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Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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1answer
42 views

How can I convert HGI coordinates to J2000 coordinates?

I am looking at old mission data such at this one from Vega 1. The coordinate system provided is HGI. Most of the data I have is in ICRF/J2000.0, as JPL Horizons provides. How can I convert between ...
9
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2answers
266 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). ...
1
vote
1answer
91 views

Ideal shape for a long, skinny reaction mass for LEO to cis-lunar and beyond? (a “space rail gun”)

@Ingolifs' answer got me procrastinating thinking further. What you're asking about is a mass driver space station in low earth orbit that a payload or ship launched from earth can dock with, and ...
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0answers
43 views

How to make satellite ground track on map?

I want to make satellite ground track using matlab. I already had a position of satellite(ECI) by solving Kepler equation. And I know convert ECI to ECEF using rotation determinant. Then I convert ...
5
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2answers
422 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 ...
0
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1answer
147 views

Analytical approximations for the shapes of these plots of ISP vs nozzle diameter? (Raptor engine)

A comment on the currently unanswered question Is there an authoritative catalog for rocket launch identifiers? led me to this Github page which links to r/spacexlounge on which currently appears a ...
2
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0answers
32 views

What is this perturbation effect called, and what would be an analytical expression for the resulting eccentricity oscillations?

In this answer I've generated some oscillations of eccentricity by being somewhat naughty and propagating a TLE for +/-250 days from epoch. I believe it's okay for the purposes of demonstrating gross ...
4
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2answers
184 views

What exactly means universal variable x and z?

I have been studying by myself Orbital Mechanics, and when solving the Lambert Problem, it's common to use the universal variable approach. I understand the algorithm, but I haven't found any book ...
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0answers
53 views

Simplified WGS84 or EGM2008 model for rapid computations?

I am searching for a simplified model of EGM2008 that can be used for rapid, repetitive computations. It needs to be valid only up to GPS altitudes. I'm thinking such a model already exists but cannot ...
123
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2answers
17k 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. ...
2
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1answer
292 views

What is the terminal velocity on Mars?

How would I calculate the terminal velocity of Mars? What is the terminal velocity of a balloon entering Mars' atmosphere? Would a balloon pop if dropped from space?
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1answer
128 views

How can articulated spacecraft components increase the risk of gimbal lock?

This answer has got me baffled. In theory, yes. In practice, no. The problem is gimbal lock... There seems to be some risk of inducing gimbal lock if a spacecraft component of sufficient mass is ...
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0answers
131 views

What is the optimal angle for a solar-sail deorbit towards the Sun when radial thrust is included?

In this answer there's a geometrical derivation of the optimal angle of a solar sail to deorbit a spacecraft into the Sun. The naive answer is 45° which would direct the reflected light directly ...
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0answers
74 views

Lighting an area of the Earth Surface from LEO

What does the math look like to calculate the size of an object in LEO that when angeled off the sun, would illuminate a part of the Earth's surface? For example, if I wanted to 'lightup' a square ...
4
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2answers
133 views

What is the functional form for r(t) for a solar-sail deorbit into the Sun?

This answer describes a spacecraft in a heliocentric orbit using a solar sail to "deorbit" into the Sun by reflecting sunlight into roughly the prograde direction. In comments it was proposed that ...
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0answers
38 views

Aerobraking on Mars

I am looking for a calculator or formulas that can be used to calculate the aerobraking on Mars mission. I have found some data on previous Mars flights where aerobraking was used and all my current ...
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0answers
43 views

Can Lissajous orbits have stable/unstable manifolds?

The question Did DSCOVR travel “along the stable manifold of it's future SE L1 Halo orbit” to get there? is specific to DSCOVR's trajectory from Earth to its primarily heliocentric orbit near Sun-...
3
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1answer
127 views

Calculating the Earth Geopotential effect

I'm calculating the acceleration due to the harmonics in ECEF frame. Gravity potential in spherical harmonics is shown here (I just removed "$1+$", as considering only harmonics effect). $U_{har}=\...
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2answers
141 views

How to prepare for an interview for a Pure Mathematics position at ISRO's Space Application Center?

I have completed a Master's in Pure Mathematics. I have read topics like Abstract Algebra, Linear Algebra, Functional Analysis, Topology, Cryptography, Course in C++, etc. Recently I found an ...
2
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0answers
79 views

Software (besides Ltool) to simulate CR3BP (three body) problems?

Are there any other softwares for CR3BP besides Ltool? As I've learnt here Ltool is not generally available to the public. I would like to simulate three body orbits within the standard circular ...
7
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1answer
221 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 ...
3
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0answers
52 views

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?
2
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1answer
91 views

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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2answers
1k views

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 ...
5
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1answer
230 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 ...
2
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1answer
156 views

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 ...
3
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1answer
396 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 ...
3
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1answer
186 views

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 ...
4
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1answer
195 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....
5
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1answer
167 views

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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0answers
54 views

How many kinds of closed or periodic orbits are there in the circular restricted three-body problem?

Question: 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 there is ...
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0answers
80 views

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 ...
4
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2answers
191 views

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 ...
3
votes
1answer
102 views

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
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1answer
86 views

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 ...
5
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2answers
155 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 ...
2
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3answers
306 views

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 ...
1
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1answer
179 views

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 ...
3
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3answers
128 views

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
1answer
226 views

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 ...
4
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0answers
123 views

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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0answers
66 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 ...
5
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1answer
136 views

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 ...
23
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2answers
5k 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 ...
3
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0answers
282 views

“Deep space” corrections in SGP4; how does it 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 ...
6
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1answer
313 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 ...
5
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1answer
545 views

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) ...
3
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1answer
1k views

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 ...
6
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0answers
409 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....
3
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2answers
605 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 ...