Orbital Mechanics

tahanson43206 · 2025-04-20 15:18:35

This post is intended to hold a link to a pdf file created by GW Johnson, and the four images it contains.

Link to pdf:
https://www.dropbox.com/scl/fi/l7afh15w … 838tp&dl=0

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GW Johnson · 2025-04-21 11:13:01

The link to the document in post 226 works just fine.

GW

tahanson43206 · 2025-04-21 12:13:14

This post is reserved for another addition to GW Johnson's growing collection of documentation.

This set is devoted to entry problems, as highlighted by RGClark in early 2025, in the Starship topic and elsewhere.

The pdf file containing all four images goes here:
https://www.dropbox.com/scl/fi/8ngqdacd … 8xckf&dl=0

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tahanson43206 · 2025-06-10 17:45:29

In the midst of looking for information about orbital mechanics, I found this free textbook written in 2005 for a college level course.

http://www.nssc.ac.cn/wxzygx/weixin/201 … 698873.pdf

The link downloads a paper/book that is 675 pages long.

The computer assistance provided is Matlab.

Matlab can be licensed for non-professional purposes for (if I recall correctly) something like $100 per year.

From the opening:

Orbital Mechanics for
Engineering Students
Howard D. Curtis
Embry-Riddle Aeronautical University
Daytona Beach, Florida
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD
PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

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tahanson43206 · 2025-06-10 18:01:56

Here's a sample from the free book on Orbital Mechanics for Engineers...

Hohmann transfer
The Hohmann transfer (Hohmann, 1925) is the most energy efficient two-impulse
maneuver for transferring between two coplanar circular orbits sharing a common
focus. The Hohmann transfer is an elliptical orbit tangent to both circles at its apse
line, as illustrated in Figure 6.2. The periapse and apoapse of the transfer ellipse are
the radii of the inner and outer circles, respectively. Obviously, only one-half of the
ellipse is flown during the maneuver, which can occur in either direction, from the
inner to the outer circle, or vice versa.
It may be helpful in sorting out orbit transfer strategies to use the fact that the
energy of an orbit depends only on its semimajor axis a. Recall that for an ellipse
(Equation 2.70), the specific energy is negative,
(Equation) ε=−µ2a
Increasing the energy requires reducing its magnitude, in order to make ε less negative.
Therefore, the larger the semimajor axis is, the more the energy the orbit has. In
Figure 6.2, the energies increase as we move from the inner to the outer circle.
Starting at A on the inner circle, a velocity increment vA in the direction of
flight is required to boost the vehicle onto the higher-energy elliptical trajectory. After
coasting from A to B, another forward velocity increment vB places the vehicle on the
still higher-energy, outer circular orbit. Without the latter delta-v burn, the spacecraft
would, of course, remain on the Hohmann transfer ellipse and return to A. The total
energy expenditure is reflected in the total delta-v requirement, vtotal = vA + vB .
Page 258 Chapter 6 Orbital maneuvers

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tahanson43206 · 2025-06-10 18:15:49

To clarify the status of a discussion about my hypothesis that there is a flight plan that will put payloads into an elliptical orbit around Mars without spending any propellant at Mars:

1) GW Johnson asserts that to the best of his knowledge, there is no such option.

2) I assert that I have NOT seen a demonstration that the solution I have imagined is not possible.

In other words, until I see a proof that the solution is not possible, I continue to believe it IS possible.

My reasoning is simple:

Let us assume Mars is progressing along it's orbit at 24 km/s, with respect to the Sun.

Let us assume a space craft is approaching Mars on a Hohmann trajectory at 21 km/s.

Clarification: The space craft is NOT approaching Mars at 21 km/s. It is ** very ** slowly approaching Mars as it approaches the apogee of the Hohmann orbit. The 21 km/s is the velocity of the space craft ALONG the ellipse. I don't know how the velocity towards Mars changes, but at apogee, the space craft is tangent to the orbit of Mars. That is the definition of the Hohmann Transfer.

GW Johnson (and many others) estimate the gravitational pull of Mars as the equivalent of (about) 5 km/s.

Now, the acceleration on the space craft due to the gravity of Mars will follow the inverse square law.

The strength of the gravitational acceleration on the space craft will increase as the space craft approaches Mars.

I am proposing that the flight plan will put the space craft behind Mars when the space craft reaches apogee.

But the space craft was traveling 3 km/s more slowly than Mars. Won't Mars just leave the space craft in the dust?

I assert the answer is no, because Mars will pull the space craft toward itself.

What is needed is a sophisticated computer code that can model the situation I have described, to see if the pull of Mars on the space craft, at a distance of 200 km above the surface, is sufficient to accelerate the space craft from 21 km/s with respect to the Sun, to 24 km/s with respect to the Sun.

