# Improved "500 Rule" Star Trail Calculator

## To compute maximum shutter time to prevent star trails on a fixed mount, like for wide angle Milky Way photos

The "500 Rule" was a simple guide (in older days for 35 mm film) calculating the maximum allowable exposure time of star photos due to the Earth’s rotation making star trails when camera is on a fixed tripod. Longer exposure causes the earth rotation to make longer star trails instead of dots. Greater focal length does magnify the Earth’s rotation. But the original 500 Rule is Not about other sensor sizes, nor about computing a useful exposure time. The "Rule" is only about a maximum exposure time for minimum and acceptable trail length due to the Earth's rotation.

500 Rule:  Maximum Seconds = 500 / focal length

That was all there was to the original rule. The 500 Rule assumes 35 mm film popular at the time, or today for a Full Frame 1x crop factor. However, for other sensor sizes, the same allowable movement and corresponding Rule Number is computed as:

Maximum Seconds = 500 / (Crop Factor x focal length)

The Rule will be more usable (longer exposure) when using a short lens for a wide sky view than for an enlargement of a smaller area. You will also have to use the ISO and f/stop that gives an acceptable exposure, and that gets much harder. The 500 Rule definitely does limit actual photographic exposure time. Some people use it as the 600 Rule because it specifies more time. Others use it as the 400 Rule for less star trails but less exposure. For example, 500 Rule / (1.6 crop factor x 12 mm lens) = 26 seconds maximum time for perhaps acceptable absence of trail blur, but 26 seconds may need ISO 4000 f/2.8 for a minimal exposure of stars. A short lens works better for this (a short lens is less magnification of the Earth's rotation). A wide angle sky view with a short lens is a better view of the Milky Way anyway (because it is full sky large up there). And don't miss seeing  below  about an easy and inexpensive DIY rotation tracker.

But even when factoring in the crop factor (which is sensor size), the original Rule ignores all other factors. But if the shutter exposure is longer than a second or two, then the trail length is also longer than zero. Even photographing the Moon requires sunlight exposures because it moves (but it is illuminated by the same Sun that the Earth sees). However in today's cameras, a trail 5 or 6 pixels long is usually not noticeable unless viewing is enlarged more than usual. This calculator computes for sensor size and CoC and trail length, but the Rule does not specify a desired exposure. For exposure of stars in a dark sky on a fixed mount, using a 14 mm lens on a 1x sensor, I’d suggest planning to start at least about f/2.8 ISO 3200 and 30 seconds, which would be Rule 420 and 1.02X CoC and EV -2 (but 500 Rule allows 36 seconds, but which is only 1/4 EV more). 30 seconds is a little more motion, so will be best with a short lens (like 14 mm) for a lessor enlargement. For a 1.5x or 1.6x crop camera, a 9 mm lens would be about the same result, about Rule 260 to 290. Be aware that the Moon is not a star, but is illuminated by our own Sun, and a Full moon itself needs more nearly the same as other exposures in sunlight, which the moon is.

An example of Crop Factors shows the calculator concept here. There's a lot here, and you should notice some relationships in the data of this example.

CropSensor
mm
Lens
mm
Field
of View
Pixels
W×H
The
Rule
Seconds
for Rule
Earth
Rotation
CoC
Mult
Trail_Length
mm    Pixels
1x36×241890° wide6000×400050027.780.116058°1.22x0.03656.08
2x18×12990° wide6000×400050055.560.232115°2.43x0.036512.2
2x18×12990° wide6000×400025027.780.116058°1.22x0.01826.08
2x18×12990° wide6000×400020622.890.095632°1x0.0155.01
1.5x24×161290° wide6000×4000333.33327.780.116058°1.22x0.02436.08
1.6x22.5×1511.2590° wide6000×4000312.527.780.116058°1.22x0.02286.08
• First row is a full frame 1x camera and 500 Rule, like the old 35 mm film days.

Second row is a 2x crop camera (called a Four Thirds model), and an Unadjusted 500 Rule (still 500, not the best choice, twice worse).

Third row properly adjusts Crop 2x to Rule 250.

Fourth row is Rule 412 adjusted to Rule 206 for crop factor 2x, which limits trail length to the normal 1x CoC limit as used to compute Depth of Field minimum acceptable sharpness.

