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Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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49 views

How to find instantaneous mass of a central body that results in a given osculating orbit?

This question was underspecified, so I'm going to try again. Osculating orbital elements are Keplerian elements of a hypothetical orbit around a specified center which would be tangent to a specified ...
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1answer
75 views

Analytical treatment of relative positions between satellites?

I have been searching the internet for estimation of relative position between two satellites. However, all sources seem to be pointing to methods used for proximity operations, where the math ...
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0answers
92 views

Why does SpaceIL's Beresheet landing graphics show m\s instead of m/s?

I just noticed that the software displayed in the video of the landing had the velocity and acceleration units with a backslash (m\s) instead of the usual convention of a forward slash (m/s). Out of ...
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2answers
78 views

Given r/a, what are the limits on the direction that an orbiting body could be moving (e.g. solid angle vs r/a)?

An orbiting object at distance $r$ and semimajor axis $a$ will move at $\sqrt{2 - \displaystyle \frac{r}{a}}$ times the speed of a circular orbit at $r$, no matter what the eccentricity or which ...
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1answer
91 views

When is it okay or at least helpful to use energy arguments in orbital mechanics?

When faced with energy-based questions about spaceflight we often comment something like orbits are about momentum, not energy and use delta-v math in answers. But in this answer to How big a nuke ...
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1answer
82 views

How can Earth-Centered Inertial (ECI) coordinates be properly called “inertial” if the Earth's orbit is always accelerating towards the Sun?

I've just realized (again) that I don't understand anything, something that often happens to me after reading a @DavidHammen answer. How can Earth-Centered Inertial (ECI) coordinates be inertial if ...
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2answers
165 views

At a height of 100 km, what speed do you need to be going to escape Earth's gravity?

I understand that the escape velocity of Earth is 11 km/s. However, Earth's gravitational sphere of influence is not infinite, so it is possible to go slower than that and still escape the sphere of ...
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2answers
248 views

What information can I get if an elliptical orbit intersecting a circular orbit of radius 𝑟 and velocity of ellipse is equal to velocity of circular?

How can I approach solving the question below? Consider an elliptical orbit intersecting a circular orbit of radius $r$ as shown below. If $v_{ellipse} = v_{circular} = v $, find: True ...
3
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1answer
130 views

How much less delta-v would it take to reach the Sun using Venus and Earth flyby's compared to direct?

@TomSpilker showed that the minimum delta-v to go from Earth orbit into the Sun is about 20.89 km/s. That's going from 200 km LEO to "perihelion at the sun's photosphere (guaranteed complete entry!)" ...
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2answers
649 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 ...
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1answer
74 views

Understanding line of sight between two satellite with oblate Earth using “scaling k-component”

I have been doing some reading on accuracy of determining line-of-sight between two satellites give a WGS84 oblate earth. The paper Rapid Determination of Satellite Visibility Periods (Alfano, Negron ...
4
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1answer
71 views

How does one impulsively null spacecraft nutation?

The Scott Manley video The Only Pilot to Shoot Down A Spacecraft - A Space Ace talks about the ASM-135 ASAT and its successful test launched from an F-15. He shows images from the page The F-15 ASAT ...
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1answer
265 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? ...
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1answer
82 views

How do you calculate the ∆v required to increase apoapsis to a certain height?

To calculate the ∆v needed to increase our apoapsis from a low orbit of 100km to 400km, we find the velocity at 100km using the semi major axis (which is 250km, in this case), and then find the ...
2
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1answer
113 views

How do you calculate gravity loss (simplified)?

Assume a rocket with constant mass and thrust. To simplify things, we can ignore aerodynamic drag, and we'll also not perform a gravity turn, hence our flight plan is as follows: -Burn radially from ...
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0answers
23 views

How can I “debounce” these TDRS satellite inclinations? (reconstruct the zero-crossings)

The plot below is from the question Why do the geosynchronous TDRS satellites have this distribution of inclinations?, and the explanation says Inclination and mean motion (rev/day) for TDRS ...
4
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1answer
793 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 ...
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0answers
44 views

Mean to Osculating conversion for non-J2 averaged elements

What is the appropriate technique to convert mean elements (true anomaly averaged) into osculating elements when the model has J2, SRP, drag and additional forces modeled. Brouwer's transformation is ...
3
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0answers
51 views

How accurate can a point in space be determined by the likes of the ICRF?