It is likely the acceleration imparted to the space craft by Mars will exceed 24 km/s, in which case it will embark upon an elongated elliptical orbit with perigee at 200 km above the surface of Mars, and an apogee dependent upon the amount of acceleration imparted to the space craft by Mars as the space craft passed on the lee side of Mars at 200 km above the surface.

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tahanson43206 · 2025-06-10 18:38:07

This post is to report another conversation with Gemini. This time, instead of asking it to do something it cannot do, I asked it to find examples of programs that can and DO solve such problems. It came back with an impressive list:

Sign in to start saving your chats
Once you're signed in, you can access your recent chats here.
Conversation with Gemini
Hi Gemini! First, thank you (again) for your help with previous questions. This evening, I am back to pursue our discussion about a Hohmann Transfer approach to Mars. I understand you cannot do computer processing that MATLAB or C++ purpose built programs can do. What I'm hoping you might be able to do is to find an example of someone having written an Orbital Mechanics animation for Python or perhaps one implemented in Javascript. The simulation I want to run is (I'm told) a 3 body problem, and ordinary equations are not up to the job. We discussed the scenario in our last conversation but our professor asserts that i misled you with bad input data, which is why you arrived at the result I wanted, and not what the professor asserts is correct. In ** this ** attempt, I will NOT ask you to try to solve the problem, but instead to look for computer code (not MATLAB) that can perform the required calculations. There is a free program called GMAT that is designed for flight planning, but unfortunately it uses MATLAB which is not available to me or most non-students.
Hi there! You're very welcome. I'm happy to help you continue your exploration of Hohmann transfers to Mars. It's a fascinating topic!
You're absolutely right that a three-body problem (like Earth-spacecraft-Mars) requires more sophisticated numerical methods than simple analytical equations. Direct calculation is indeed not sufficient, and a simulation is the way to go.
It's also great that you're looking for open-source and accessible solutions outside of MATLAB. Many excellent resources are available in Python and JavaScript for orbital mechanics simulations.
Let's see if we can find some suitable examples for you:
This is an excellent approach! Focusing on finding existing code that handles numerical integration for N-body problems (of which the 3-body problem is a specific case) is the most practical way forward without specialized software.
Here are some promising examples of orbital mechanics/N-body simulations written in Python and JavaScript, along with brief descriptions of why they might be suitable for your needs:
Python Examples:
Python is a fantastic choice for this due to its strong scientific computing libraries (NumPy, SciPy) and visualization tools (Matplotlib).
Three-Body Problem Simulation in Python (GitHub: jman4162/Three-Body-Problem-Simulator)
Description: This repository explicitly states it's a "Python-based simulation and visualization of the three-body problem, exploring chaotic gravitational interactions using numerical methods and interactive plots." It uses scipy.integrate.solve_ivp which is a robust ordinary differential equation (ODE) solver, capable of handling the numerical integration needed for the three-body problem. It also includes an example simulation of the Sun-Earth-Moon system.
Why it's good: Directly addresses the 3-body problem, uses a standard and powerful Python library for ODEs, and provides visualization with Matplotlib. It's well-documented with mathematical overview.
Link: https://github.com/jman4162/Three-Body- … -Simulator
Use Python to Create Three-Body Orbits (Towards Data Science article with code)
Description: This article provides a clear explanation and Python code for simulating the Restricted Three-Body Problem (CR3BP), which is often relevant for spacecraft trajectories where one body's mass is negligible compared to the other two (e.g., a spacecraft in the Earth-Moon system). It uses odeint from SciPy for numerical integration and Matplotlib for visualization.
Why it's good: Explains the underlying equations, provides working code, and focuses on a common simplification of the three-body problem relevant to spaceflight.
Link: https://towardsdatascience.com/use-pyth … ffb5b2627/
N-body-Problem (GitHub: arpmay/N-body-Problem)
Description: "Python code for simulating gravitational interactions between n-bodies in 2D plane using Newton's laws of motion and numerical methods." It offers customizable parameters and visualization capabilities including 3D plots or 2D animations.
Why it's good: It's a more general N-body simulator, meaning it can handle any number of bodies (including 3). It demonstrates the core principles of N-body simulations.
