<p>What Kepler (and others before and after him) wanted to do is predict where a planet would be. To do this we need some set up:</p>
<p>First we want a coordinate system.  This is a system of axes: x, y and z, at right angles to each other.  And it should be an inertial coordinate system, so Newton's laws work. This means that the axes should not be rotating.  And we will be able to choose a coordinate system that is not moving relative to the centre of mass of the objects.  Along with the coordinate system, we need a system of time, with some particular moment designated t=0 (and because this isn't relativity we shan't worry about clocks running at different speeds or curved space-time etc...).</p>
<p>So now we meaningfully ask &quot;where is the planet at t=3 years&quot; and understand the reply: &quot;It is at coordinates (0.6, 1.2, 0.1) AU&quot;.</p>
<p>Now it turns out that to completely describe the future motion of the planet, as it obeys Newton's laws of motion and gravity, you need 6 numbers. You could, for example, give the position and velocity at t=0.  Three coordinates for position and three for velocity would give sufficient information.</p>
<p>But this ignores facts that Kepler discovered about orbits (and Newton was able to prove from his laws of motion).  The planet moves in an ellipse. So we can describe the orbit in terms of the ellipse. Ellipses are squashed circles and there are two special points called foci that are off center along the long axis of the ellipse. The centre of mass will be at one focus.</p>
<p>First there is the size of the ellipse.  Now it turns out that the length of the long axis is important, since the period of the orbit (the year) is related to the length of the long axis.  Conventionally we speak of the &quot;semimajor axis&quot; which is the half the length of long axis, or the distance from the centre of the ellipse to the furthest point.  We label this <span class="math-container">$a$</span>, and this is the first of our six numbers.</p>
<p>Next there is the shape of the ellipse. Ellipses can be fat and circular or thin and skinny.  The nearly circular ones have foci that are close to the center. So we measure the distance from the centre to either focus: <span class="math-container">$c$</span>.  And then define the eccentricity to be <span class="math-container">$e=c/a$</span>  All ellipses with the same <span class="math-container">$e$</span> have the same shape, in the sense that they are all mathematically similar. <span class="math-container">$e$</span> is the second number.</p>
<p>Now we need to describe the orientation of the ellipse in space. To do this we come back to our coordinate system.  The x-y plane is a flat plane with the origin at the focus of the ellipse (remember that we chose coordinates with the origin at the the centre of mass). The ellipse won't be in the x-y plane but sometimes be above (in the sense of positive z-coordinate) and sometimes below this plane.</p>
<p>The two planes: the plane of the ellipse and the x-y plane, will be at some angle to each other.  We call this angle the inclination. This is <span class="math-container">$i$</span> and is the third number.  An inclination of 0 means the planet orbits in the x-y plane. An inclination of 90 means the orbital plane is perpendicular to the x-y plane. And an inclination of 180 means it orbits in the x-y plane on a retrograde orbit (to define retrograde we need some definition of positive and negative angles. Conventionally negative angles are taken to be clockwise).</p>
<p>To describe directions in the x-y plane, we will use angles.  The x-axis will be at 0 degrees.  And then we measure the angle from the x-axis, from 0 to 360 degrees.  There will be a point in the x-y plane where the orbit crosses it, and where the planet's motion causes it to move from below the plane to above the plane.  This is called the ascending node.  It is a fixed point and the direction of this fixed point is called the longitude of the ascending node, <span class="math-container">$\Omega$</span>.  The inclination and <span class="math-container">$\Omega$</span> together define the plane that the planet orbits in.</p>
<p>We next look at the location where the planet is closest to the focus. This is called the periapse. This is an angle in the orbital plane.  It is the angle between the ascending node and the direction of periapse. This is the argument of periapse and is labelled <span class="math-container">$\omega$</span>.</p>
<p>These five numbers define the shape and orientation of the ellipse. They have been chosen partly for convenience, and partly for historical reasons. But the principle reason is that each describes a different aspect of the orbit.  Unlike the state vectors where the three position coordinates are the same kind of value, each orbital element is distinct and meaningful.</p>
<p>The sixth element is simply the location of the planet at time <span class="math-container">$t=0$</span>. This is an angle in the orbital plane, and there are different ways to describe this angle.  The simplest is the angle between the periapse direction and the planet.  This is the True Anomaly <span class="math-container">$\nu$</span>  (nothing to do with anomalous data!) But sometimes you might see the Mean Anomaly.  This looks like an angle, but is actually a measure of time.  0 means &quot;at the periapse&quot; and 360 means &quot;one orbital period (year) later&quot;, so 90 means &quot;one quarter of a <em>year</em> through the orbit&quot;.  Because the planet doesn't move at a constant speed, a mean anomaly of 90 doesn't correspond to a true anomaly of 90 degrees.</p>
<p><strong>Those are the six orbital elements:</strong></p>
<p><span class="math-container">$a$</span> size of ellipse<br />
<span class="math-container">$e$</span> shape of ellipse<br />
<span class="math-container">$i$</span>, <span class="math-container">$\Omega$</span> position of orbital plane relative to x-y plane<br />
<span class="math-container">$\omega$</span> orientation of ellipse within orbital plane<br />
<span class="math-container">$\nu$</span> position of planet on ellipse</p>
<p>It is a standard problem in celestial mechanics to convert from 6 orbital elements to the state vectors of position and velocity, and convert back from state vectors to orbital elements.  It is also a standard problem to take the six orbital elements, which give the true (or mean) anomaly at time 0, and calculate the true anomaly at an other time.  But the maths involved is rather involved: See <a href="https://web.archive.org/web/20170810015111/http://ccar.colorado.edu/asen5070/handouts/kep2cart_2002.doc" rel="nofollow noreferrer">https://web.archive.org/web/20170810015111/http://ccar.colorado.edu/asen5070/handouts/kep2cart_2002.doc</a> or <a href="https://web.archive.org/web/20160418175843/https://ccar.colorado.edu/asen5070/handouts/cart2kep2002.pdf" rel="nofollow noreferrer">https://web.archive.org/web/20160418175843/https://ccar.colorado.edu/asen5070/handouts/cart2kep2002.pdf</a> to go the other way</p>
<p>For the solar system, it is conventional to take the x-direction to be in the direction where the plane of the equator crosses the plane of the Earth's orbit around the sun. And the y-direction to be perpendicular to it, in the plane of the Earth's orbit (so by definition the Earth has zero inclination). This points towards the constellation of &quot;Pisces&quot; but it does slowly move and 2000 years ago it pointed towards &quot;Aries&quot; and so is called &quot;the first point of Aries&quot;, or (since the sun is at the position in spring) the vernal equinox, <span class="math-container">$\gamma$</span>.</p>
<p>No discussion of orbital elements is complete without the Wikipedia image which describes them.</p>
<p><a href="https://i.sstatic.net/P9519.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/P9519.png" alt="enter image description here" /></a></p>
<p>By Lasunncty at the English Wikipedia, CC BY-SA 3.0, <a href="https://commons.wikimedia.org/w/index.php?curid=8971052" rel="nofollow noreferrer">https://commons.wikimedia.org/w/index.php?curid=8971052</a></p>

