Star trail length calculator

Enter the exposure duration in seconds and the object's declination, and it returns the ideal angular trail length in degrees. The calculation uses Earth's mean sidereal rotation rate and the cosine projection for declination, making it useful for planning night-sky photographs, comparing exposure choices, and checking expected motion across an imaging sensor.

What the calculated trail length represents

Earth turns once relative to the distant stars in a sidereal day, so stars appear to travel across a fixed camera's field throughout an exposure. The calculator reports the angular length of that ideal apparent motion in degrees. A target on the celestial equator traces the largest circle and receives the full sidereal rotation rate. As declination approaches either celestial pole, its circle around the pole becomes smaller, so its angular trail becomes shorter for the same exposure. Positive and negative declinations at equal magnitudes produce equal lengths because the cosine projection is symmetric. This result describes angular sky motion, not a distance measured in pixels and not the object's physical movement through space. Converting the answer to a sensor length requires the image scale, usually expressed in arcseconds per pixel. The model assumes a stationary, perfectly fixed optical system and ideal sidereal motion. Tracking, polar alignment error, atmospheric refraction, field rotation, lens distortion, and a star's proper motion are outside its scope.

How exposure and declination enter the formula

The calculation uses L = t × (360 / 86164.0905) × cos(δ). Here, L is angular trail length in degrees, t is exposure duration in seconds, 86164.0905 seconds is the adopted mean sidereal day, and δ is declination converted from degrees to radians for the cosine. Exposure scales the answer directly: doubling the exposure doubles the ideal trail length. Declination supplies the geometric projection: zero degrees keeps the complete rate, 60 degrees reduces it by half, and either pole reduces it to zero in the ideal model. The calculator accepts finite declinations from -90 through +90 degrees, inclusive. Exposure must be finite and cannot be negative, because a negative duration has no physical meaning; an exposure of zero is valid and returns a zero-length trail. Results are rounded to twelve decimal places for stable browser and API output. The rotation-rate field is also returned so the adopted constant is explicit and results can be audited or reused in a larger imaging calculation.

Using the result for astrophotography planning

Use the angular length to compare candidate exposure times before a fixed-tripod photograph or an intentionally trailed composition. To estimate the trail in arcseconds, multiply the returned degree value by 3,600. If the camera system's plate scale is known in arcseconds per pixel, divide that arcsecond length by the plate scale to estimate the trail across the sensor in pixels. This helps assess whether motion will remain below a chosen sharpness tolerance or become visibly elongated. Remember that the formula depends only on exposure and declination; focal length, pixel pitch, and sensor dimensions do not change the angular motion, though they strongly affect how that motion appears in an image. Very wide lenses may make short trails inconspicuous, while long focal lengths reveal the same angular displacement across many more pixels. For tracked mounts, the result represents the motion the mount is intended to cancel rather than the residual trail. Each automated API calculation costs $0.002, while the deterministic calculation can also run locally in the browser.