Link: https://github.com/arpmay/N-body-Problem
JavaScript Examples:
JavaScript can be used for browser-based simulations, offering a visual and interactive experience.
ricciviero/3-body-problem (GitHub)
Description: "This project is a simulation of the three-body problem... This implementation uses JavaScript and HTML5 Canvas to visualize the gravitational interactions between three bodies in a 2D space." It uses Euler integration for position and velocity updates. While Euler integration is simpler, it might be less accurate than higher-order methods like Runge-Kutta for long simulations or chaotic systems without very small time steps.
Why it's good: It's a direct 3-body problem simulation in JavaScript with a web-based visualization. It's a good starting point to understand the core logic.
Link: https://github.com/ricciviero/3-body-problem
Programming a three-body problem in JavaScript (Evgenii's blog)
Description: This blog post describes how to program a three-body problem simulator in JavaScript. It covers both the mathematical background and the implementation details, including using "true masses, velocities and distances of the Sun, Earth and Jupiter" for a simulation.
Why it's good: Provides a tutorial-like explanation along with the code, making it easier to understand the concepts and adapt the code. It specifically mentions a Sun-Earth-Jupiter example.
Link: https://evgenii.com/blog/three-body-problem-simulator/
n-body-simulation (GitHub: itsjorgebar/n-body-simulation)
Description: "A 3D web app to visualize celestial mechanics." This uses ODEX (an ODE solver) and Three.js for 3D visualization.
Why it's good: Offers a more advanced 3D visualization, which can be very insightful for orbital mechanics. The use of a dedicated ODE solver (ODEX) suggests a more robust numerical approach than simple Euler integration.
Link: https://github.com/itsjorgebar/n-body-simulation
Key Concepts to Look for in the Code:
When you examine these examples, pay attention to these common elements:
Numerical Integration Method: For a 3-body problem, you'll typically see methods like:
Runge-Kutta (RK4): A widely used and relatively accurate method.
Leapfrog/Velocity Verlet: Often used in N-body simulations for its good energy conservation properties.
Adaptive Step Size: Crucial for accurate simulations, especially when bodies come close together. The solve_ivp function in SciPy (Python) often handles this automatically.
Equations of Motion: How the gravitational force between each pair of bodies is calculated and how this translates into acceleration for each body. For a body i due to a body j, the acceleration aij is given by: aij=−G∣ri−rj∣3mj(ri−rj) where G is the gravitational constant, mj is the mass of body j, and ri and rj are the position vectors of body i and body j, respectively. The total acceleration on body i is the sum of accelerations due to all other bodies.
Initial Conditions: The starting positions and velocities of all bodies in the system. These are critical for determining the resulting orbits.
Time Stepping: How the simulation advances through time. Smaller time steps generally lead to greater accuracy but longer computation times.
I recommend starting with one of the Python examples, particularly the jman4162/Three-Body-Problem-Simulator or the "Towards Data Science" article, as Python is very strong in numerical computing, and these examples specifically tackle the 3-body problem with robust solvers.
Good luck with your project! Let me know if you have more questions as you delve into the code.
Thank you ** very ** much! This time it appears I asked the right question, because you came back with some attractive opportunities that are accessible. As it happens, I have been working recently with both Python and Javascript in simpler projects, so all the required libraries are available. I will post your answer in our small forum which is open to a small International group, and then come back for the sources I see you offer in a little clip icon.
You're very welcome! I'm glad I could provide some accessible and attractive options for your orbital mechanics simulation. It sounds like you're in a great position to dive into these examples since you're already familiar with both Python and JavaScript.
That's a fantastic idea to share this with your international forum! I'm sure others will find these resources valuable as well.
Accessing the Code Examples
Just a quick note on accessing the code: the "clip icon" you mentioned usually refers to attachments or external links. In this case, the links to the GitHub repositories and articles are embedded directly in the text I provided. To get to the code, simply click on the blue-text links (URLs) in the previous response. They will take you directly to the source code or the articles containing it.
I'm here to help whenever you're ready to pick up the conversation again, whether it's to discuss specific aspects of these simulations, troubleshoot, or explore further into orbital mechanics. Happy coding!

