Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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1answer
79 views

Implementation of NASA Breakup model Area to Mass distribution

In the research "NASA's new breakup model of evolve 4.0" the implementation for the area to mass distribution of fragments produced by collisions and explosions is provided. The distribution ...
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108 views

Kepler prediction problem, fundamentals of astrodynamics

I needed help with problem 4.3 of Fundamentals of Astrodynamics by Roger R. Bate, Donald D. Mueller, Jerry E. White (dover, archive.org) Given that $r_0 = (I+j)$ and $v_0 = 2k$ find $r$ and $v$ for ...
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4answers
357 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)
2
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2answers
135 views

Are there some three-body orbits that can't be escaped? Can we know without propagating forever?

update: Searching for "choreographies" I found this Physics SE question which is related but different because it asks only if periodic solutions can be proven to be periodic numerically and my ...
21
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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 ...
6
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1answer
229 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 ...
3
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1answer
81 views

How do you calculate the percentage of light-speed a ship with a certain delta-v can achieve, accounting for relativistic effects?

I was doing some back-of-a-postage-stamp calculations on high-isp interstellar spacecraft earlier and realised something. If I design a spaceship which according to the rocket equation has a very high ...
3
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2answers
334 views

Why is the bend angle of a hyperbolic trajectory giving different results?

The bend angle ($\delta$) is the reflex angle between the two asymptotes of a hyperbolic trajectory. From this Stack Exchange answer, the formula to calculate the bend angle is: $$ \delta = 2 \sin^{-...
3
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1answer
99 views

Different results for Orbital Period?

I am making a hypothetical solar system with random masses and distances (but still physically and mathematically accurate). I want to calculate the trajectory of a spacecraft, and to do that, I need ...
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0answers
54 views

How to get lunar L, B, C parameters from the Moon's 3x3 rotation matrix from the Python package Skyfield?

Answers to What are the “Moon L, B, C” angles shown in this solar eclipse simulation? explain that L and B (or "l" and "b") are the "selenographic coordinates of the Moon's apparent center" (latitude ...
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2answers
119 views

Is a polar geosynchronous orbit described by Viviani's curve?

This answer to Is there a name for the great circle where latitude and longitude are equal? says (in part): More generally, a clélie is the name given to any spherical curve where the longitude $\...
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1answer
207 views

How did NASA calculate 22 million “rocket horsepower” for the SLS Booster?

This answer to Does NASA really report the power of rockets in horsepower? cites NASA news item Space Launch System Booster Aimed and Ready to Fire which says "What's impressive about this test is ...
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1answer
240 views

Are hyperbolic trigonometric functions used to calculate hyperbolic orbits?

The following comment is really intriguing! I would say, "Yes, The equations for going from Mean Anomaly to Eccentric Anomaly to True Anomaly are indeed different for hyperbolic orbits than for ...
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51 views

Help on Rocket Injector Equation

Hello can someone explain me how can I get the value of (6340) because when I try to solve the given values on the book I get (6423.980075) when I round it off it is still not the right answer can ...
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3answers
329 views

How to solve this exercise in orbital mechanics; the moon stops in its orbit, what happens next?

As an exercise in orbital mechanics, suppose the moon were to suddenly stop orbiting Earth, and instead at time $t=0$ begin moving in the same direction as Earth so that their relative velocity were ...
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2answers
217 views

What does Earth's J₂₂ moment look like?

Wikipedia's Geopotential model give's the form of the potential component linked to $J_2$ as $$u = J_2\frac{P^0_2(\sin \theta)}{r^3} = J_2\frac{1}{r^3}\frac{1}{2} \left(3 \sin^2\theta-1 \right) = ...
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1answer
88 views

Do Lagrange points still exist if there is significant radiation pressure on the third body from the first?

From this answer: To obtain the distance to L1, find the smallest value of $r$ such that $$\frac{M_2}{r_1^2} + \frac{M_1}{R^2} - \frac{r_1(M_1+M_2)}{R^3} - \frac{M_1}{(R-r_1)^2} = 0.$$ To obtain the ...
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1answer
53 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
110 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
96 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
87 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
99 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
187 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
290 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
364 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 $, find: True anomaly ($\...
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1answer
160 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
749 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
127 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 ...
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1answer
74 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
302 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
149 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 ...
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1answer
250 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
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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 ...
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1answer
1k 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
56 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 ...
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0answers
53 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 ...
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3answers
8k 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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83 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/...
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2answers
143 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
183 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
54 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 ...
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1answer
69 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 ...
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2answers
98 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
80 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
204 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
170 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
130 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
91 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
75 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 ...
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2answers
358 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). ...