Rows 5 & 6 are the other 1.5x and 1.6x crops, with Rule adjusted for crop factor (500 / crop factor). I used the six significant digits just so the Earth's rotation could fully show to be the same exact value (but yes, I do know crop factor and focal length are Not normally more precise than a couple of digits).

• The field named Seconds is Not about the photographic exposure, which is not determined by the Rule (but the longer the better for astrophotography, within reason). Instead Seconds is the Rule's time limit for the focal length that magnifies the Earth's rotational movement.
• Trail length: The rotation Trail length is the length dimension in mm on the sensor, and is also given as pixels on the sensor (pixels are not fractional, so round up). The trail is shorter if Rule is adjusted for smaller sensors, which however require corresponding greater enlargement to viewing size, so it comes out the same visual result again. The pixels (of same 6000x4000 number) are smaller on the smaller sensor.
• Lens mm (Focal Length): Cropped sensors routinely use a shorter equivalent focal length (shorter than the 1x sensor's focal length) to maintain the same Field of View size in both cameras, which is observed in this example too. Milky Way pictures will need a pretty wide view, and the shorter lens helps the Rule time limit.
• The 500 Rule divided by crop factor ( = 250 Rule here with 2x Crop) will give the same result as a 1x sensor and the 500 Rule (the trail is shorter on the smaller sensor, but the additional viewing enlargement result is the same). But the shorter focal length for the same Field of View used by a Crop Factor with an unadjusted 500 Rule (Second row again) will allow a longer maximum Rule exposure than is realistic, but that will be a longer rotation and a longer star trail blur, so the image quality will be less.
• It turns out that the 412 Rule adjusted for crop factor ( = 206 Rule here with 2x Crop) will limit the trail length to no more than the size of the maximum allowable CoC limit that also computes the Depth of Field range (acceptable Depth of Field stops when at Coc reaches 1x). That seems useful for stars too, because the Earth's rotation blurs the star dot. See the Crop Factor of many sensors and films,

We can also use CoC to judge star rotation blur trail motion length too. As evidence of that suitability, the original 500 Rule was a close approximation, computing 1.22x CoC for full frame 35 mm film sensors. Some users always used the 400 Rule (0.97x CoC) to improve sharpness. I'm suggesting Rule 412 is 1x CoC. And some like to use Rule 600 for more seconds of exposure, which is 1.42x CoC, a little outside the best sharpness. But (also depending on f/stop and ISO), star exposure may need more exposure time than the Rules allow on fixed mounts.

More than you may want to know:

Actually, specifications of sensor dimensions, crop factors, focal lengths, CoC values, are all normally rounded values. So the computing precision is slightly vague, but the 1x CoC computes as Rule 411.47. But 412 is close enough, and a sharper result than 500.

The relationship of CoC and star trails is that the factors of CoC, sensor size, crop factor and viewing enlargement are all determined by the sensor diagonal mm. The star trail is on the sensor dimension too. We may think sensor size is unimportant in todays digital, because digital doesn't have film grain to show. But sensor size is still quite important, because (regardless of the noise content of smaller sensors), the sensor diagonal determines how much external enlargement is required to view the DOF conventional standard 8×10 inch print. Enlargement determines how well we can see any detail or any blur. If you enlarge the sensor size 15x to view it, that viewing size divides the sensor resolution number by 15. Which is still usually adequate, except printing much larger than 8x10 inches can suffer.

CoC is Circle of Confusion, the diameter of an out-of-focus "point" (a hypothetical point of zero size, but seen when not at the focused distance), which is determined in lens calculations. The CoC number we see published is a maximum CoC size limit determining what is seen as sharp, or not, which is routinely used in Depth of Field to calculate the DOF near and far span extents. This CoC limit also depends on sensor size, with smaller CoC on smaller sensors, because it is calculated as its size on the sensor, but is seen in the enlarged viewing size of a standard 8×10 inch print.

CoC mm = (sensor diagonal mm / a constant determined by judging the blur size, but an exact blur size is very hard to judge). My calculator is using the CoC Divisor of 1442 which computes the standard CoC = 0.03 mm for full frame sensors (same as 35 mm film size). This CoC is sometimes seen as 0.0288 mm (from constant of 1500, which is another guess about sharpness), but it is 0.03 mm here, and routinely at most other places.

If this CoC notion were 0.96x CoC instead of 1x, it would compare same as if the 0.0288 mm CoC, which is then 395 Rule (for a full frame sensor). Note that a Rule is seconds × focal length, which involves both.

