Help with orbital mechanics for a school project

[SlabGizor117]

So, I'm working on a school project essay on none other than orbital mechanics and basic maneuvering - Terms, mechanics, maneuvering, types of orbits, etc.

Here's my notes so far on terms:

Orbital terms:

*Apoapsis – Highest point in an elliptical orbit or parabolic trajectory, opposite of Periapsis

*Periapsis – Lowest point in an elliptical orbit or hyperbolic trajectory, opposite of Apoapsis

Types of trajectories:

*Parabolic – A suborbital trajectory where one must accelerate to achieve orbit, and the vertex is the Apoapsis

*Circular – A theoretical, perfectly circular orbit in which there is neither an Apoapsis nor Periapsis. Also, “Circular†is used to this degree of whole number calculations: a 150.0km by 150.0km orbit is called Circular, but no measurements are more precise than the meter

*Elliptical – The exact definition of every stable orbit, where there is an Apoapsis and a Periapsis

*Hyperbolic – A hyperbola-shaped interplanetary trajectory in which the vertex of the hyperbola is the Periapsis and one must slow down to achieve orbit

So most of you who, unlike me, know anything about orbital mechanics past terms like semi-major axis, Oberth Effect, and Sidereal Period, probably already know what's wrong: Parabolic and Hyperbolic. I kinda flew by the seat of my pants explaining Parabolic and Hyperbolic because just from looking you can tell that a Sub-orbital trajectory is parabolic, and from light research I know that a hyperbolic trajectory is interplanetary. Here's where I cut corners:

"Parabolic trajectories are(only) sub-orbital trajectories in which the vertex is an Apoapsis and one must speed up to achieve orbit."

"Hyperbolic trajectories are(only) interplanetary trajectories through the SOI of a planet in which the vertex is a Periapsis and one must slow down to achieve orbit."

So, I looked up some pictures of a Parabolic, Circular, Elliptical and Hyperbolic trajectories all together such as this:

http://www.insight3d.com/resources/educational-alliance-program/astro-primer/primer63.htm

and along with everything else, none of them showed a sub-orbital trajectory as parabolic. It seems the only difference between parabolic and hyperbolic trajectories is eccentricity, which can't be right, and it also shows that a parabolic trajectory could be interplanetary too.

I was pretty proud of my "seat-of-my-pants" definitions and this kinda wrecks it. :/

Let me know the correct way to define these terms, I'd love to see how my definition compares to what it really is.

Thanks for reading!

-Slab

Edited February 3, 2015 by SlabGizor117

[Laie]

Try this one:

http://www.braeunig.us/space/orbmech.htm

A sub-orbital trajectory is still elliptic, with the center of gravity at one of it's focal points, even if most of the ellipse is below ground. It may look like a parabola; if you flatten the ground it may actually *be* a parabola, or at any rate that's what artillerists assume and it seems to work well for them.

[MarvinKitFox]

*Apoapsis – Highest point in an elliptical orbit or parabolic trajectory, opposite of Periapsis

I kinda flew by the seat of my pants explaining Parabolic and Hyperbolic because just from looking you can tell that a Sub-orbital trajectory is parabolic, and from light research I know that a hyperbolic trajectory is interplanetary. Here's where I cut corners:

Some errorfixes for you.

A parabolic or hyperbolic orbit will not have an Apoapsis. (technically, a parabolic trajectory/orbit has an Apoapis, but the value of it is Infinity)

A sub-orbital trajectory is NOT parabolic! It is an ellipse, with the periapsis somewhere under the ground surface!

Try this as a starting point:

Orbit: an elliptical trajectory, with Periapsis(closest) and Apoapsis(furthest) points. If it heads out on this trajectory, it will return. And repeat. Orbits are cyclic.

Circular orbit: Just another elliptical trajectory, which happens to have Periapsis and Apoapsis as exactly the same value (yes, a circle is an ellipse, just a special one)

Hyperbolic trajectory: if it heads out on this trajectory, it will not return.

Parabolic trajectory: a variant of hyperbolic trajectory, where it will *almost* return. That path where the apogee is *at* infinite distance. Anything at all slower than this will be an orbit.

Parabolic trajectories do not exist in the real world, as even the most infinitestimal deviation will change it to either hyperbolic or elliptical.

However, many comets, etc.. have orbits/trajectories that are *very close* to parabolic, as they started as virtually-stationary objects at enormous distances from the sun. A rough but close approximation of the "infinite" apoapsis of a parabolic path.

[OhioBob]

The conical sections are all related to the eccentricity of the orbit.