CC BY-SA 4.0
2021-04-14T13:03:28.703Z
Actually, I can get the calculations of orbital elements from Wikipedia, right?
Questioner
CC BY-SA 4.0
2021-04-14T13:05:14.837Z
Orbital elements are like a giant celestial coordinate system. If I got it wrong, please point out my mistakes.
Questioner
CC BY-SA 4.0
2021-04-14T14:01:01.877Z
They are *not* like a giant celestical coordinate system. They are like what I described in my answer.
James K
CC BY-SA 4.0
2021-04-14T14:08:02.033Z
Orbital elements are way of describing the size, shape, position and orientation of an ellipse in space, and the position of a body on that ellipse.  The state vectors are based off the coordinate system.
James K
CC BY-SA 4.0
2021-04-14T14:15:41.677Z
Okay. Also, I'm having trouble opening the links you attached to your post. If you can, please edit your post, and, at the simplest, just copy the information on the webpage, and paste it to your post.
Questioner
CC BY-SA 4.0
2021-04-14T21:53:42.603Z
@Questioner do you need it typed, single-sided, double-spaced?
uhoh
CC BY-SA 4.0
2021-04-14T23:12:27.960Z
I don't know what that means.
Questioner
CC BY-SA 4.0
2021-04-15T15:46:48.547Z
@Questioner the links in the answer are to Microsoft Word documents which is why they may not open right away. Your browser should offer you a place to download them and then you can open them in MS Office/Word or a free alternative such as LibreOffice
astrosnapper
CC BY-SA 4.0
2021-04-15T17:29:16.540Z
Also, does the direction of the ascending node refer to the direction the planet is approaching the ascending node?
Questioner
CC BY-SA 4.0
2021-04-26T17:09:25.857Z
Nevermind. I found out what it is. Thanks for the help, @JamesK!
Questioner
CC BY-SA 4.0
2021-04-13T20:31:38.540Z
2021-04-13T23:15:21.530Z
2021-04-13T23:15:21.530Z
James K
<p>So, I've been doing endless research, and I've found a bit of information, all in different places, that is. Now, I'm getting really confused about it; I can't keep up with it all anymore, so, now, it's time I release it here. What are the Keplerian Elements, and how do you calculate them?</p>
<p>I'm particularly interested in getting an explanation of angular orbital elements, such as the longitude of the ascending node (as well as the ascending node itself), the angular momentum, the argument of perihelion, the mean and true anomaly, and some other variables like the Orbital Momentum Vector.</p>
<p>In an answer, I want a feasibly understandable explanation of what the orbital elements above are, and equations describing how to calculate them. If there are any variables that require further research to grasp, please do the same for them. Afterwards, at the end of a response, please link the source where you got it from (if there is any), but if you got it from in-person courses, which have no link, please ignore this part, and do not try to do treacherous research to try and find one.</p>
<p>Hoping to find an answer soon, thanks in advance!</p>