The mention of CoC here is NOT about Depth of Field. Depth of Field at Infinity focus for stars is mighty large, extending back to the lens Hyperfocal distance. A short lens at widest aperture likely extends DOF back to only a few feet, or a longer lens may be a couple of thousand feet, depending on focal length and aperture. So Depth of Field is not a concern for focus at Infinity for stars, unless including foreground objects. Depth of Field calculators will show Hyperfocal, and you can enter 999999 feet there for the Infinity distance (189 miles). CoC is the size of mis-focus blur, and is the well known measure of minimum acceptable image sharpness (the same 1x CoC limit of minimum acceptable sharpness that determines the Depth of Field range extents, when Depth of Field sharpness is conventionally viewed in a standard 8×10 inch print size). CoC is used here only as a known value to compare a star trail's rotational blur.

So Rule 412 (as adjusted for crop factor) limits the rotation trail blur to be no more length than the same size as the Depth of Field CoC limit that we think of as the minimum extremes for sharpness in a normal photo due to mis-focus. Depth of Field is computed as mis-focus blur less than 1x CoC is still sharp, but greater than 1x CoC is not. Depth of Field calculations draw a very firm line there, but that boundary is not real critical (no detectable difference just either side of it). So it might be 0.8x CoC or 1.2x CoC, but 2x CoC might be more concern.

• If the sensor had half the number of pixels (like 3000x2000) but the same sensor mm size, then the trail length is the same mm, but half the number of the larger pixels. Note that 5 or 6 pixels out of 6000 is 0.1%, and will still appear adequately sharp to us (unless we are really critical or viewing it considerably enlarged). Again, 1x CoC is the same DOF blur limit that we still consider to be a sharp print. I do suggest that the multiplier of the CoC limit is the better guide of sharpness than length of star trail.
• There is more info below, and more of these comparisons at page bottom.

CoC can be a few pixels in size, but our large megapixels today mean we normally can’t see that in an 8x10 inch viewing. Except however, if you are going to enlarge the viewing size of the image much larger than 8x10 inches, then you may want to look at blur length in pixels, and this calculator computes that too.

Regarding exposure, while there are great numbers of fainter stars, my notion is that f/2.8 at 30 seconds at ISO 3200 is a decent but rather minimal exposure in a dark sky with a short lens (meaning for about a 12 to 20 mm lens). However, 30 seconds with a longer lens will have objectionable star trails. The Exposure Calculator will compute Equivalent exposures of that, or of other values.

Determine Rise and Set times (your location)
for Sun, Moon, planets, and other major objects:

This calculator below offers other more specific goal-oriented methods to examine the trailing problem, offering visibility of the situation. It is NOT about proper exposure, but it does compute with actual sensor and pixel size. We can compute the exposure time that limits the star trail length to a specific length in digital pixels, or to a specific multiple of the Circle Of Confusion (the standard Depth of Field CoC guide of blurriness). Stars actually are point sources, so comparing to this existing standard for blurriness is very reasonable. However (on a fixed mount), any method involves a very big trade-off between enough exposure time for the stars, or less exposure time to reduce the Earth's motion blur.

If interested in the long blur trails to make the circular trails around the North Star, it is simply 360 degrees in 24 hours, which is 15 degrees per hour. But FWIW, say for 2 hours, you can enter time as 120 minutes x 60 = 7200 seconds to compute 30 degrees, or 6 hours x 60 x x 60 = 21600 seconds, to compute 90 degrees. You can ignore all the other numbers for that use.

There are major issues when trying to determine the sensor size of compact or phone cameras. Also with mixed formats (both video and still photo images from the same camera). These Sensor Size issues are summarized at Issues determining sensor size. Please see that if any issues.

## Milky Way Star Motion Trail on a fixed mount

Select one of four ways to specify camera sensor size
1 Sensor size
Width:

mm
Sensor size
Height:

mm
The larger number will be called the Width
2 Crop Factor x For Options 2 or 3:

Aspect Ratio

3 Focal length of this lens mm
Equivalent focal length
used on 35 mm film
mm
4 Film and
Sensor Size Description

Mostly this is Film Size. The CCD 1/x inch number
is a rather vague number (the size is WxH mm)
 Focal Length mm Sensor Width pixels For a goal of: Seconds of time A new Rule Pixels of trail × CoC blur Star Declination degrees Sidereal rate Solar rate Lunar rate

Focal length is the actual real focal length of your lens, NOT any Equivalent Focal Length.

