How to plot this orbit?

Question

I have these data for some orbits, without any explanation about meaning of each datum:

{
  "objects": [
    {
      "name": "Earth",
      "id": "solar.planet.earth",
      "elements": {
        "q": 0.98322073104899
        "a": 0.99920537923658
        "node": 207.00538100359,
        "e": 0.0159973600220234 
        "peri": 256.950444448438,
        "incl": 0.0038067895470298  
        "T": 2459218.90577274
      },
      "position": {
        "y": 0.9297659392321,
        "z": -4.515675204152e-05,
        "x": 0.3275590103186
      }
    },
    {
      "position": {
        "z": 0.06029318307615,
        "x": -0.4584994822692,
        "y": 1.094698818492
      },
      "elements": {
        "q": 0.806176191073582,
        "a": 1.0323483836211,
        "node": 75.5720068963318,
        "e": 0.219085142318127,
        "incl": 4.80809622578363,
        "peri": 179.397909345983,
        "T": 2459318.63483092
      },
      "id": "solar.minorplanet.98943",
      "name": "2001 CC21"
    },
    {
      "name": "1998 KY26",
      "id": "solar.minorplanet.1998KY26",
      "elements": {
        "T": 2459001.17727408,
        "incl": 1.48102189788285,
        "peri": 209.372033608463,
        "e": 0.201828518064676,
        "node": 84.3664587661475,
        "a": 1.23285180453922,
        "q": 0.984027151835704
      },
      "position": {
        "y": 1.407810076987,
        "z": -0.002082322485705,
        "x": 0.2198001273522
      }
    },
    {
      "name": "Hayabusa 2",
      "id": "solar.spacecraft.hayabusa2",
      "elements": {
        "T": 2459212.01173911,
        "incl": 5.99763881195244,
        "peri": 205.214139025915,
        "e": 0.187058473659886,
        "node": 253.856900249799,
        "q": 0.970678629676518,
        "a": 1.19403253275367
      },
      "position": {
        "z": 0.006301058519122,
        "x": 0.3323137692169,
        "y": 0.932379875888
      }
    }
  ],
  "TT": 2459186.10708546,
  "UTC": "2020-12-02T14:33:03"
}

I am trying to plot them in 3d in this JSFiddle, but I am proceeding randomly.... can anybody help?

The final result should be something like the one visible in the page which use these data (and in this page in English), but I want to represent it in 3d.

The accepted values for spacekit library are:

• initialValues.a Semimajor axis

• initialValues.e Eccentricity

• initialValues.i Inclination

• initialValues.epoch Epoch in JD

• initialValues.period Period in days

• initialValues.ma Mean anomaly

• initialValues.n Mean motion

• initialValues.L Mean longitude

• initialValues.om Longitude of Ascending Node

• initialValues.w Argument of Perihelion

• initialValues.wBar Longitude of Perihelion

Ok for incl, peri and node, I can make a guess for "a" ( initialValues.a - Semimajor axis ), "e" (initialValues.e - Eccentricity), but what about "q" and "peri"?

• initialValues.a Semimajor axis ---> "a"

• initialValues.e Eccentricity ---> "e"

• initialValues.i Inclination ---> "incl"

• initialValues.epoch Epoch in JD ---> "T"

• initialValues.period Period in days --->?

• initialValues.ma Mean anomaly --->?

• initialValues.n Mean motion --->?

• initialValues.L Mean longitude --->?

• initialValues.om Longitude of Ascending Node ---> "node"

• initialValues.w Argument of Perihelion ---> "peri"?

• initialValues.wBar Longitude of Perihelion ---> "peri"?

I also retrieved data from NASA Horizons for Earth to compare them to available ones, but I can't anyway figure out what "q" and "peri" match to:

• JD 2459218.905775463,
• A.D. 2021-Jan-04 09:44:19.0000,
• EC Eccentricity, e 0.01637
• QR Periapsis distance, q (km) 147092887 km
• IN Inclination w.r.t X-Y plane, i (degrees) 0.00285
• OM Longitude of Ascending Node, OMEGA, (degrees) 177
• W Argument of Perifocus, w (degrees) 284.4
• Tp Time of periapsis (Julian Day Number) 2459216 km
• N Mean motion, n (degrees/sec) 0.0001141
• MA Mean anomaly, M (degrees) 2.15
• TA True anomaly, nu (degrees) 2.22
• A Semi-major axis, a (km) 149541385 km
• AD Apoapsis distance (km) 151989883 km
• PR Sidereal orbit period (sec) 31540276 km

---

• "q": 0.98322073104899, Periapsis distance, q (AU) = 147087727 km
• "a": 0.99920537923658, Semi-major axis, a (AU) = 149478997 km
• "node": 207.00538100359,
• "e": 0.0159973600220234, Eccentricity, e
• "peri": 256.950444448438,
• "incl": 0.0038067895470298, Inclination w.r.t X-Y plane, i (degrees)
• "T": 2459218.90577274

Answer 1

Work in progress, new figures will follow.