Circular: e = 0

Elliptical: 0 < e < 1

Parabolic: e = 1

Hyberbolic: e > 1

A circular orbit is just a special case of the elliptical orbit. If a spacecraft is traveling at less than escape velocity its orbit will be elliptical. If a spacecraft traveling at exactly escape velocity its orbit is parabolic. And if a spacecraft is traveling faster than escape velocity its orbit is hyperbolic. A suborbital trajectory is actually a elliptical orbit that has its periapsis inside the planet.

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It may look like a parabola; if you flatten the ground it may actually *be* a parabola, or at any rate that's what artillerists assume and it seems to work well for them.

That's correct, if you assume a gravity vector that is normal to a flat surface, then the trajectory is a parabola. Over short distances this is often a close enough approximation. In the real world, however, the ground is curved and the gravity vector points toward the center of curvature. This makes the trajectory a segment of an ellipse.

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Try this one:

http://www.braeunig.us/space/orbmech.htm

Thanks for the reference. That's my web page.

[SlabGizor117]

Awesome!

First, let me say that although the fancy words are a bit confusing, this really helps. Thank you so much for the corrections, what you're saying makes alot more sense.

But, my essay is kinda a "Orbital mechanics for dummies" paper and if I can't say that a sub-orbital trajectory is a parabola, my only other choice is to say that

"If you assume a gravity vector that is normal to a flat surface, then the trajectory is a parabola. Over short distances this is often a close enough approximation. In the real world, however, the ground is curved and the gravity vector points toward the center of curvature. This makes the trajectory a segment of an ellipse."

So, with all of this new information, I'm having a hard time putting it together into a clear thought and understanding that I can make simple.

Maybe a little help along the lines of what MarvinKitFox did to kinda redo things and explain them better.

I don't mean write my essay for me, but like I said I'm having a hard time understanding this clearly.

Basically, I don't know what I want XD a bullet point list of corrections I guess, but not something that will, like I said, write my essay for me, just something to get it clear in my head.

Thanks so much for the help!

-Slab

[Raptor9]

Other good wiki articles about orbital properties is the Molniya orbit, Tundra orbit and argument of periapsis. When I heard Molniya orbits were going to be in 0.90 contracts, I had a look, which led to more study of various orbital properties...kept going down the rabbit hole until an hour later I snapped out of it

[SlabGizor117]

Some errorfixes for you.

Orbit: an elliptical trajectory, with Periapsis(closest) and Apoapsis(furthest) points. If it heads out on this trajectory, it will return. And repeat. Orbits are cyclic.

Circular orbit: Just another elliptical trajectory, which happens to have Periapsis and Apoapsis as exactly the same value (yes, a circle is an ellipse, just a special one)

Well, considering that an Apoapsis is defined as the apsis which is farthest from the center of attraction,

and the Periapsis is defined as the point in the path of an orbiting body at which it is nearest to the body that it orbits,

In a theoretically perfectly circular orbit in which the Apoapsis is 150.0000000000000000km and the Periapsis is 150.0000000000000000km,

there IS no Periapsis or Apoapsis considering there IS no higher or lower point in the orbit.

But, if the Periapsis is 150.0000000000000000km and the Apoapsis is 150.0000000000000001km, it's an ellipse.

Correct?

-Slab

[Laie]

if I can't say that a sub-orbital trajectory is a parabola

The defining element of "sub-orbital" is not the shape, but that the surface is getting in the way. If the body you're orbiting was only small enough (iow: point-sized), then there would be no suborbital trajectories.

[Sirrobert]

but no measurements are more precise than the meter

What do you mean by that? It's a very confusing sentence in this context

[Laie]

In a theoretically perfectly circular orbit [...], there IS no Periapsis or Apoapsis considering there IS no higher or lower point in the orbit.

But, if the Periapsis is 150.0000000000000000km and the Apoapsis is 150.0000000000000001km, it's an ellipse.

Nitpick: a circle is a special case of ellipse, like a cube is a special case of cuboid.

And yes, if "periapsis altitude" = "apoapsis altitude" then there is no point in distinguishing the two, much less in trying to find out where they are. You need to select one point as periapsis in order to properly define an orbit, but you're free to chose as you will.

[SlabGizor117]

What do you mean by that? It's a very confusing sentence in this context

I didn't do a good job of explaining this in my notes because at the time they were only for me to read and understand, but I planned to elaborate in my actual essay.

Originally, before I realized I was wrong, I defined a circular orbit as:

"A theoretical, perfectly circular orbit in which there is neither an Apoapsis nor Periapsis."

I say theoretical because aside from theory, there is no way to get a perfectly circular orbit as you could continue "zooming in" to smaller and smaller measurements where the orbit was not perfectly circular, to infinity, even. Like I said in the question on having neither a Periapsis or an Apoapsis in a perfectly circular orbit, from my understanding, a perfectly circular orbit is:

150.0000000000000000km by 150.0000000000000000km.