Options 1-4 are four ways to specify camera sensor size.

• Option 1 - Best accuracy is when entering actual exact sensor dimensions (width and height in mm, from camera specifications).
• Option 2 - Or, if you can determine the accurate sensor crop factor, sensor size can be computed. For mixed cases (of still and movie formats in the same camera), please see Issues determining sensor size.
• Option 3 - You can specify both Equivalent Focal Length and its actual matching real Focal Length to compute the sensor size. If any questions about this, please see Issues determining sensor size.
• Option 4 - You may be able to select one of the general sensor descriptions. This is written for pixels, but if selecting a film size, then "Sensor Width in Pixels" will be made blank (not applicable, but CoC is still there). Millimeter results apply to the negative size, NOT to an enlarged print. Or if the film was scanned at 100% size (and not cropped and not resampled), enter it as Option 1, digital of dimensions of the full frame scanned image size (which is shown here for film sizes).

Film sizes should be accurate in the list, but the "1/x inch" sensor numbers require a little more. Did I mention to please see Issues determining sensor size?

Aspect Ratio computes Sensor Size and CoC and Megapixels, but it does not affect the “Rule” numbers. A red warning may be shown if the 3:2 or 4:3 Aspect Ratio you specify is not commonly appropriate for the Crop Factor computed with Option 2 or 3 sensor sizes (not shown for 16:9 video options). It means that likely the correct aspect ratio was not selected, which is an easy oversight. It can be ignored if it is actually OK (you could please let me know about the facts of that situation). It makes the standard assumption that crop factors less than 2x should be 3:2, or larger than 3x should be 4:3, which are the normal expected and required values. So this warning is not shown for One-inch or Four Thirds sensors (2x and 2.7x crops), because many of those recent cameras provide a menu to select any of 1:1, 3:2, 4:3, or 16:9 formats.

Pixels of Sensor Width is the largest dimension in pixels of the sensor image the camera took (if 6000x4000, then 6000 is the width here, even if you hold it vertical on end). Pixels are only used in calculations in pixels (not used at all in film cases). Only sensor size, focal length and time (and Not pixels) compute other lengths in mm or degrees. Megapixels are shown so you can check that your sensor size dimensions are entered correctly, and megapixels within a few tenths is adequate accuracy. However camera specifications do round off specified values of sensor size in mm, aspect ratio, and focal length, all of those affect exact precision.

This Rule calculation is NOT about exposure or ISO or aperture. It's only about focal length and sensor size and the Earths motion causing star trails if on a fixed mount. You will still need proper exposure, and for a dark sky and a short wide lens, I'd say start at ISO 3200, f/2.8, and about 30 seconds (you may like the Milky Way exposure to be longer, and manual Bulb shutter can be longer). Since the earth rotates, some trail is inevitable without a tracking mount, and this will be a serious conflict with adequate exposure.

The 500 Rule was for 35 mm film size (called FF for Full Frame here), and it does not account for other sensor sizes, and it is NOT about exposure. Using a cropped sensor must adjust for crop factor, either Rule / crop factor, or Rule / (focal length x crop factor) is the same thing. Then this difference makes any printed Rule number be ambiguous without explanation of how it applies to sensor size.

However, this calculator needs the real focal length, so if specifying a Rule here, adjust it for crop factor by dividing the Rule instead of multiplying the focal length.

Nomenclature:  The "(FF 500): 333" Rule nomenclature used here is "(Full Frame): cropped" Rule (only shown with cropped sensors). Cropped sensors complicate the "Rule". This is an attempt to show rules for both sensors to clarify which is meant. If it shows (FF 500):333, then it's a 333 Rule when adjusted for your selected cropped sensor (500 / 1.5 crop = 333), but it would be a 500 Rule result for a full frame sensor. "Full Frame" conventionally means the standard 35 mm film size comparison. If selecting a full frame sensor here, this conversion part is not necessary, and is not shown. If you specified a Rule, then that is what you get, but both versions are shown, trying to be clear.