How accurately could a 3D point in open space (i.e. not on an object), say at an elevation of 30,000m above sea level, be located or modelled using the likes of the International Celestial Reference ...
28
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3answers
7k 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?
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0answers
64 views

Can we calculate elliptical properties in a spherical coordinate system?

Can we calculate the elliptical properties like the semi-major axis and eccentricity using the state vectors (position and velocity) in the spherical coordinate system? http://www.braeunig.us/space/...
2
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2answers
130 views

What is the formula for Legendre Polynomials from EGM96 calculated in F447.f program?

The F447.f program is designed to calculate the geoid undulation for EGM96. I'm trying to understand how it accomplish this calculation. I stuck in ...
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1answer
129 views

How were the geodetic and geocentric latitudes of the Space Shuttle defined and calculated?

The image below is cropped from https://i.stack.imgur.com/4XFaJ.jpg as described in this answer. It shows a geographic state of a space shuttle at a given instant in time. There is only one value for ...
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1answer
52 views

What do the axis labels mean in this plot (apparently from Sutton 8th edition)?

In the Analytical Graphics Inc. video Low-Thrust Space Trajectory Design and Optimization the speaker Pradipto Ghosh of AGI shows a slide at around 07:00 shown ...
2
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1answer
59 views

What Methods are Used for Satellite to Site Visibility Calculation?

In the paper, Rapid Satellite-to-Site Visibility Determination Based on Self-Adaptive Interpolation Technique it is claimed that "Satellite-to-site visibility periods are typically determined by the ...
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2answers
82 views

How to find instantaneous positions of a central body of specified mass that results in an osculating orbit?

If I understand correctly, osculating orbital elements such as those posted in JPL's Horizons represent the mathematical osculating (tangential or "kissing") Keplerian orbit about a specified location ...
4
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1answer
69 views

Mechanics of Sun-Synchronous Orbit & North Korea's KMS-4

QUICK QUESTION: Is North Korea's KMS-4 in a proper Sun-synchronous orbit? https://www.n2yo.com/?s=41332 2016-009A FOLLOW-UP QUESTION: How do I tell? I want to understand the mechanics. RESEARCH:...
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1answer
197 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 ...
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1answer
141 views

Help with my tensor tension; how to derive and calculate this rigid body gravity gradient torque?

Tensors make me tense. Imagine a long thin rod in a circular orbit. The gravity gradient will produce a net torque on the rod whenever it is not oriented parallel, or perpendicular to the radius ...
3
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1answer
124 views

Help with Scott Manley's rigid body dynamics narration

I've just watched Scott Manley's video SpaceX's Water Landing Reveals Rocket "Secrets" (or, What We Learned from CRS-16) (again) and there's some rigid body dynamics in there where I'm not sure I am ...
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0answers
89 views

What is wrong with my orbit sim equations? How can I fix them?

Edit: Partial question: The object now moves in an ellipse like it's supposed to, but it orbits around a different point than I intend. How can I make it orbit around a particular object at a point? ...
3
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1answer
66 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
341 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). ...
2
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2answers
164 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
212 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 ...
6
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2answers
709 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 ...
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1answer
282 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 ...
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0answers
105 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
321 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
92 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 ...
134
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2answers
20k 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
510 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
140 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
146 views

How to get from the perturbative acceleration formula to the acceleration on the 3 axis in the case of the oblate Earth, considering j2?

If we start from the pertrubative acceleration formula, how do we get the expression for the acceleration for the 3 axis written under that form. I've used the partial derivative for the gradient ...
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0answers
169 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
79 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
167 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 ...
2
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0answers
76 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 ...
2
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0answers
98 views

Do 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
159 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}=\...