And, further, from my understanding, if you had an orbit that was:

150.0000000000000001km by 150.0000000000000000m, then it wasn't circular.

All that to say, people still use the term circular when measuring an orbit by 150.0km by 150.0km. It may be that your true orbit is 150.01 by 150.00, but like I said, nobody practically measures their orbit more precisely than the meter, and so even if it is, like I said, 150.01 by 150.00km, it's still called circular because we measure it as 150.0 by 150.0.

Hope that helps!

-Slab

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So I still need help with putting all this together into an easy to understand list of definitions, because I can't list all the nitpicks and technicalities in an "Orbital mechanics for dummies" essay. How exactly would you define the revised list of terms?

[cantab]

Over distances where it's meaningful to talk of orbital mechanics at all, as opposed to ballistics, a sub-orbital trajectory is an ellipse.

[Raptor9]

@SlabGizor117

Honestly, if it's for a school project like an essay, I would only get bogged down into the hardcore technicalities if that's what the project calls for; or in the case of presenting your essay to a group or audience, consider who you're going to be speaking to. Your target audience has to be taken into account. Trust me, if there is someone especially knowledgeable in the group, they'll just ask a question regarding specifics, in which case you can go into greater detail regarding that question. Keep it thorough, but keep it within the scope of your project. Just my two cents.

[RSwordsman]

target audience

Truth. Unless it's a class full of Kerbal nerds, most of them won't give a quarter crap about orbits. Getting into specifics like argument of periapsis (sort of hard to fathom for the casual listener) is likely to make a lot of people shut the information out. I'd focus more on inclination, prograde vs. retrograde, and the relationship of velocity to the proximity of one or the other apsis. At least it might reduce the notion that somebody gets to space by going straight up.

Edited February 2, 2015 by RSwordsman

[Kryxal]

Well, you CAN get to space by going straight up ... it's just that you're probably not staying there.

[Newt]

I think that it seems usually helpful to start, not with the ideas of periapsis and apoapsis, ecentricity and inclination, but more with ideas of how actually it works in terms familiar to people-- maybe you could start with some anecdotal concept like Newton's Cannon, and use that to expand from someone hurling a ball and having it land on the ground, to a satellite orbiting the Earth. Then get into technical details, if that suits the essay. But, probably, a this sounds like you might just need the concepts of how it works, without all the jargon.

[SlabGizor117]

Yeah, like I said I'm not gonna make it complicated but if my original definitions are wrong, then I need to correct that, but I don't have a good understanding of how to correct "A parabolic trajectory with the vertex as an apoapsis" for a sub-orbital trajectory into the fact that it's actually an ellipse without getting complicated, and that's what I need help with is trying to figure out for myself how exactly this works and how to make it simple.

[OhioBob]

but I don't have a good understanding of how to correct "A parabolic trajectory with the vertex as an apoapsis" for a sub-orbital trajectory into the fact that it's actually an ellipse without getting complicated, and that's what I need help with is trying to figure out for myself how exactly this works and how to make it simple.

Perhaps all you need is a diagram...

The above is an illustration of a suborbital orbit. The orbit is an ellipse that intersects Earth's surface. It is suborbital because the body crashes into Earth's surface before it can complete an orbit. The part the we can track above ground is a segment of the ellipse.

[EtherDragon]

So most of you who, unlike me, know anything about orbital mechanics past terms like semi-major axis, Oberth Effect, and Sidereal Period ...

Thanks for reading!

-Slab

Hey, since you're getting a great understanding of Parabolic and Hyperbolic trajectories (calling them orbits is a little misleading...) One thing to consider with Parabolic trajectories, is that, similar to a circular orbit, they are more mathematical constructs than they are physical reality. Mathematically, it's possible to engineer a Parabolic orbit/trajectory (yes it's both). With eccentricity = 1 a parabolic trajectory can maintain an orbit ... just barely. If you so much as sneeze in the wrong direction you're going Hyperbolic.

Also, one generalization you can make. Hyperbolic trajectories don't necessarily mean you're going interplanetary. A hyperbolic trajectory around the Moon, just means that you fly by the moon; you are probably staying within orbit of the planet.

With that knowledge, you are armed with enough info to know about the Semi-Major Axis and Sidereal Period.

Semi-Major Axis: Quite simply, this is the average distance from the center of mass for the Periapsis and Apoapsis.

Maths: Sma = (P+A)/2

If your orbit is circular, then the Sma would be equal to the radius.

Sidereal Period: Simply put - the Sidereal Period is the Orbital Period. It is the time taken for an object to complete a full orbit when viewed from a static frame of reference.

