NPF rule vs 500 rule: untracked exposure limits on modern camera sensors

Where the 500 rule came from

No original publication of the 500 rule has turned up. The earliest statement we found is on Dick Hutchinson’s aurora page from Circle, Alaska, written in the film era: divide 600 by the lens focal length to get the number of seconds you can expose before stars begin to trail. Frédéric Michaud of the Société Astronomique du Havre traces the 500 version in his notes behind the NPF rule. He also cites a 1997 photographic sky atlas, which we have not checked.

Michaud shows where 500 comes from. On 35 mm film, a circle of confusion of 0.029 mm was the accepted size of a “point”. Keeping a star’s drift within it gives t ≈ 13,713 × 0.029 ÷ (f × crop × cos δ), which is 398 to 562 divided by f × crop for declinations from 0° to 45°. The middle of that range is 500.

That origin explains the rule’s flaw on digital cameras. At the celestial equator the 500 rule always allows a trail of about 36 µm on a full-frame sensor, and about 24 µm on an APS-C sensor with a 1.53 crop factor, whatever the focal length (computed). The rule ignores pixel size, so the trail stays 36 µm whether the pixels are 4 µm or 6 µm.

The NPF rule

Michaud published the NPF rule, short for Nightscape Photography Formula, after developing it in 2010 for untracked photography of the night sky with camera lenses. He states that it does not suit telescope or planetary work. He asks that any implementation credit him and the SAH, and both forms below are his. The untracked exposure calculator computes them side by side with the 500 rule. The complete form is:

t (s) = k × (16.9 N + 0.10 f + 13.7 p) ÷ (f × cos δ)

N is the f-number, f the focal length in millimeters, p the pixel pitch in micrometers, and δ the lowest declination in the frame, meaning the one closest to the celestial equator (0 if unknown). k sets the tolerance: at k = 1 a star moves by one radius and stays round, at k = 2 it moves by one diameter and the trail is slightly perceptible, at k = 3 the trail is perceptible.

The coefficients come from a model of the star’s size and speed on the sensor. Michaud adds three blur terms: diffraction measured out to the second dark ring, which lies between the first and second bright rings (a diameter of 4.47 λN, at 550 nm), seeing of 3″ projected at the focal length, and 2 pixels for Bayer demosaicing. The star drifts at f × cos δ ÷ 13,713 per second, where 13,713 is the sidereal day of 86,164 s divided by 2π. The IERS gives the conventional sidereal day as 86,164.09 s, which gives the same 13,713 (computed). Multiplying out reproduces 16.9, 13.7 and 0.10 (computed).

The simplified rule assumes k = 2, a star on the celestial equator and no seeing:

t (s) = (35 N + 30 p) ÷ f

With k = 2 the complete rule’s coefficients become 33.8 and 27.4; Michaud’s simplified 35 and 30 sit just above them. The seeing term drops out because it is small: it adds only 0.1 × k seconds at any focal length (computed).

Pixel pitch on real bodies

The Nikon Z 8 and Nikon Z6III share a 35.9 × 23.9 mm sensor, with 8256 and 6048 pixels across. Nikon does not publish a pitch; dividing gives 4.35 µm and 5.94 µm. The Fujifilm X-T5 has a 23.5 mm wide APS-C sensor and 7728 pixels across, 3.04 µm (all pitches derived). With two lens settings, at δ = 0 (computed):

Body and lens	500 rule	500-rule trail	NPF simplified	NPF complete, k = 2
Z 8, 20 mm f/1.8	25.0 s	8.4 px	9.7 s	9.2 s
Z6III, 20 mm f/1.8	25.0 s	6.1 px	12.1 s	11.4 s
X-T5, 20 mm f/1.8	16.3 s	7.8 px	7.7 s	7.4 s
Z 8, 14 mm f/2.8	35.7 s	8.4 px	16.3 s	15.5 s
Z6III, 14 mm f/2.8	35.7 s	6.1 px	19.7 s	18.6 s
X-T5, 14 mm f/2.8	23.3 s	7.8 px	13.5 s	12.9 s

The 500 rule gives the two Nikons identical times because their sensors match in size. The NPF rule gives the Z 8, with its smaller pixels, about 20 % less. The f-number matters too: stopping down to f/2.8 enlarges the diffraction spot, so the NPF rule tolerates more drift. The X-T5 also differs in its color filter: Fujifilm describes an X-Trans sensor, while Michaud’s demosaicing term assumes a Bayer array. For the trail column, the calculator’s check is 15.041″/s × t × cos δ ÷ image scale, with the sidereal rate from the IERS day (1,296,000″ ÷ 86,164.09 s, computed).

Declination

Stars near the celestial poles cover less sky per second, by the factor cos δ. Using the lowest declination in the frame keeps every star within the chosen k. For the Z 8 at 20 mm f/1.8 with k = 2 (computed):

Lowest declination in frame	Factor	Exposure
0°	1.00	9.2 s
30°	1.15	10.6 s
50°	1.56	14.3 s
60°	2.00	18.4 s

Michaud recommends the complete rule when you frame a known region, such as the Big Dipper or the area around Polaris, and δ = 0 for wide Milky Way frames, where many stars near the celestial equator are in the picture.

Limits of both rules

Michaud describes the NPF rule as neither systematically pessimistic nor optimistic for an average system: about 3″ of seeing, a real lens with moderate aberrations and a Bayer sensor. With sub-arcsecond seeing, a lens near the diffraction limit and a sensor without a low-pass filter, stars can be smaller than the model assumes, and the allowed drift should shrink in proportion. His strictest variant limits drift to one pixel, t ≈ 14 p ÷ (f cos δ), or 3.0 s for the Z 8 at 20 mm (computed), and he calls it unrealistically short. The 500 rule’s only model is the film-era circle of confusion. The DSLR and mirrorless sensor table lists pixel pitches for the bodies in our data, and the field of view guide explains why crop factor changes framing but not the drift speed on the sensor.