CC BY-SA 4.0
2021-04-13T17:50:43.980Z
Google is your friend: "Keplerian Elements" https://marine.rutgers.edu/cool/education/class/paul/orbits.html and https://www.amsat.org/keplerian-elements-tutorial/
eshaya
CC BY-SA 4.0
2021-04-13T17:52:05.360Z
I've tried that link, even before I posted this question.
Questioner
CC BY-SA 4.0
2021-04-13T18:02:31.567Z
The sources do a phenomenal job of explaining what the Keplerian Elements are, but doesn't give me any calculations of them.
Questioner
CC BY-SA 4.0
2021-04-13T19:38:25.947Z
Answering this question well is probably the equivalent of an introductory course on orbital mechanics.  The calculations aren't widely posted, because they aren't very easy. There are many ways to calculate them, including starting with {az, el, time} observations, then estimating state vectors, and then orbital elements.  Here is a paper with a "simple" method that also includes an example at the end: http://articles.adsabs.harvard.edu/pdf/1981A%26A...102...59N .
Connor Garcia
CC BY-SA 4.0
2021-04-13T23:34:00.737Z
@ConnorGarcia that's a pretty cool paper! It's the complete recipe for taking 3 or more Earth-based observations of a heliocentric orbiting body and constructing the 3D position vectors $\mathbf{E}$ and Earth observer unit vectors $\mathbf{e}$ via analytical propagation (Eq. 16, see also [1](https://space.stackexchange.com/a/43237/12102), [2](https://space.stackexchange.com/a/43738/12102)). That's where the fitting is done. Then *ex post facto* the cartesian positions are finally converted to Keplerian elements via Eqs. 1-5.  They even give an example at the end. *Sweet!*
uhoh
CC BY-SA 4.0
2021-04-14T17:13:06.413Z
I agree. That PDF contributed in answering my question greatly.
Questioner
CC BY-SA 4.0
2021-04-27T12:45:34.893Z
I have made a follow-up question about *Launch* Elements. Here's the link:
[**Launch Elements**](https://astronomy.stackexchange.com/questions/43573/launch-elements)
Questioner
CC BY-SA 4.0
2021-04-27T18:33:49.463Z
@ConnorGarcia, you are right, the orbital element calculations aren't widely available to the public, and the ones that I do find involve advanced math that I am not capable of understanding. Now, though, I've found out I don't need those calculations, they're simple aspects of an orbit, that they're values are ones you can see with your eyes.
Questioner
<astrophysics><orbital-elements><celestial-mechanics>
Keplerian Elements
Questioner
# Keplerian Elements

(Asked by: username_0 on 2021-04-13)

So, I've been doing endless research, and I've found a bit of information, all in different places, that is. Now, I'm getting really confused about it; I can't keep up with it all anymore, so, now, it's time I release it here. What are the Keplerian Elements, and how do you calculate them?

I'm particularly interested in getting an explanation of angular orbital elements, such as the longitude of the ascending node (as well as the ascending node itself), the angular momentum, the argument of perihelion, the mean and true anomaly, and some other variables like the Orbital Momentum Vector.

In an answer, I want a feasibly understandable explanation of what the orbital elements above are, and equations describing how to calculate them. If there are any variables that require further research to grasp, please do the same for them. Afterwards, at the end of a response, please link the source where you got it from (if there is any), but if you got it from in-person courses, which have no link, please ignore this part, and do not try to do treacherous research to try and find one.

Hoping to find an answer soon, thanks in advance!