[OhioBob]

Semi-Major Axis: Quite simply, this is the average distance from the center of mass for the Periapsis and Apoapsis.

Maths: Sma = (P+A)/2.

That's the semi-major axis of an elliptical orbit. A hyperbolic trajectory also has a semi-major axis, though it is a bit more difficult to visualize and understand. In the diagram below the semi-major axis is the line segment labeled "a", which, for a hyperbola, is a negative number.

[SlabGizor117]

Ok, so thank you for the help, now what would you change

"*Parabolic – A suborbital trajectory where one must accelerate to achieve orbit, and the vertex is the Apoapsis

*Circular – A theoretical, perfectly circular orbit in which there is neither an Apoapsis nor Periapsis. Also, “Circular†is used to this degree of whole number calculations: a 150.0km by 150.0km orbit is called Circular, but no measurements are more precise than the meter

*Elliptical – The exact definition of every stable orbit, where there is an Apoapsis and a Periapsis

*Hyperbolic – A parabola-shaped interplanetary trajectory in which the vertex of the parabola is the Periapsis and one must slow down to achieve orbit"

to make it correct?

I mean, this all makes sense but what exactly in these definitions is incorrect?

And how would you change them to make them correct?

Like I said, you're not writing my essay for me, these are only notes to help me write it.

Thanks!

-Slab

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Also, I know Hyperbolic trajectories aren't strictly interplanetary, I plan to use a hyperbolic trajectory around the moon to illustrate the same for my essay.

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A better way to say this is, I know now that a suborbital trajectory is an ellipse, but is it flat out wrong to call it a parabolic trajectory?

[cantab]

I think it's confusing, because a parabolic orbit more normally refers to one that escapes - but only just. Like a circular orbit, it's an ideal that's never possible in real life - real orbits are either elliptical or hyperbolic. Most non-periodic comets, for example, are on near-parabolic orbits.

I think the order to explain things will depend on your focus. If you're talking about satellites in Earth orbit, I'd start by considering an ideal circular orbit, then move on to "what if it's at this height but it's going a bit faster" and discuss real elliptical orbits. If on the other hand you start by considering the planets and moons, you probably want to start by talking about how the orbits are ellipses - almost circular ellipses, but ellipses nonetheless.

Edited February 2, 2015 by cantab

[SlabGizor117]

Yes but you're not answering my question exactly lol, what exactly is wrong in my definitions, and what should they say instead?

[Red Iron Crown]

A parabolic trajectory isn't suborbital, it's (barely) an escape trajectory.

Ellipse = All suborbital and closed orbital trajectories (circular is a special case of elliptical), not enough energy to escape.

Parabola = Exactly enough energy to escape.

Hyperbola = More than enough energy to escape.

You might also look into defining the six orbital elements:

Semi-major axis

Eccentricity

Inclination

Longitude of ascending node

Argument of periapsis

Mean anomaly at epoch

Those six values uniquely define an orbit.

[OhioBob]

*Parabolic – A suborbital trajectory where one must accelerate to achieve orbit, and the vertex is the Apoapsis

*Circular – A theoretical, perfectly circular orbit in which there is neither an Apoapsis nor Periapsis. Also, “Circular†is used to this degree of whole number calculations: a 150.0km by 150.0km orbit is called Circular, but no measurements are more precise than the meter

*Elliptical – The exact definition of every stable orbit, where there is an Apoapsis and a Periapsis

*Hyperbolic – A parabola-shaped interplanetary trajectory in which the vertex of the parabola is the Periapsis and one must slow down to achieve orbit

Think of it this way. When we first start an ascent the orbit is elliptical and suborbital. It is suborbital because the rocket hasn't yet achieved enough velocity to raise it's periapsis above the surface of the Earth. As the rocket ascends and gains horizontal velocity, the periapsis rises. Once the periapsis has risen high enough above the surface that complete orbits can be made without atmospheric drag bringing the rocket back down again, orbit has been achieved. As the rocket continues to gain speed it will push its apoapsis farther and farther away from the planet. When the rocket reaches exactly the escape velocity, the orbit will be parabolic for an infinitesimally small instant. Just as escape velocity is exceeded the orbit becomes hyperbolic.

Suborbital and parabolic have nothing to do with each other - you need disassociate those two terms from each other. You should delete your current definition of a parabolic orbit, move it to between elliptical and hyperbolic, a give it a correct description. Furthermore, a hyperbolic orbit has the shape of a hyperbola, it is not parabola-shaped. Hyperbolas and parabolas are two different conic sections.

The conic sections and orbital elements are defined here: http://www.braeunig.us/space/orbmech.htm

Edited February 2, 2015 by OhioBob