When the inclined orbit crosses the reference plane, why does it go "up" and "down" twice?
The intersection point is through the reference plane, not through the earth.
Planets and Small Solar System Objects
Imagine an oblique path around the sun, intersecting a reference plane passing through the sun. A celestial body moving from the south side to the north side of the plane is called the ascending node, and vice versa is called the descending node. Knowing the orbital parameters, we can convert this direction into elapsed time.
Guess first
When a celestial body passes through the ascending node of the ecliptic, does that mean it hits the earth?
Show answer and explanation
No, just across the reference plane — The earth is not necessarily in the direction of the intersection, nor is it necessarily in the same position as the celestial body.
Then connect experience and science
First determine which reference plane to use
The intersection points belong to the ecliptic equinoxes specified by the orbital roots. If you switch to another reference plane, the intersection point may be slightly different.
The reference plane must be consistent with the root number
Find the direction through the plane
The angle measured from the ascending node to the perihelion along the direction of motion is ω, so the two intersections correspond to a fixed true perihelion angle.
Rising: v=−ω; falling: v=180°−ω
Change direction to time
The ellipse first calculates E from v, then calculates M and divides it by the average motion; the parabola uses the Barker form and does not force the ellipse formula to apply.
Ellipse: t=T+M/n
Understanding the boundaries of orbital approximation
These formulas assume perturbation-free orbits. To obtain a more accurate true intersection, one needs to calculate the yellow latitude nearby and find the zero point.
The intersection point does not mean a collision with the earth
Expand the calculation
ν升 = −ω；ν降 = 180°−ω E = 2atan2(√(1−e)sin(ν/2), √(1+e)cos(ν/2)) M = E−e sin E n = k/a^(3/2) t交点 = Tperihelion + M/n
Expand the elliptical branches above. When calculating M and n, use radians and radians/day, and choose the time branch of the target intersection point of this lesson; the Barker equation should be used instead for the parabola, and e=1 cannot be substituted into the ellipse formula.
give it a try
Compare the two intersection points of Halley's Comet: they are not at the moment of perihelion symmetry, nor are they at the same heliocentric distance.
No need to memorize symbols
- ω / v / E / M
- Perihelion argument / true perihelion angle / partial perihelion angle / mean perihelion angle
- T
- The time of perihelion passage, not the number of centuries
- n
- Average exercise, degrees/day
- q / a / e
- Perihelion distance/semi-major axis/eccentricity
Algorithm, accuracy and scope of application
Implement 39.1–39.4 ellipse algorithm, parabolic Barker form and three examples from the original book. The ellipse uses atan2 to find the signed E, M=E−e sinE is calculated in radians, and then converted to degrees to divide the average motion; select the nearest equivalent events before and after perihelion to avoid mistakenly moving the ascending node by a whole cycle. The parabola has no finite distance/time at v=±180°, and the return is invalid. The graphical representation of the inclination angle only refers to passing through the plane and does not participate in the calculation; the Helin-Roman representation of the inclination angle is not the root number provided in the original book. Actuarial calculation of yellow latitude zero point without adding perturbation and arbitrary date.
my understanding check
How does the ascending node cross the reference plane?
from south side to north side — This is the definition of an ascending node; a descending node goes in the opposite direction.
Can ellipses and parabolas use the same time formula without distinction?
cannot — An ellipse has mean motion and a period; a parabola uses another relationship.
If the reference equinoxes are changed for the roots in this lesson, does the intersection point still necessarily belong to the original ecliptic plane?
Not necessarily — The intersection point shall be defined relative to the reference plane specified by the orbital root number.
First find which direction of which plane the track crosses, and then use the track type to correctly convert the direction into time.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 39 chapter
Planets and Small Solar System Objects · All interactive courses