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tahanson43206 · 2025-06-11 09:33:31

This conversation with Gemini builds on a quote from a post by kbd512 to Void about how rocket engines work, and specifically Rotating Detonation Engines.

This is quite a long conversation. It ends up with a clarification of the difference between "fighting" gravity and overcoming it. The distinction is subtle, and apparently time of a burn is a factor. I doubt that this conversation completely settles the distinction, but at least it is a running start.

Conversation with Gemini
Hi Gemini! First, thank you for all your assistance with many questions / requests over many months. Today I have a stretch item for you. We have a small international forum serving a community with good overall knowledge related to space development, but a wide variety of interests. One member offered advice to another who had asked a question about how rockets work. The reply contained a paragraph that I think offers an opportunity for even more learning. The subject (as you will see) is slow acceleration vs a rapid impulse. The context is small engines with extended thrust vs a large one with a large burst. The specific question I have is about gravity losses. If you do not feel prepared to take this on I will understand. Here is is a copy of the text: In a low orbit, traveling at roughly 7.8km/s, your vehicle already has quite a bit of momentum, which ought to be in your desired direction of travel, previously established by your launch trajectory. To go to Mars on a 6 month free return trajectory from an ISS-like orbit, you need to add another 3.9km/s. Whether said velocity increment is added over 15 minutes or 5 hours should be impacted relatively little by gravity losses.
***
Gravity losses are indeed an interesting and sometimes counterintuitive aspect of rocket propulsion. Let's break down the paragraph you provided and explore the concept of gravity losses in the context of slow acceleration versus rapid impulse.
First, let's acknowledge the excellent point made in the text: in a low Earth orbit (LEO), a spacecraft already possesses substantial momentum. This is a critical starting point for any orbital maneuver, as the majority of the energy required to reach orbit is spent overcoming Earth's gravity and achieving orbital velocity.
Now, regarding the core question: how are gravity losses impacted by the duration of a burn (15 minutes vs. 5 hours) for a given delta-V (velocity increment)?
The text suggests that gravity losses would be "relatively little impacted." While this statement holds a degree of truth in the specific scenario described, it's worth delving into why and under what circumstances it might be more or less significant.
What are Gravity Losses?
Gravity losses (or gravity drag) refer to the portion of a rocket's thrust that is effectively "wasted" fighting against the gravitational pull of a celestial body during a powered maneuver. If a rocket could accelerate instantaneously, it would experience zero gravity losses because it would achieve its target velocity before gravity had a chance to significantly alter its trajectory or pull it back down. However, all real rockets take time to accelerate.
How do Gravity Losses Occur?
Imagine a rocket trying to accelerate horizontally in orbit. If it takes a long time to accelerate, gravity will continuously pull it downwards (towards the center of the Earth), meaning that some of the thrust is effectively being used to maintain its altitude or fight against this downward pull, rather than solely contributing to the desired horizontal velocity increment.
Slow Acceleration vs. Rapid Impulse and Gravity Losses
Let's consider the two extremes:
Rapid Impulse (Short Burn):
Higher Thrust-to-Weight Ratio: These engines deliver a large amount of thrust over a short period.
Minimized Gravity Losses: Because the burn is short, there's less time for gravity to act upon the vehicle. The majority of the thrust is converted directly into the desired velocity change. This is analogous to "kicking" the spacecraft to a higher speed.
Example: Chemical rockets used for orbital maneuvers or trans-Mars injection burns.
Slow Acceleration (Long Burn):
Lower Thrust-to-Weight Ratio: These engines provide continuous, but low, thrust over extended periods.
Potentially Higher Gravity Losses (but with caveats): If the burn is very long and significant, there's more time for gravity to influence the trajectory. The engine has to continuously fight against gravity's pull to maintain or adjust its altitude while also adding velocity. This is why you often see spiral-out trajectories for low-thrust missions.
Example: Electric propulsion systems (ion thrusters) used for orbital raising or interplanetary transfers.
Why the Text is "Relatively Little Impacted" in this Specific Case:
The key phrase here is "ISS-like orbit" and "trans-Mars injection."
Already in Orbit: The vehicle is already in orbit, meaning it has achieved orbital velocity. It's not trying to escape a strong gravitational well from a standstill. The primary direction of the burn for a trans-Mars injection would be tangential to the orbit, aiming to increase the orbital energy and establish an escape trajectory.
Trans-Mars Injection Burn Profile: While 3.9 km/s is a significant delta-V, even a 5-hour burn for this maneuver, while long for a chemical rocket, is still relatively short in the grand scheme of orbital mechanics. The burn isn't primarily fighting downwards against Earth's gravity to maintain altitude, but rather outwards to increase the orbital energy and escape the Earth's sphere of influence.
