When two stars orbit around each other, why do the orbits seen through a telescope become flattened?
The true orbit has an inclination; what the telescope gives you is its projection on the vertical plane of view.
Like slowly turning a ring from the front to the side, it will become flatter and flatter. The orbits of binary stars also undergo such a projection; just because they look flat on the screen, it does not mean that the real orbits are equally flat.
Guess first
If a ring looks flat from the side, does that mean it has been squashed?
Show answer and explanation
No, just projection — Viewing direction changes the 2D appearance.
Then connect experience and science
Select the time first, then calculate the real position
The period and periapsis time determine the mean periapsis angle; Kepler's equations convert this into true orbital position.
M=360(t−T)/P；M=E−e sinE
Put the track into your field of vision
Orbital tilt compresses one direction. When viewed from the side, the entire visual orbit even degenerates into a line.
Orbital transverse component multiplied by cos i
Report position using angular distance and position angle
The main star is at the origin, and the companion star uses ρ to indicate how far apart it is, and θ to indicate which way it is facing; θ=0 points to the north, and 90° points to the east.
ρ=hypot(N,E)；θ=atan2(E,N)
Don’t mistake the apparent orbit for the real orbit
Original book example 57.a The real e is about 0.276, and the apparent orbit e′ is about 0.860; the projection changes the shape.
e′ is determined by the true e, the inclination angle i and the periapsis direction w
Expand the calculation
M = 2π(t−T)/P E − e sin E = M r = a(1−e cos E) N = r[cos Ω cos(ν+ω) − sin Ω sin(ν+ω) cos i] EEast = r[sin Ω cos(ν+ω) + cos Ω sin(ν+ω) cos i] ρ = √(N²+EEast²)；θ = atan2(EEast,N)
The first line and Kepler's equations are in radians. N goes north, E east goes east, E east is not the periapsis angle E; ρ and a are both arc seconds, and θ is the position angle from north to east.
give it a try
First read it according to the angle of the original book, and then change it to "front" and "side". The eccentricity of the protostar remains unchanged, but its apparent orbit changes.
No need to memorize symbols
- P / T
- Revolutionary period and periapsis year
- a / ρ
- True orbit semi-major axis arc seconds / projected separation arc seconds
- i / Ω / w
- Inclination angle, ascending node position angle, periapsis direction
- θ
- Position angle from north to east
Algorithm, accuracy and scope of application
Covers Chapter 57 position angle, angular separation, apparent orbit eccentricity and η CrB, γ Vir original book examples/exercises. The real orbit equation uses r, ν, and then projects into the north/east components; avoid division where cos(θ−Ω) is 0. The viewing angle can be switched as a model experiment; changing the viewing mode does not mean a sudden change in the real orbit.
my understanding check
What is the unit of distance ρ between stars in this lesson?
arc seconds — It is an angular separation in the sky.
What does the apparent orbit look like when the inclination angle is 90°?
a line — The vertical line of sight projection flattens one dimension.
When the projections of the companion star and the main star coincide with each other, can a unique position angle be given?
cannot — Directional angle degradation at zero separation.
The real orbit of the binary star is not the same as the apparent orbit in front of you; the inclination and orientation must be taken into account when calculating the position.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 57 chapter