If using a "Rule", it divides Rule by Focal Length to get Seconds. But those seconds won't give the same star trail effect on a smaller cropped sensor. If the same lens, the Earth rotates at the same rate across the pixels, but the cropped sensor is smaller. The star would cross the smaller sensor faster, so there is greater blur. And cropped DOF CoC is smaller for same reason. So if hoping for same limited blur effect when using a crop sensor, fewer seconds are necessary. This calculator does not know if you already divided your rule to compensate, so it doesn't divide your stated rule. If using "Rule", you must divide that "Rule" by your crop factor first, and the calculator should guide you with that as you go.

But the "Rule" is just one choice here, you can instead just use one of the three other option methods offered (seconds or pixels or X CoC), which have a more defined goal, and do not use a "Rule" as input (they will also show the computed Rule number, and also the 500 Rule result).

Sidereal rate is the default implementation, which is the precise rate of rotation to follow the stars, which is 0.27% different than Solar clock rate. This difference is important over minutes, and magnification (i.e., focal length) increases it tremendously. But if the focal length is short so the magnification is low (for example, full sky Milky Way pictures with 1x crop factor and a wide angle lens), it may have little effect on short fixed mount exposures up to about 25 seconds. Solar days are 24 hours or 86400 seconds for 360 degrees movement of the Sun. Sidereal days are 23 hours, 56 minutes, 4.0916 seconds, or 86164.0905 seconds for 360 degrees movement of the stars (because the Earth moves along its orbit during that day). Adding the Leap Year day makes them agree long term.
Sidereal motion is 15.04107 arcseconds per second of time.

Lunar rate is the Moons motion (eastward retrograde motion, slower than the Earths rotation), loses about 1/2 degree per hour against the stars (27.3 days per lunar revolution).
Lunar motion is 14.685 arcseconds per second of time.

Solar rate is the Suns motion, which is our regular clock time.
Solar motion is 15 arcseconds per second of time.

Stellar Declination of 0° computes the maximum length blur which occurs at the celestial equator, and so 0° declination is suggested as a general case to include the entire frame view. The Field of View angle is shown, and can be quite wide for a wide angle lens. Rotation blur is zero precisely at the pole at 90° declination (stars rotate around the pole, which is approximately 0.7° from the North star Polaris). Plus and minus declination compute the same blur numbers, around either pole. Directly above our head is celestial declination equal to our Earth latitude. Technically, star rates right at the horizon are also additionally affected by angular refraction, which varies.

Seconds of time goal entered can be entered as a fraction, like 1/60 or 1/250 second (maybe for Lunar or Solar rates). But stars will like a minimum of at least 20 or 30 seconds (at ISO 3200 and f/2.8)

Degrees computed is the length of the blur trail, in rotational degrees. When extremely small, it might be shown in a format like 1e-7. This -7 means move the decimal point 7 places right, to be 0.0000001 degrees. This might be seen if declination is near the poles, or for a very fast shutter speed.

Results. Seconds is Exposure time (computed by the Rule, it is NOT about correct exposure). Degrees is the star rotation in that time, at selected rate. Trail length is also the motion blur computed in mm, pixels and in terms of CoC, these are measured at the sensor plate.

The modes Seconds and Rule must use the specified seconds, so the blur trail varies accordingly.
The modes Pixels and X CoC must hold at the specified blur length, and so these computed seconds will vary.

Arcseconds per pixel and pixels per degree and can aid stellar angular measurements from the photo image. But note the measurement is limited to spans of "small angles", not more than about 10 degrees (but even 15° is still fairly close). Telescopes and even 200 mm telephotos only see a small angle, but wide angle lenses see much more. The "flat" dimension in mm or pixels (in an image) is the tangent of the angle, and tangent is NOT a linear function for larger angles (but accuracy is OK for angles less than 10 degrees). This method is astronomy's plate scale (stellar arc-seconds per flat plate mm) which measures small angles on flat plates (60 arcseconds per arcminute, 60 arcminutes per arcdegree). This dimensional property works to measure Small angles on any images that are not cropped or resampled, if you can enter the sensor and focal length properties used then. Remember, not exceeding 10 degrees.