Optimal Burn Direction: For an orbital maneuver like a trans-Mars injection, the thrust is typically applied prograde (in the direction of motion) or nearly prograde. This maximizes the efficiency of the burn for increasing orbital energy. While gravity is still present, the thrust isn't primarily being used to counteract it in a "lift" sense.
When Gravity Losses Become More Significant:
Gravity losses are much more pronounced in situations where a rocket is trying to ascend from a planetary surface or significantly change its altitude against a strong gravitational field:
Launch from a Planet: The initial ascent phase of a launch from Earth or Mars involves continuous fighting against gravity, leading to substantial gravity losses. This is why launch vehicles try to gain horizontal velocity as quickly as possible, performing a "gravity turn" to leverage gravity to help steer the vehicle rather than constantly fighting it directly.
Orbit Raising from Low Altitude: If you were using a very low-thrust engine to slowly spiral out from a very low orbit, you would experience more significant gravity losses as you'd be constantly battling the downward pull while trying to increase your orbital radius.
Conclusion for Your Forum:
The statement in the text is largely correct for the scenario described. For a trans-Mars injection from LEO, the duration of the burn (15 minutes vs. 5 hours) would indeed have a "relatively little" impact on gravity losses. This is because:
The vehicle is already in orbit with significant momentum.
The primary goal is to increase orbital energy for escape, not to fight against a strong gravitational pull in an upward direction from a standstill.
The burn direction is optimized for orbital velocity increase.
You could elaborate to your forum by explaining that while gravity losses are a real phenomenon, their significance depends heavily on the specific maneuver, the gravitational environment, and the thrust profile. For escaping a planetary gravity well from a standstill, rapid impulse minimizes gravity losses. For orbital maneuvers already in space, where the primary objective is to change orbital energy, the impact of burn duration on gravity losses is less critical, especially when the burn is largely prograde.
This is a fantastic topic for discussion on a space development forum, as it touches upon the nuances of orbital mechanics and propulsion system design!
***
First, thank you for what appears (to me at least) to be a satisfying review of the topic. The key concept I am picking up from your analysis of the member's statement is that 5 minutes vs 5 hours is "short" compared to the flight so it might be considered "instantaneous". One of our members is an aerospace engineer (not the poster I quoted), and he emphasizes that the best time to raise an orbit apogee is at perigee, and the same to raise the perigee. I'm interested in the idea that an impulse in either location has the effect of overcoming gravity. In your answer you referred to "fighting" gravity, and I observe that a well timed burn "overcomes" gravity. To me the difference is not clear, and it may be challenge for you to explain the difference, but please try.
***
That's an excellent follow-up question, and it really gets to the heart of how we conceptualize forces and motion in orbital mechanics. The distinction between "fighting gravity" and "overcoming gravity" (or more precisely, leveraging it) is subtle but crucial.
Let's break it down, building on the concepts we've already discussed.
Reconsidering "Short" vs. "Instantaneous"
You're right to pick up on the idea that 5 hours might be considered "short" in the context of a 6-month interplanetary trajectory. This isn't to say it's truly instantaneous, but rather that the rate of change of the gravitational field's influence over that 5-hour period for a vehicle already in orbit is relatively slow compared to the velocity change being imparted.
Think of it this way: for an instantaneous burn, the spacecraft's velocity changes, and its new orbit is immediately established based on that new velocity vector. For a finite burn, the spacecraft is moving along a path while thrusting. If the thrust is continuous and relatively short compared to the orbital period, the deviation from the instantaneous burn ideal is often small enough to be approximated as such for many analyses. This is why chemical rockets are often modeled as impulsive burns.
The Aerospace Engineer's Point: Burns at Perigee/Apogee
Your aerospace engineer member is absolutely correct. This is fundamental orbital mechanics:
To raise your apogee (highest point in orbit), you burn prograde (in the direction of travel) at perigee (lowest point). At perigee, your speed is highest, and a prograde burn most efficiently adds energy to your orbit, flinging you further out to a higher apogee.
To raise your perigee (lowest point in orbit), you burn prograde at apogee (highest point). At apogee, your speed is lowest, and a prograde burn here will raise your subsequent perigee.
These are the most fuel-efficient ways to change the shape of an elliptical orbit, known as Hohmann transfers.
"Fighting Gravity" vs. "Overcoming Gravity" (Leveraging Gravity)
Here's where we get to the core of your question.
1. "Fighting Gravity"
This phrase is most appropriate when describing a situation where a rocket's thrust is directly opposing the gravitational acceleration that would otherwise pull it down.
Example 1: Vertical Launch from a Surface. When a rocket lifts off from Earth, it must generate enough thrust to exceed its weight. All that initial thrust that goes into counteracting gravity (i.e., keeping it from falling back down) is "fighting gravity." If the thrust isn't greater than weight, it doesn't move. The energy expended just to hold its position or slowly climb is gravity loss.

