Knowing how flat an orbit is is not enough to tell us where a planet is?
It's like placing a tilted ring: the shape, orientation and where the object goes are all explained.
Planets and Small Solar System Objects
Tracks of the same shape can be placed flat, tilted, or in other directions. To give a spatial position, the word "ellipse" is not enough. This lesson puts the six orbital roots under your control.
Guess first
If two orbits have the same shape, do they necessarily coincide in space?
Show answer and explanation
Not necessarily, the orientation may be different — Shape and spatial orientation are different parameters.
Then connect experience and science
Determine the shape first
The semimajor axis determines the scale, and the eccentricity determines how flat the ellipse is. The sun is at the focus, not the center of the ellipse.
a, e: scale and shape
Then lift the track plane
The inclination angle is the angle between the orbital plane and the reference ecliptic plane; the ascending node determines the orientation of the intersection of the two planes in the reference plane.
i, Ω: plane orientation
Rotate the perihelion in the plane
The perihelion argument is measured from the ascending node along the orbital plane. The perihelion longitude is Ω+ω, do not confuse it with the argument.
ϖ=Ω+ω
Finally, explain where to go
The mean periapsis angle gives the time phase in the orbit. Puller's equation is first solved, and then the plane position is rotated to three-dimensional coordinates.
M=L−ϖ；E−e sinE=M
Expand the calculation
ϖ = Ω + ω M = L − ϖ E − e sin E = M r = a(1−e cos E) x = r[cos Ω cos(ω+ν) − sin Ω sin(ω+ν) cos i] y = r[sin Ω cos(ω+ν) + cos Ω sin(ω+ν) cos i] z = r sin(ω+ν) sin i
Ω is the yellow longitude of the ascending node, and ω is the argument of perihelion; the sum of the two is ϖ. Solve Pler's equation in radians, and use the rotation formula to find trigonometric functions in the specified angular units.
give it a try
First adjust the inclination to 90°, then turn to the ascending node and observe the projection; the sun is still in focus.
No need to memorize symbols
- a / e
- Semi-major axis AU/eccentricity
- i / Ω
- Inclination / ascending node celestial longitude
- ω / M
- Perihelion argument / mean perihelion angle
Algorithm, accuracy and scope of application
Chapter 31: Definition of orbital radicals, combined with Chapter 30/33 Kepler solution and heliocentric three-dimensional rotation. M is different from the true periapsis angle; the sun is at the focus, and the spatial position is obtained by Rz(Ω)Rx(i)Rz(ω) acting on the orbital plane vector. All angles are relative to the specified reference ecliptic, and the ephemeris model is not changed by dragging.
my understanding check
Where is the sun in the ellipse?
a focus — Elliptical orbit with the sun at focus.
What is the relationship between perihelion longitude and argument?
ϖ=Ω+ω — Both are measured along the reference plane and orbital plane respectively.
Can the mean near-point angle M be directly regarded as the true direction ν?
Generally not, you have to solve Puller's equation — The time phase differs from the true direction in non-circular orbits.
The shape of the orbit, the orientation of the plane, the direction of perihelion, and the time phase are all described separately.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 31 chapter
Planets and Small Solar System Objects · All interactive courses