For planned future images too. Wikipedia says the Sun's angular size is 31′31″ to 32′33″ (about 0.53 degrees) and the Moon's size is 29′20″ to 34′6″ (0.488 to 0.568 degrees). The distance to the elliptical orbits varies slightly with position, but both diameter sizes are roughly about 0.5 degrees. So this pixels per degree number will tell you what size of those to expect on your sensor and lens. Lens focal lengths are rounded, so it will help to know your lenses actual focal length accurately too. My tests of the calculator for the full moon diameter, with rounded 600 mm and 2000 mm lenses both showed about 0.54 degree, which is within tolerance (at the "largest full moon" periods recently). Here's a Moon perigee and apogee calendar. The expected 0.5 degree size will fill half of that pixels per degree number of pixels. The moon is illuminated by the Sun, the Full moon and is roughly near normal daylight exposure, but exposure varies depending on phase and orbit position. At apogee position, the moon is 14% smaller and about 1/3 EV dimmer than at perigee. A quarter phase moon (side lighted) will need +2 or +3 EV more exposure than Full. The Moon's average reflectivity (albedo) is 12%, so the photo should look middle gray dark, not white.

Circular polar star trails: If using an interval timer with one second spacing, the calculator's one second setting can compute the gap size in pixels between shots (or just use wide angle). Specify the declination of your widest view, because rotation blur is zero at exactly 90° declination at the pole. Note that a camera shutter speed of 30 seconds is actually 32 seconds, so that timer interval needs to be 33 seconds, so that it does not skip shots.

But this calculator is NOT about long circular star trails (those are circular and rotate 15 degrees per hour). This goal is to compute movement of the Earth's rotation on a fixed mount

Milky Way pictures are normally improved with substantial post processing to add dazzle. There are many ideas, see Google: How to Edit Milky Way photos.

### Measuring the blur trail

At left was to be M42 in Orion. Exposure was 20 seconds, ISO 3200, full frame. And f/2.8, 200 mm, so it is (20 x 200) = 4000 Rule full frame, which is rather too much on a fixed tripod. 😊 This trail length is 9.7x CoC mm. On a fixed mount, a 20 mm lens would 1/10 as much trail length as 200 mm, but it would be a wide sky view, which is better for Milky Way pictures.

The enlargement at right is a 100% crop of the nebula area, near frame bottom edge. The calculator says, for 20 seconds (35.9 mm sensor width, 7360 pixels width, 1x crop factor), the result is 59.4 pixels of trail. The nebula is at -5.4° declination, but that small amount has negligible effect. Computing zero degrees declination would seem a better general case for the entire field of view which can often include zero declination. The center picture at left shows it to be somewhat vague, but as rotated, I measure it as 59.5 pixels, so my own measurement could be off by at least the one pixel.

When this image is rotated to be horizontal at left, and enlarged more to see the pixels, I count the overall trail to be about 69x10 pixels (a tight marquee crop shows the count, or it can be actually cropped and examined). This does NOT imply the star size is 10 pixels, because at this scale (many light years of astronomical distance), the stars are actually seen as zero star diameter, but the camera sees them larger. This visible blob is simply photo anomalies (notice size varies with brightness). But don't count the entire trail. Star blobs are round, so if the width of the trail line is 10 pixels, the additional blur trail of a round star must be 69-10 = 59 pixels extension which agrees with the calculation. This path addition is what is calculated. Counting pixels is a bit arbitrary, due to focus, and image aliasing, and the sensor exposes a few Bayer pixels for every image RGB pixel. There's always a few more neighboring pixels, but the additional length due to rotation is what is calculated.

Turn VR Off on the tripod. VR (Vibration Reduction) is Nikon's image stabilization method. It is Not intended for tripods, and worse, in extremely dark conditions, VR can cause a visible red streak seen in the star image (guessing it is an infrared-ish internal reflection, very dim but visible in this darkest scene). Image stabilization can help to correct camera shake, but VR can have no effect on subject motion. Unfortunately, I failed to turn VR off for this one of Andromeda (at left) with a Nikon D800 and 70-200 mm VR lens at 200 mm f/2.8 and 20 seconds at ISO 3200. See a better one below.

Also note the vignetting (dark corners). That's going to be worse in a wide angle lens. The Adobe Camera Raw editor (ACR) has a great tool to fix vignetting with one click (Lens Profile Corrections), but it was not bothered to do on this one.