**Tags:** astrophysics, orbital-elements, celestial-mechanics
> **Comments:**
> **username_1**: Google is your friend: "Keplerian Elements" https://marine.rutgers.edu/cool/education/class/paul/orbits.html and https://www.amsat.org/keplerian-elements-tutorial/  
> **username_0**: I've tried that link, even before I posted this question.  
> **username_0**: The sources do a phenomenal job of explaining what the Keplerian Elements are, but doesn't give me any calculations of them.  
> **username_2**: Answering this question well is probably the equivalent of an introductory course on orbital mechanics.  The calculations aren't widely posted, because they aren't very easy. There are many ways to calculate them, including starting with {az, el, time} observations, then estimating state vectors, and then orbital elements.  Here is a paper with a "simple" method that also includes an example at the end: http://articles.adsabs.harvard.edu/pdf/1981A%26A...102...59N .  
> **username_3**: @ConnorGarcia that's a pretty cool paper! It's the complete recipe for taking 3 or more Earth-based observations of a heliocentric orbiting body and constructing the 3D position vectors $\mathbf{E}$ and Earth observer unit vectors $\mathbf{e}$ via analytical propagation (Eq. 16, see also [1](https://space.stackexchange.com/a/43237/12102), [2](https://space.stackexchange.com/a/43738/12102)). That's where the fitting is done. Then *ex post facto* the cartesian positions are finally converted to Keplerian elements via Eqs. 1-5.  They even give an example at the end. *Sweet!*  
> **username_0**: I agree. That PDF contributed in answering my question greatly.  
> **username_0**: I have made a follow-up question about *Launch* Elements. Here's the link: [**Launch Elements**](https://astronomy.stackexchange.com/questions/43573/launch-elements)  
> **username_0**: @ConnorGarcia, you are right, the orbital element calculations aren't widely available to the public, and the ones that I do find involve advanced math that I am not capable of understanding. Now, though, I've found out I don't need those calculations, they're simple aspects of an orbit, that they're values are ones you can see with your eyes.  

---

## 1 answers

### Answer by username_4 on 2021-04-13. Score: 5 | Accepted answer

What Kepler (and others before and after him) wanted to do is predict where a planet would be. To do this we need some set up:

First we want a coordinate system. This is a system of axes: x, y and z, at right angles to each other. And it should be an inertial coordinate system, so Newton's laws work. This means that the axes should not be rotating. And we will be able to choose a coordinate system that is not moving relative to the centre of mass of the objects. Along with the coordinate system, we need a system of time, with some particular moment designated t=0 (and because this isn't relativity we shan't worry about clocks running at different speeds or curved space-time etc...).

So now we meaningfully ask "where is the planet at t=3 years" and understand the reply: "It is at coordinates (0.6, 1.2, 0.1) AU".

Now it turns out that to completely describe the future motion of the planet, as it obeys Newton's laws of motion and gravity, you need 6 numbers. You could, for example, give the position and velocity at t=0. Three coordinates for position and three for velocity would give sufficient information.

But this ignores facts that Kepler discovered about orbits (and Newton was able to prove from his laws of motion). The planet moves in an ellipse. So we can describe the orbit in terms of the ellipse. Ellipses are squashed circles and there are two special points called foci that are off center along the long axis of the ellipse. The centre of mass will be at one focus.

First there is the size of the ellipse. Now it turns out that the length of the long axis is important, since the period of the orbit (the year) is related to the length of the long axis. Conventionally we speak of the "semimajor axis" which is the half the length of long axis, or the distance from the centre of the ellipse to the furthest point. We label this $a$, and this is the first of our six numbers.

Next there is the shape of the ellipse. Ellipses can be fat and circular or thin and skinny. The nearly circular ones have foci that are close to the center. So we measure the distance from the centre to either focus: $c$. And then define the eccentricity to be $e=c/a$ All ellipses with the same $e$ have the same shape, in the sense that they are all mathematically similar. $e$ is the second number.

Now we need to describe the orientation of the ellipse in space. To do this we come back to our coordinate system. The x-y plane is a flat plane with the origin at the focus of the ellipse (remember that we chose coordinates with the origin at the the centre of mass). The ellipse won't be in the x-y plane but sometimes be above (in the sense of positive z-coordinate) and sometimes below this plane.

The two planes: the plane of the ellipse and the x-y plane, will be at some angle to each other. We call this angle the inclination. This is $i$ and is the third number. An inclination of 0 means the planet orbits in the x-y plane. An inclination of 90 means the orbital plane is perpendicular to the x-y plane. And an inclination of 180 means it orbits in the x-y plane on a retrograde orbit (to define retrograde we need some definition of positive and negative angles. Conventionally negative angles are taken to be clockwise).