I took the orbit data of Hayabusa2 we got to plot the magenta ellipse. I used the initialValues.a Semimajor axis and initialValues.e Eccentricity. The two red dots are the foci of the ellipse.

The blue dot is the one position of Hayabusa2 we got, only x and y were used.

Sun is at the center x = 0 and y = 0, of course one focus point is there too.

Then I rotated the ellipse by the node": 253.856900249799 angle to get the green angle. The one and only position of Hayabusa (blue dot) is now on the rotated ellipsis but not on the magenta ellipse.

I added the axes, the lines in magenta are the axes of the ellipse (magenta too) aligned to the coordinate system. The green lines are the rotated axes belonging to the green ellipse. The 'x' markers in magenta are the foci of the unrotated ellipse. The x marker at the origin is one focus of the green rotated ellipse too. The black arc shows the node angle. I added some annotations.

In the next step this 2D plane should be inserted to the 3D plot. The Sun should be at the center for 2D and 3D plot. Hayabusa2 should be at the given xyz position. The "incl": 0.0038067895470298, Inclination w.r.t X-Y plane, i (degrees) should be correct too.

But we need 3 points to define the correct orientation of the plane. We may look if the Earth orbit may be used for some additional information. But we should have a time when both Earth and Hayabusa are at the same location. Crossing the Earth orbit alone would not help when Earth is far away from that point when Hayabusa is there. The two small bodies should be close to the Hayabusa orbit too.

The 3D plot to include the 2D plot above.

Answer 2

You need to learn a little about the many ways in which orbit data can be represented, starting from a basic tutorial like https://en.wikipedia.org/wiki/Orbital_elements

One of the important things to keep in mind is there are no more than six independent numbers out of that set, but not just any six can be chosen. If you input values for too many, things will go wrong because they won't be consistent with each other. For example, if you have semi-major axis $a$ and eccentricity $e$, then by definition periapsis distance is $a(1-e)$ and apoapsis distance is $a(1+e)$, so specify no more than two of those four or you invite problems. Similarly, mean motion, period, and semi-major axis are all redundant with each other, assuming you know which body is being orbited (because the conversion involves its mass). True anomaly, mean anomaly, and time since perigee passage or other reference epoch are another set from which you should choose at most one to be an input.

The traditional order of the Keplerian orbital elements is semi-major axis $a$, eccentricity $e$, inclination $i$, right ascension of the ascending node (RAAN) $\Omega$, argument of periapsis $\omega$, and true anomaly $\nu$. Be aware that some of these definitions break down if some of the others take on certain values. For example, if inclination is exactly zero, then there isn't an ascending node so it doesn't have an $\Omega$, but that still specifies the orbit plane uniquely. Similarly, if eccentricity is exactly zero, then there isn't a periapsis (all points are the same distance from the center, because the ellipse with $e=0$ is a circle), so you need to define $\varpi$ rather than $\omega$ to say where to start measuring $\nu=0$.

Answer 3

It looks like there is not a possible answer to this question: the available data are not enough to plot a 3d orbit. Although there are apparently 6 orbital elements, they are actually just 5; indeed these three data are provided:

 - "e": 0.187058473659886
 - "a": 1.19403253275367
 - "q": 0.970678629676518

But:

 periapsis distance = a * ( 1 - e) = 1.19403253275367 * (1 - 0.187058473659886 ) 
 = 0,970678629676520

So as a matter of fact we have only 5 orbital elements:

 - "incl": 5.99763881195244,
 - "peri": 205.214139025915,
 - "e": 0.187058473659886,
 - "node": 253.856900249799,
 - ("q": 0.970678629676518,)  <<<====== can be calculated from a and e
 - "a": 1.19403253275367

They are not enough to plot 3d orbit; the original page using these data only plots orbits in 2d; by reversing engineering the source code we can see the "q" parameter is not used at all:

---

key: "calcEllipse",    
value: function calcEllipse(object) {


////// "a" and "e"//////
var cx = -factor * object.elements.a * object.elements.e;     
var cy = 0;    
var rx = factor * object.elements.a;    
var ry = factor * object.elements.a * Math.sqrt(1 - Math.pow(object.elements.e, 2));    


////// "node", "incl" and "peri" //////
var transform = "\n              rotate(".concat(-1.0 * object.elements.node, ")\n    
scale(1.0, ").concat(Math.cos(object.elements.incl / 180 * Math.PI), ")\n    
rotate(").concat(-1.0 * object.elements.peri, ")");  
return {    
cx: cx,    
cy: cy,    
rx: rx,    
ry: ry,   
transform: transform

Passing just 5 parameters to spacekit.js library results in error:

NaN position value - you may have bad or incomplete data in the following ephemeris:` 
a: 0.937550412197935, 
e: 0.140879661549663, 
i: 0.0748349916816544, 
om: 1.2865260962475196,
w: 2.0673222342313817`