Has anyone made an attempt to enumerate and classify/categorize them? Notes: If ...
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1 vote
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### Will these equations give me the approximate apparent RA and Dec using positions from Horizons?

In this answer @DavidHammen showed me how to get coordinates from JPL's Horizons using the Earth's equatorial plane as a reference plane by selecting ...
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### How to intuitively explain that tri-elliptic transfers are not better than bi-elliptic transfers?

The question How to intuitively explain that reaching geostationary orbits consume more Delta V than escape orbits? includes the graphic below, which illustrates that when transferring between ...
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### What are the symbol definitions in the equations for finding orbital elements?

I've been looking at the answer to this question on how to convert position and velocity to orbital elements, but some of the symbols are not defined and there are no reference links for those symbols....
1 vote
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### What does the phrase "topological point of view" mean when applied to two-body orbits in this context?

A recent question includes the passage: From a topological point of view, it seems likely that any rigid body would have to admit at least one stable low orbit, regardless of mass concentrations. I ...
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### 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 ...
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### How to calculate cone angle between two satellites given their look angles?

I have the azimuth, elevation and distance of 2 satellites relative to a ground station, defined by latitude and longitude, as I've discussed in my previous question. I'm using the satellites' TLEs ...
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### Equation for orbital period around oblate bodies, based on J2?

In this answer I point out that the period of items (ring particles, moons, spacecraft, etc.) around an oblate body will not scale exactly as $a^{3/2}$ because the closer you are to the planet, the ...
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### Interpolate the ephemerides of the Earth

I have a table of state vectors generated from ephemerides of the Earth with time steps of 1 minute. Now I would like to interpolate this to a finer time increments in order to use it as part of an ...
1 vote
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### How to plot a Clohessy Wiltshire trajectory on MATLAB?

So I essentially want to plot the change in trajectory as I perform a maneuver using the clohessy wiltshire equations to chase a target. I have a for loop for 10,000s and I would like to calculate ...
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### Mathematical reasoning behind pe=pa leading to maximum thrust

I've been looking at both the thrust and thrust coefficient equations (below) and I understand that they will be optimum in a vacuum when $P_a=0$. But, I see in textbooks that $p_e=p_a$ corresponds to ...
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### 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 ...
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### Thrust for solar sail equation derivation

In the new SMAD (Chapter 18 Space Mission Engineering and Analysis) they have the equation for thrust of a solar sail and a reduced version. Variables are described in this question. I would like to ...
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### 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 ...
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### How do SDP4's "Deep space" corrections to SGP4 account for the Sun's and Moon's gravity?

The Simplified Perturbations model SGP4 is used to calculate Earth satellite state vectors (position and velocity) using standard ephemeris data encoded as TLEs (Two Line Elements). According to ...
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### 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 ...
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### How to derive the polar form of the equations of motion?

As part of an optimal control problem (see linked problem), I need the polar form of the equations of motion (EOM) defining the orbit of a spacecraft. In e.g. Bryson and Ho (1969) and Vallado (2007) ...
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### How to calculate the current position of a satellite?

I want to write a SIMULINK model that calculates the position visible of satellites for the current time. As input for the model I have the coordinates of a location, the current time and the ...
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### Trying to understand the two "holes" in this alt/az plot of the ISS

I'm plotting the position of satellites in the sky using Skyfield. If I project the .altaz() data on to a rectangle (in a similar fashion to the way ground track ...
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### What is a "synodic frame"? Can one be defined for an elliptical orbit?

A description of a "synodic frame" was one of the main ingredients in this excellent answer to a question about an animation of a funky-looking orbit that several different and distinctly interesting ...
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### 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 ...
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### How to rotate orbital ellipse when radial velocity is added to the orbital velocity

I am trying to plot an orbit and am having some success. I have directly calculated the orbital parameters so I am able to plot the ellipse. I am also plotting a path of the orbit as a sanity check. ...
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### 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 ...
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### 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 ...
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### Formula in "Hidden Figures" Movie

In the scene where Katherine Johnson/Taraji Henson has to calculate "Go / No-Go," which I take to mean the operational go-no-go orbital-insertion criteria. She calculates a value for R, using this ...
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### Deriving the changes in Keplerian Elements induced by small impulses

I'm trying to make derive a concise table of the effects on each Keplerian orbital element for a small impulse in the Radial, Tangential, or Perpendicular direction at periapsis or apoapsis, good to ...
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### 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 ...
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### 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 ...
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### 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 ...
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### How to calculate the trajectory needed to enter geostationary orbit?

Im building a rocket simulation engine and I would like to try and calculate the components of acceleration required to get the rocket into geostationary orbit. For example, the yellow and green lines ...
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### 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 ...
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### 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 ...
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### Calculating dV to raise apoapsis at an arbitrary point of an orbit

This is similar to another StackExchange post (right down to the Kerbal Space Program mentions), but with a different approach. The question I'm trying to solve is thus: Given current orbital state ...
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