<p>As opposed to what @user21 said, the longitude of periastron does depend on the longitude of ascending node (I was surprised by that myself).</p>
<p>In orbital dynamics, there exist the <strong>argument</strong> of periastron (symbol ω) as well as the <strong>longitude</strong> of periastron (symbol π [or ϖ but may lead to confusion) ), which is equal to the argument of periastron (ω) plus the longitude of ascending node (Ω). In other words:</p>
<p>π = ω + Ω</p>
<p>Now, the <strong>argument</strong> of periastron is equal to:</p>
<p>ω = arccos <span class="math-container">$\frac{\mathbf{n}\ ·\ \mathbf{e}} {|\mathbf{n}||\mathbf{e}|}$</span></p>
<p>If <em>e<sub>z</sub></em> &lt; 0 then ω → 2π − ω.</p>
<p>where:</p>
<p><span class="math-container">$\mathbf{n}$</span> is a vector pointing towards the ascending node (i.e. the <em>z</em>-component of <span class="math-container">$\mathbf{n}$</span> is zero), and</p>
<p><span class="math-container">$\mathbf{e}$</span> is the eccentricity vector (a vector pointing towards the periastron).</p>
<p>This formula can be found at <a href="https://en.wikipedia.org/wiki/Argument_of_periapsis" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Argument_of_periapsis</a> (I call it instead the “argument of periastron,” as Meeus says: “The word ‘periapse’, used by some authors, is incorrect. The word <em>perihelion</em> means the point of the orbit that is closest to the Sun [from the Greek <em>peri</em> = near + <em>helios</em> = Sun]. Similarly, <em>perigee</em> is the point closest to the Earth [<em>ge</em> = Earth]. Therefore, ‘periapse’ would really mean the point closest to the apse; but this is ridiculous, because what is meant is the apse itself!”)</p>
<p>So, using the above formula, you should be able to figure out the longitude of the ascending node for each of your hypothetical bodies.</p>
<p>Have fun!</p>

CC BY-SA 4.0
2020-11-24T21:33:03.913Z
Note that the equation π = ω + Ω represents a compound angle; If the Orbital Plane has non-zero Orbital Inclination, you cannot just add the values of the Longitude of the Ascending Node and the Argument of Periapsis to get the Longitude of Periapsis; you're going to have to do some more complicated spherical trigonometry.
notovny
CC BY-SA 4.0
2020-11-25T01:39:38.097Z
It may be a compound angle, but that’s the definition of “longitude of periastron,” as can be seen here https://en.wikipedia.org/wiki/Longitude_of_the_periapsis and in any and all other references on the subject you may find.
Pierre Paquette
CC BY-SA 4.0
2020-11-24T01:06:09.590Z
2020-11-24T01:06:09.590Z
Pierre Paquette
<p>I'm writing a program to randomly generate star systems and I have been having a difficult time figuring this out. I've found the equations I need to get the heliocentric equatorial coordinates, but I haven't been able to figure out how to determine the longitude of perihelion. Is it just the opposite of the longitude of the ascending node or is it a completely different variable?</p>

<p>It might just me, but the names for this stuff is definitely making it more confusing than it should be.</p>

<p>Here is all my notes:</p>

<p>** = power</p>

<p>P = planet's period</p>

<p>a = semi-major axis</p>

<p>b = semi-minor axis</p>

<p>M = mean anomaly</p>

<p>E = Eccentric anomaly</p>

<p>e = eccentricity of orbit</p>

<p>v = true anomaly</p>

<p>r = radial distance</p>

<p>A = longitude of asending node</p>

<p>w = longitude of perihelion</p>

<p>i = inclination</p>

<p>t = time</p>

<p>(r, v) = polar coordinates</p>

<p>(x, y, z) = Heliocentric Ecliptic coordinates</p>

<p>(X, Y, Z) = Heliocentric Equalorial coordinates</p>

<p>eccentricity</p>

<p>e = sqrt(1 - (b ** 2 / a ** 2))</p>

<p>Period</p>

<p>P = sqrt(a ** 3)</p>

<p>Kepler's Equation</p>

<p>M = E - e * sin(E)</p>

<p>Mean Anomaly
use as starting point for eccentric anomaly</p>

<p>M = (2 * pi * t) / P</p>

<p>True Anomaly</p>

<p>v = 2 * atan(sqrt((1 + e) / (1 - e)) * tan(E / 2))</p>

<p>Radial Distance</p>

<p>r = (a(1 - e ** 2)) / (1 + e * cos(v))</p>

<p>Heliocentric Ecliptic coordinates</p>

<p>x = r(cos(A) * cos(w + v) - sin(A) * sin(w + v) * cos(i)</p>

<p>y = r(sin(A) * cos(w + v) + cos(A) * sin(w + v) * cos(i)</p>

<p>z = r * sin(w + v) * sin(i)</p>

<p>Heliocentric Equalorial coordinates</p>

<p>X = x</p>

<p>Y = y * cos(i) - z * sin(i)</p>

<p>Z = y * sin(i) + z * cos(i)</p>

CC BY-SA 3.0
2015-12-11T15:04:25.173Z
Ok but what is it?
Eegxeta
CC BY-SA 3.0
2015-12-11T17:27:20.137Z
Right but I don't know what it is. Does it affect the orientation of the orbit? What units is it in? I kinda need to know what it is as it has enough significance to be in the equations.
Eegxeta
CC BY-SA 3.0
2015-12-11T23:03:30.047Z
I found a definition. The units would be degrees right?
Eegxeta
<orbit><planet>
How to get the longitude of the perihelion to find the Heliocentric Equatorial coordinates of a planet
Eegxeta
# How to get the longitude of the perihelion to find the Heliocentric Equatorial coordinates of a planet