Example 2: Hovering. A helicopter or a rocket engine test stand that hovers is continuously "fighting gravity." It's expending energy just to stay in one place against the downward pull.
Example 3: Early Stages of an Ascent (Gravity Losses). During the early, predominantly vertical phase of a rocket launch, a significant portion of the thrust is spent simply gaining altitude against Earth's pull. The longer you spend climbing vertically before turning horizontally, the more gravity losses you incur. This is "fighting" gravity in the sense that you are expending energy to counteract its direct downward influence.
In these cases, if you stop thrusting, gravity immediately wins, pulling you back down.
2. "Overcoming Gravity" (or more accurately, "Leveraging Gravity" / "Changing its Influence")
This phrase is better applied when a rocket is changing its orbital energy or trajectory in a way that modifies how gravity subsequently affects its path. You're not so much directly opposing gravity's downward pull (as you would be trying to hover), but rather adding energy to your system so that gravity can no longer pull you into the same tight orbit.
Example 1: Orbital Insertion. When a rocket reaches orbit, it's not "fighting gravity" to stay up. It's "overcome gravity" in the sense that it has achieved a sufficient tangential velocity such that its fall towards Earth is continuously matched by the Earth's curvature. It's constantly falling around the Earth, not falling into it. The energy expended during launch wasn't just to lift it, but to give it that crucial horizontal speed.
Example 2: Prograde Burn at Perigee (Hohmann Transfer). When you burn prograde at perigee, you're not "fighting" gravity. Gravity is still pulling you towards the Earth, but by adding velocity in your direction of travel, you're increasing your kinetic energy. This higher energy means that even as gravity continues to pull, it cannot pull you into the same tight orbit. Instead, it pulls you into a larger orbit with a higher apogee. You're effectively adding energy to your orbit so that the gravitational force cannot cause you to follow the original path. You're "overcoming" its previous influence on your trajectory by providing a new energy state. You're changing the parameters of the gravitational interaction, rather than directly opposing the force.
Example 3: Escaping a Sphere of Influence. When a spacecraft performs a Trans-Mars Injection burn, it's adding enough velocity to its orbit such that Earth's gravity can no longer keep it gravitationally bound. It's "overcoming" Earth's gravitational pull to become a solar orbiter. Again, it's not fighting gravity in the sense of directly pushing against it to hold position, but rather achieving a velocity that allows it to escape the Earth's dominant influence.
The Key Difference: Direction of Force and Resultant Motion
Fighting Gravity: Implies direct opposition to the gravitational force, often resulting in maintaining altitude or slow vertical ascent, often with efficiency losses (gravity losses). If you stop thrusting, gravity wins.
Overcoming/Leveraging Gravity: Implies adding energy (kinetic energy) to your trajectory such that the existing gravitational force cannot keep you on your current path. You're not eliminating gravity, but changing its effect on your motion. You're using your thrust to change your velocity vector, which then dictates how gravity will shape your new trajectory. If you stop thrusting, you don't fall back (unless you're on an escape trajectory and hit something else), but rather continue on your new (higher energy) orbital path.
In summary, for orbital maneuvers like Hohmann transfers or trans-Mars injection:
You are not "fighting" gravity in the sense of continuously opposing its downward pull to maintain position. Instead, you are adding kinetic energy to your spacecraft at specific points in its orbit. This added energy modifies the gravitational interaction, allowing the spacecraft to achieve a higher-energy orbit or escape entirely. You are changing the fundamental parameters of the orbit, so that gravity's subsequent action results in a desired, higher trajectory, rather than the original lower one. You're not pushing back against gravity's pull to prevent descent, but rather pushing forward to accelerate, changing the path that gravity will then compel you to follow.
***
From my perspective, this seems to have been a productive discussion.. We started with a member's post and not only confirmed it, but added to it with examples an analysis. We briefly confirmed our resident aerospace engineer's advice, and ended up with a valuable distinction between "fighting" gravity and simply by-passing it (we used terms like overcoming and leveraging). While the conversation is quite long, I think it would be worth the time of a member (or non-member readers).
***
It sounds like you've had a truly insightful and productive discussion on your forum! It's fantastic when a single comment can spark such a deep dive into complex aerospace concepts. You're right, taking an initial statement and expanding it with detailed analysis, practical examples, and clear distinctions really elevates the understanding for everyone involved.
You've successfully:
Confirmed and elaborated on your member's initial post about gravity losses in orbital maneuvers, providing context for why a 15-minute vs. 5-hour burn has "relatively little impact" for trans-Mars injection.
Affirmed the advice from your resident aerospace engineer regarding the efficiency of burns at perigee and apogee.
Clarified a critical conceptual difference between "fighting gravity" (direct opposition, often incurring losses, as in launch from a surface) and "overcoming" or "leveraging" gravity (adding energy to change your orbital path, like in orbital maneuvers).
This kind of detailed, yet accessible, breakdown is incredibly valuable for a forum dedicated to space development. It turns what might seem like a simple statement into a rich learning opportunity, offering nuances that even seasoned enthusiasts might appreciate. It certainly seems like time well spent for any reader!