Find a location with a fairly dark sky, at least a hundred miles from any large city, which is difficult in the eastern USA (see Dark Site Finder). I drove 1.5 hours to the Texas Astronomical Society of Dallas club dark sky site near Atoka Oklahoma. The site is 20 miles from an 18000 population town, which glow is visible on the low horizon, but the sky was dark. And choose a date with no moon visible. See Google Celestial and Planet rise and set times

Camera settings for stars are:

• VR OFF   (turn image stabilization off on the tripod)
• Auto focus OFF, use Manual focus (Auto focus may work on the Moon, but not on stars)
• MANUAL camera exposure of f/stop and shutter
• Auto ISO OFF, but like exposure, set the fixed ISO you want to use.

ISO 3200, f/2.8, 30 seconds can work in a dark sky, but longer can be much better. A tracking mount will make all the difference (see below about an inexpensive DIY solution).

Manual focus is required, to prevent every shutter button from seeking new focus. Actually seeing stars in the viewfinder is difficult or impossible, but focusing in Live View at high screen magnification (zoom preview) is the way to easily focus on a bright star. Old lenses had mechanical stops so they could not be turned past infinity, but many/most newer lenses will go past infinity, so rotating it until it stops is probably no longer useful. Setting the focus first to the middle of the infinity mark is not a bad try, the depth of field should be helpful. Be aware that the least touch later might change the lens. If zooming changes focus on your lenses, you'll need to recheck focus then.

Or, if the scene also includes some land objects, you may want to consider focusing at the hyperfocal distance (see calculator).

Live View Mode can allow manual focus. Pick a bright star or planet to be in your field, and then zoom in on the rear Live View LCD preview to magnify it greatly in Live View. You are using high ISO and a lens very near wide open. You might see some stars then, which is great, focus to make them be the smallest brightest spots. Or nothing may show at all, a blank black screen, but when you slowly move focus past the right point, a star might appear, and then suddenly disappear. This can be very touchy. If as you move focus, if you see a bright dot momentarily appear, go back and look for it, very slowly. You may have to re-aim to another brighter star to focus, and then move it back where you want it. When you can see the stars, that's close, but manually focus for the smallest but brightest dot made by the star. This sounds harder than it is, you just have to understand the plan.

### 500 Rule for photos of Milky Way Stars

The Earth rotates. A reasonably long star exposure on a tripod or other fixed mount will show an elongated star trail (blur) instead of a pin point dot. A longer lens seriously magnifies this movement, so usually a short wide lens is used for the Milky Way (also to show a lot of sky). However, there are trade-offs, blur for exposure.

There's a rule of thumb for Milky Way photography called the 500 Rule. This idea from 35 mm film days says 500 / focal length = seconds is the maximum exposure time still retaining sharp round stars (if using a fixed mount). But sensor size also affects magnification, and for a cropped sensor, this rule is 500 / (focal length x crop factor). However, this does not take megapixels into account, and more megapixels will show more pixels of trail, but Not more mm of trail length. The 500 Rule is intended to be a compromise of the most exposure time vs. the least blur trail size, and typically Rule values from 400 to 600 are tried.

Why would we use one Rule vs another? For image quality to be less affected by the blur trails caused by the Earth's rotation when camera is on a fixed mount. The purpose of this calculator is to give a reasonable expectation about what to expect from the blur trails. How many pixels long are the trails? However, it's a serious trade-off, because stars require quite a few seconds of exposure, and the Earth does rotate.

This calculator computes the fixed mount star trail blur based on your focal length, sensor size, pixel dimensions of your image, and shutter exposure time.
The resulting blur trail length is shown in degrees, mm, and in pixels of length. Pixels might mean the most to you, but there is added depth here of comparing this blur to the standard DOF CoC maximum acceptable limits for blur that we already use.
To plan your star session, you can choose to enter a shutter time, or a new Rule, or to limit the star blur trail to X pixels long, or to limit it's size on the sensor to be X times the CoC diameter (relative to the normal DOF limit).

The mm length of the elongated pixel trail is compared to standard Depth of Field CoC limit in mm, to judge how much it matters. The Depth of Field definition is an existing standard where the limit of 1x CoC diameter is the boundary where our eyes decide fuzzy instead of sharp enough (but it becomes much more evident at larger enlargements). A 24 megapixel sensor computes motion across 1x CoC mm as about 5 pixels of trail, regardless of crop factor or lens focal length (but is affected slightly by Aspect Ratio, and by megapixels). A trail length of a few pixels may not always be quite as bad as it may sound, but many pixels will be a problem.