(Asked by: username_0 on 2015-12-11)

I'm writing a program to randomly generate star systems and I have been having a difficult time figuring this out. I've found the equations I need to get the heliocentric equatorial coordinates, but I haven't been able to figure out how to determine the longitude of perihelion. Is it just the opposite of the longitude of the ascending node or is it a completely different variable?

It might just me, but the names for this stuff is definitely making it more confusing than it should be.

Here is all my notes:

\*\* = power

P = planet's period

a = semi-major axis

b = semi-minor axis

M = mean anomaly

E = Eccentric anomaly

e = eccentricity of orbit

v = true anomaly

r = radial distance

A = longitude of asending node

w = longitude of perihelion

i = inclination

t = time

(r, v) = polar coordinates

(x, y, z) = Heliocentric Ecliptic coordinates

(X, Y, Z) = Heliocentric Equalorial coordinates

eccentricity

e = sqrt(1 - (b \*\* 2 / a \*\* 2))

Period

P = sqrt(a \*\* 3)

Kepler's Equation

M = E - e \* sin(E)

Mean Anomaly
use as starting point for eccentric anomaly

M = (2 \* pi \* t) / P

True Anomaly

v = 2 \* atan(sqrt((1 + e) / (1 - e)) \* tan(E / 2))

Radial Distance

r = (a(1 - e \*\* 2)) / (1 + e \* cos(v))

Heliocentric Ecliptic coordinates

x = r(cos(A) \* cos(w + v) - sin(A) \* sin(w + v) \* cos(i)

y = r(sin(A) \* cos(w + v) + cos(A) \* sin(w + v) \* cos(i)

z = r \* sin(w + v) \* sin(i)

Heliocentric Equalorial coordinates

X = x

Y = y \* cos(i) - z \* sin(i)

Z = y \* sin(i) + z \* cos(i)

**Tags:** orbit, planet
> **Comments:**
> **username_0**: Ok but what is it?  
> **username_0**: Right but I don't know what it is. Does it affect the orientation of the orbit? What units is it in? I kinda need to know what it is as it has enough significance to be in the equations.  
> **username_0**: I found a definition. The units would be degrees right?  

---

## 1 answers

### Answer by username_1 on 2020-11-24. Score: 2

As opposed to what @user21 said, the longitude of periastron does depend on the longitude of ascending node (I was surprised by that myself).

In orbital dynamics, there exist the **argument** of periastron (symbol ω) as well as the **longitude** of periastron (symbol π [or ϖ but may lead to confusion) ), which is equal to the argument of periastron (ω) plus the longitude of ascending node (Ω). In other words:

π = ω + Ω

Now, the **argument** of periastron is equal to:

ω = arccos $\frac{\mathbf{n}\ ·\ \mathbf{e}} {|\mathbf{n}||\mathbf{e}|}$

If *ez* < 0 then ω → 2π − ω.

where:

$\mathbf{n}$ is a vector pointing towards the ascending node (i.e. the *z*-component of $\mathbf{n}$ is zero), and

$\mathbf{e}$ is the eccentricity vector (a vector pointing towards the periastron).

This formula can be found at <https://en.wikipedia.org/wiki/Argument_of_periapsis> (I call it instead the “argument of periastron,” as Meeus says: “The word ‘periapse’, used by some authors, is incorrect. The word *perihelion* means the point of the orbit that is closest to the Sun [from the Greek *peri* = near + *helios* = Sun]. Similarly, *perigee* is the point closest to the Earth [*ge* = Earth]. Therefore, ‘periapse’ would really mean the point closest to the apse; but this is ridiculous, because what is meant is the apse itself!”)

So, using the above formula, you should be able to figure out the longitude of the ascending node for each of your hypothetical bodies.

Have fun!

> **Comments:**
> **username_2**: Note that the equation π = ω + Ω represents a compound angle; If the Orbital Plane has non-zero Orbital Inclination, you cannot just add the values of the Longitude of the Ascending Node and the Argument of Periapsis to get the Longitude of Periapsis; you're going to have to do some more complicated spherical trigonometry.  
> **username_1**: It may be a compound angle, but that’s the definition of “longitude of periastron,” as can be seen here https://en.wikipedia.org/wiki/Longitude_of_the_periapsis and in any and all other references on the subject you may find.  

---