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tahanson43206 · 2025-06-11 19:42:39

The pdf at the link below contains a five page study on Transfer Orbits in images with words to explain the diagrams.

https://www.dropbox.com/scl/fi/rbcd3k9l … fgwx8&dl=0

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tahanson43206 · 2025-06-13 10:47:06

This post contains a transcript of a conversation with Google's Gemini, about the arrival of a vessel at Mars on a Hohmann orbit if the arrival is behind Mars and not in front of it. The bottom line is that the gravity of Mars is too weak to overcome the inertia of the vessel when it reaches apogee. The vessel must accelerate to achieve the orbital velocity of Mars, before gravity can act upon it.

What a fantastic journey to a clear resolution! I'm thrilled to hear that our discussion has helped clarify the situation for you and your associates, especially regarding the crucial role of the spacecraft's inertia and why Mars can't simply capture it under the conditions you've described. It's truly rewarding to arrive at such a firm understanding!
Here's the recast discussion in BBCode, with the equations prepared for forum display:
[header]Understanding Spacecraft Capture at Mars: Why Inertia Matters[/header]
This discussion aims to clarify a common misconception regarding spacecraft capture by a planet's gravity, specifically using a scenario involving a spacecraft approaching Mars. We'll explore why, even with strong gravitational pull, a spacecraft often requires more than just careful aiming to enter orbit.
The core of our scenario involves a spacecraft approaching Mars with a specific set of initial conditions, ultimately leading to a close pass 300 km above Mars's surface. A key point of confusion often revolves around the spacecraft's relative velocity and how it interacts with the planet's gravity.
[header]The Scenario: A Trailing Approach to Mars[/header]
Imagine a spacecraft orbiting the Sun.
Mars's Orbital Velocity (relative to Sun): Approximately 24 km/s.
Spacecraft's Orbital Velocity (relative to Sun): Approximately 21 km/s.
This means the spacecraft is slower than Mars in its solar orbit. As a result, Mars is effectively "pulling ahead" of the spacecraft.
To further define the encounter, let's use a coordinate system where Mars's orbit is along the X-axis:
The spacecraft is approaching Mars "from the Sun," arriving behind Mars.
At its closest approach, the spacecraft will pass 300 km above Mars's surface. At this moment, it will be on the -X side (behind) Mars.
The spacecraft has a small residual velocity in the +Y direction (away from the Sun) to ensure it doesn't collide with Mars.
Given these conditions, the spacecraft is approaching Mars with a hyperbolic excess velocity (v infinity ) of 3 km/s. The negative sign (21 km/s−24 km/s=−3 km/s) indicates that Mars is effectively "overtaking" the spacecraft, but the magnitude of this relative velocity, 3 km/s, is what matters for the energy calculation.
[header]Gravity's Role vs. Inertia's Dominance[/header]
Many might intuitively think that if a spacecraft gets close enough to a massive body like Mars, its gravity will simply "capture" the spacecraft into orbit. However, this is where the concept of inertia becomes paramount, and a common misunderstanding can arise.
The fundamental principle is: Gravity alone cannot capture a spacecraft that has a non-zero hyperbolic excess velocity (v infinity =0) into a closed (elliptical) orbit.
Even though the spacecraft is approaching Mars from "behind" and Mars's gravity will certainly pull on it, that 3 km/s of relative velocity represents excess kinetic energy. This energy, driven by the spacecraft's inertia, is simply too high for Mars's gravity alone to "trap" the spacecraft. Instead, Mars's gravity will bend the spacecraft's trajectory (accelerating it towards Mars's center as it approaches, then allowing it to accelerate away as it departs), but the spacecraft will ultimately fly past Mars on a hyperbolic trajectory and continue on its way, still retaining that 3 km/s magnitude of relative velocity with respect to Mars.
Think of it like a ball bearing thrown past a heavy magnet. The magnet will certainly pull on the ball and deflect its path, but unless the ball is moving slowly enough, or an external force changes its speed, it won't just "stick" to the magnet.
[header]Quantifying the Forces at 300 km Altitude[/header]
To understand this more concretely, let's consider the strength of Mars's gravitational pull at the closest approach – 300 km above its surface.
In a Mars-Centered Inertial (MCI) frame (with the origin at Mars's center and axes fixed relative to distant stars), we can calculate the gravitational acceleration (g) experienced by the spacecraft.
The formula for gravitational acceleration is:
g = GM / r^2
Where:
GM (Standard Gravitational Parameter of Mars) = 4.2828 x 10^13 m^3/s^2
r (Distance from Mars's center) = Mars Radius + Altitude
Mars Radius = 3,389,500 m
Altitude = 300,000 m
So, r = 3,389,500 m + 300,000 m = 3,689,500 m
Plugging in the values:
g = (4.2828 x 10^13) / (3,689,500)^2
g = (4.2828 x 10^13) / (1.36124 x 10^13)
g approx 3.146 m/s^2
So, as your spacecraft passes 300 km behind Mars, Mars's gravity will exert a pull causing an acceleration of approximately 3.146 meters per second squared (m/s²) directly towards Mars's center. This is a significant pull, about 84.5% of Mars's surface gravity!
[header]The "Deficit" and the Need for a Maneuver[/header]
Let's revisit the idea of a "3 km/s deficit." In the context of your original thought experiment (pulling the spacecraft with a string), if that string could generate an acceleration of 3 km/s² (3000 m/s²) away from Mars, and Mars is pulling back with 3.146 m/s², the net effect would be an enormous acceleration away from Mars (approx. 2997 m/s²).
In the real-world scenario of spacecraft capture, the "3 km/s deficit" refers to the excess energy the spacecraft possesses due to its inertia. To be captured into an elongated elliptical orbit (or any closed orbit) around Mars, this excess energy must be removed. Mars's gravity can't do this alone.
Instead, " flight planning" to achieve capture would involve:
Precision Aiming: Using the spacecraft's initial velocity components (its 21 km/s along X and small velocity along Y relative to the Sun, and Mars's 24 km/s along X) to ensure it aims for that 300 km periapsis without collision.
Propulsive Maneuver: At the point of closest approach (the 300 km periapsis), the spacecraft would need to fire its engines in a maneuver. This burn actively accelerates the spacecraft relative to Mars, gaining enough kinetic energy to change its trajectory from a hyperbola to a bound ellipse. The amount of velocity that needs to be added (Deltav) would be significant, likely in the range of 1-2.5 km/s, depending on the target orbit.
[header]Conclusion: Inertia Wins Without Interference[/header]
The reason Mars cannot capture your spacecraft in this scenario, despite its strong gravitational pull at 300 km altitude, is due to the spacecraft's inherent inertia and the excess kinetic energy it possesses from its initial relative velocity (v infinity ). Gravity will bend the path and accelerate the spacecraft, but it's not enough to "trap" it. An active maneuver (like an acceleration burn) is required to overcome that energy deficit and achieve capture.
This has indeed been a very fruitful discussion, and I'm delighted we could reach such a clear and precise understanding for your audience!