To describe directions in the x-y plane, we will use angles. The x-axis will be at 0 degrees. And then we measure the angle from the x-axis, from 0 to 360 degrees. There will be a point in the x-y plane where the orbit crosses it, and where the planet's motion causes it to move from below the plane to above the plane. This is called the ascending node. It is a fixed point and the direction of this fixed point is called the longitude of the ascending node, $\Omega$. The inclination and $\Omega$ together define the plane that the planet orbits in.

We next look at the location where the planet is closest to the focus. This is called the periapse. This is an angle in the orbital plane. It is the angle between the ascending node and the direction of periapse. This is the argument of periapse and is labelled $\omega$.

These five numbers define the shape and orientation of the ellipse. They have been chosen partly for convenience, and partly for historical reasons. But the principle reason is that each describes a different aspect of the orbit. Unlike the state vectors where the three position coordinates are the same kind of value, each orbital element is distinct and meaningful.

The sixth element is simply the location of the planet at time $t=0$. This is an angle in the orbital plane, and there are different ways to describe this angle. The simplest is the angle between the periapse direction and the planet. This is the True Anomaly $\nu$ (nothing to do with anomalous data!) But sometimes you might see the Mean Anomaly. This looks like an angle, but is actually a measure of time. 0 means "at the periapse" and 360 means "one orbital period (year) later", so 90 means "one quarter of a *year* through the orbit". Because the planet doesn't move at a constant speed, a mean anomaly of 90 doesn't correspond to a true anomaly of 90 degrees.

**Those are the six orbital elements:**

$a$ size of ellipse  
$e$ shape of ellipse  
$i$, $\Omega$ position of orbital plane relative to x-y plane  
$\omega$ orientation of ellipse within orbital plane  
$\nu$ position of planet on ellipse

It is a standard problem in celestial mechanics to convert from 6 orbital elements to the state vectors of position and velocity, and convert back from state vectors to orbital elements. It is also a standard problem to take the six orbital elements, which give the true (or mean) anomaly at time 0, and calculate the true anomaly at an other time. But the maths involved is rather involved: See <https://web.archive.org/web/20170810015111/http://ccar.colorado.edu/asen5070/handouts/kep2cart_2002.doc> or <https://web.archive.org/web/20160418175843/https://ccar.colorado.edu/asen5070/handouts/cart2kep2002.pdf> to go the other way

For the solar system, it is conventional to take the x-direction to be in the direction where the plane of the equator crosses the plane of the Earth's orbit around the sun. And the y-direction to be perpendicular to it, in the plane of the Earth's orbit (so by definition the Earth has zero inclination). This points towards the constellation of "Pisces" but it does slowly move and 2000 years ago it pointed towards "Aries" and so is called "the first point of Aries", or (since the sun is at the position in spring) the vernal equinox, $\gamma$.

No discussion of orbital elements is complete without the Wikipedia image which describes them.

[![enter image description here](https://i.sstatic.net/P9519.png)](https://i.sstatic.net/P9519.png)

By Lasunncty at the English Wikipedia, CC BY-SA 3.0, <https://commons.wikimedia.org/w/index.php?curid=8971052>

> **Comments:**
> **username_0**: Actually, I can get the calculations of orbital elements from Wikipedia, right?  
> **username_0**: Orbital elements are like a giant celestial coordinate system. If I got it wrong, please point out my mistakes.  
> **username_4**: They are *not* like a giant celestical coordinate system. They are like what I described in my answer.  
> **username_4**: Orbital elements are way of describing the size, shape, position and orientation of an ellipse in space, and the position of a body on that ellipse.  The state vectors are based off the coordinate system.  
> **username_0**: Okay. Also, I'm having trouble opening the links you attached to your post. If you can, please edit your post, and, at the simplest, just copy the information on the webpage, and paste it to your post.  
> **username_3**: @Questioner do you need it typed, single-sided, double-spaced?  
> **username_0**: I don't know what that means.  
> **username_5**: @Questioner the links in the answer are to Microsoft Word documents which is why they may not open right away. Your browser should offer you a place to download them and then you can open them in MS Office/Word or a free alternative such as LibreOffice  
> **username_0**: Also, does the direction of the ascending node refer to the direction the planet is approaching the ascending node?  
> **username_0**: Nevermind. I found out what it is. Thanks for the help, @JamesK!  

---

