What it does
Double stars are described by two numbers: the separation ρ, the angle between the two stars in arcseconds, and the position angle θ, the direction of the fainter star B from the brighter star A, measured from north through east. Amateur measurements of both are still collected and used to work out binary orbits.
This calculator does two jobs. Predict from an orbit takes the seven orbital elements of a binary and gives ρ and θ on any date. It draws the apparent orbit as it looks on the sky, lists a 12-step ephemeris, and, with the parallax, gives the real size of the orbit and the total mass. Measure my image lets you open your own photo, tap the two stars (each tap snaps to the centre of the star’s light), and works out ρ and θ from your camera’s pixel size, focal length and orientation.
How to use it
- Choose Predict from an orbit. Pick Sirius or η Coronae Borealis, or type the elements of another binary from the Sixth Orbit Catalog: period P, periastron time T, eccentricity e, semi-major axis a in arcseconds, inclination i, node Ω and periastron argument ω.
- Pick a date. Add the parallax in milliarcseconds to get the mass.
- Or choose Measure my image. Open a photo, tap star A, then star B, or type their pixel positions. Enter your camera’s pixel size and the telescope’s focal length, and the position angle of the image’s top (0° when north is up). Tick mirrored if a star diagonal flipped the view.
- Download the results as a PNG, PDF or Word file.
How it is worked out
For the prediction, Kepler’s equation gives the eccentric anomaly, then the radius r and true anomaly ν. The position angle follows from tan(θ − Ω) = sin(ν + ω) cos i ÷ cos(ν + ω), and the separation from ρ = r cos(ν + ω) ÷ cos(θ − Ω), as in chapter 57 of Meeus’s Astronomical Algorithms. The mass comes from Kepler’s third law in solar units: M = (a/ϖ)³ ÷ P², with a and the parallax ϖ in arcseconds and P in years. For a measurement, the image scale is 206.265 × pixel size (µm) ÷ focal length (mm) arcseconds per pixel. The angle is measured from the image’s top, adding its position angle.
Worked example
Meeus’s example for η Coronae Borealis (P = 41.623 years, T = 1934.008, e = 0.2763, a = 0.907″, i = 59.025°, Ω = 23.717°, ω = 219.907°) on 1 January 1980 gives θ = 318.4° and ρ = 0.411″, exactly as in the book.
For Sirius, with the orbit by Bond and colleagues (2017), the white dwarf Sirius B was closest to Sirius A, about 2.9″ away, at periastron in 1994. It reached its widest, about 11.3″ at θ ≈ 62°, in 2023, which is why it is now easier to see. With the parallax of 379.21 mas the orbit’s semi-major axis is 19.8 AU, and the pair weighs 3.07 solar masses.
Frequently asked questions
Why is the position angle measured towards the east, which is to the left? That is how it looks on the sky with north up: east is to the left, because we look up at the sky from below. Position angle runs north (0°), east (90°), south (180°), west (270°).
How do I find the orientation of my image? Turn the drive off and let a star drift: it moves towards the west. Measure the drift direction’s angle on the image; the top of the image is at that angle plus 270°. Or plate-solve the image.
How accurate is a measurement? A one-pixel error moves the separation by one pixel’s worth of arcseconds. Averaging several images, and calibrating the scale on a wide pair with a well-known separation, both help.
Does my image leave my device? No. It is opened and measured in your browser, and nothing is uploaded.
Where the numbers come from
- Orbit formulas: Meeus, Astronomical Algorithms, chapter 57.
- Example orbits: Sirius from Bond et al. (2017) and η CrB from Meeus's worked example (seven numbers each).
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Last updated 2026-11-08.