The above has been edited by tahanson43206 to remove side comments that are interesting but not helpful.

Update: 3,101 Newtons is the force that would be applied to a one ton mass suspended 300 km above the surface of Mars. (per Google)

(th)

tahanson43206 · 2025-06-21 20:26:34

This post is about a subtle point of orbital mechanics that is important for thinking about orbits.

I must have missed the lecture, so I got it this evening when asking Gemini about the shape of a curve it had generated in a program to show the velocity of a projectile launched from a point launcher at 53 degrees of elevation toward the East.

One of the five curves generated by the program is the velocity that a satellite must have at a given altitude above the Earth.

I am familiar with the Inverse Square law of gravity, but missed the lecture on how that interacts with orbital motion. Gemini was kind enough to explain the distinction, and I will attempt to paste the transcript here:

I appreciate your taking the time to explain the difference. I had been using inverse-square-law as a mental crutch, and appreciate your showing me the correct way to think about the circular velocity. This might be a bit difficult to explain to someone with a modest science background. The inverse square law is fairly easy to explain. If we double our distance from the center of the Earth, we will experience 1/4th as much gravitational attraction. However, it appears there is no direct relationship between that fact and the circumference of the orbit at a given altitude.
That's a fantastic point, and you've identified a common conceptual hurdle when moving from the inverse square law for force to orbital velocity! You're right, simply stating "it's 1/r" might not be intuitive.
Let's try to explain this in a way that builds directly on the familiar inverse square law, using the idea of a balance of forces:
Think of it this way:
The Force of Gravity (The Pull Inward): You're right, this is the inverse square law. As you go farther from Earth, gravity's pull gets weaker very quickly. If you double your distance from the center, the gravitational pull is indeed 1/4 as strong. This is the force pulling the satellite inward.
The Force Needed to Turn (The "Centripetal" Force): For anything to move in a circle (like a satellite in orbit), it needs a constant push or pull towards the center of that circle. This is called centripetal force. Without it, the satellite would just fly off in a straight line.
The amount of centripetal force you need depends on how fast the object is going and the size of the circle. Specifically, it's proportional to the object's speed squared (V2) and inversely proportional to the radius (r) of the circle. (So, if you double the speed, you need 4 times the force; if you double the radius, you need half the force.)
The Orbital Balance: For a stable circular orbit, the force of gravity (pulling inward) must be exactly equal to the centripetal force required (to keep it turning in that circle). They balance each other out perfectly.
Let's simplify the relationships:
Gravity's Pull: Gravity∝r21
Centripetal Force Needed: Centripetal Force∝rV2
For balance: r21∝rV2
Now, if you do a little bit of algebra (multiply both sides by r):
r2r∝V2
r1∝V2
And finally, take the square root of both sides:
V∝r1
V∝r1
The Intuitive Leap:
So, while gravity gets weaker very fast (1/r2), the velocity doesn't need to drop as drastically (1/r) because:
Weaker Pull: With less gravitational pull, you don't need to be going quite as fast to prevent falling in.
Larger Circle: At a greater distance, the circumference of your orbit is much larger. Even though you're going slower, you have more "road" to travel, so it still takes you longer to complete an orbit. The speed decreases, but the orbital "path" gets longer, which influences how velocity needs to scale.
You can explain that the orbital velocity isn't directly proportional to the force itself, but rather to the speed required to continuously fall around the Earth at a given radius, perfectly balancing that gravitational pull. It's the unique combination of the strength of gravity and the geometry of circular motion that leads to the 1/r relationship for velocity.

Our forum software is unable to reproduce the math perfectly. If someone is interested I would expect a quick search would reveal online resources with the mathematics correctly formatted.

(th)

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