### The "Rules"

• The 500 Rule result is also shown for comparison. Its star trail length (when Rule is adjusted for sensor crop factor, adjusted meaning 500 Rule for full frame, or a crop 1.5x sensor would use Rule 500/1.5 = Rule 333) which computes 1.21x CoC size.

This 1.21x CoC might be 6 pixels, but if the star is in focus, it could be interpreted as being 21% worse than the Depth of Field that we begin to call blurred. 1x CoC is considered a maximum acceptable limit of sharpness, but 21% is not greatly more. (500 / 18 mm lens = 27.8 seconds)

• We hear of the 600 Rule, which seems popular to get more seconds, but it's more rotation and less quality, about 1.45x CoC (when rule is adjusted for crop factor), allowing a little more motion blur. More time is good for exposure, however I think opinions may differ about the blur. 600 rule is only about 1/4 EV more exposure than the 500 rule, so it is of debatable value. (600 / 18 mm lens = 33 seconds.)
• And the 400 Rule is popular too, for more critical results. Again, that 400 is for full frame, but it is 0.97x CoC if rule is adjusted for crop factor. (400 / 18 mm lens = 22 seconds.) If the lens is short enough, 400 seems a good choice because ...
• NOT knocking the 400 Rule at all, but the 412 Rule (which I just now made up for this) could be of interest because 412 will compute a star trail length of 1x CoC diameter (when rule is adjusted for crop factor). CoC is a guide which is considered a maximum limit of Depth of Field sharpness, so 1x is the worst possibly acceptable still result. If the Rule is adjusted for Crop Factor, then it also is 1 X CoC then. (412 / 18 mm lens = 22.9 seconds.) The 400 Rule would be marginally better (shorter) trails than 412.

There's another chart summarizing how the rule affects these "changes" at the bottom of this page.

But for quality results (no visible blur trail), then the Rule becomes numerically small, especially with the focal length of cropped sensors, with inadequate exposure for the Milky Way stars. It's a difficult problem, because the Earth rotates.

CoC is an existing limit on acceptable blurriness in Depth of Field calculations. Here, we use CoC diameter as an existing size reference. also for the blur star trail due to the Earth's rotation motion. Depth of field is not an issue when focused on a star at infinity, but just being able to focus on a star is a major issue. Actually seeing most stars in the viewfinder is difficult or impossible, but focusing is greatly aided if in Live View Mode, and then zooming in greatly on the rear LCD preview. You may not see anything until you reach the right focus, then you can see the bright ones to focus (manually focus for the brightest but smallest dot made by the star).

The trail surely will look a couple of pixels longer than calculated, due to the stars size itself, and also the star dot straddling multiple pixels and affecting neighboring pixels. Any movement of one pixel obviously involves at least two pixels, another maybe at both of the start and end points, which calculation does not include. And misfocus blur makes the blur pixels be larger dots too. The star dot is round, so its minimum length must be at least its width, so subtract the line width from the line length to get the extended blur trail length calculated. The calculator only calculates Width and horizontal dimensions, but magnification is the same in all directions.

It may matter if the camera times the shutter, or if you time it manually. Because, the camera's nominal shutter speed of 30 seconds is actually implemented to be 32 seconds (25 = 32, and nominal 30 seconds is actually 32). That's so our concept of 2x time being exactly 2x exposure will work (has to be 1, 2, 4, 8, 16, 32 steps.) Nominal 20 and 25 seconds are more as expected. You can compute with the correct time if you expect it. See chart of actual shutter speeds.

Light gathering Power: Aperture is not a factor of star trail length. However it affects exposure of stars with even greater significance. In sunlight, we stop down and use short shutter speed to block excessive light. With stars, we struggle to gather every possible photon. Effective diameter of the aperture is focal length / fstop Number. This makes the 14 mm f/2.8 lens be a 5 mm "telescope" (slightly wider than the eye's pupil of maybe 4 mm). A 28 mm f/2.8 lens is 10 mm diameter, and a 50 mm f/2 lens is a 25 mm "telescope". Larger telescopes are good, greater light gathering power, and also greater resolving power. The Palomar telescope is 5.1 meters diameter, and McDonald Observatory has one 9.2 meters diameter. The 50 mm f/2 lens compares as 3.5 times more magnification (magnified blur trails too) than the 14 mm lens, but also five times diameter with 25 times more area and "light gathering power", which aperture difference is 2.5 x log10((25mm/5mm)2) = 3.5 magnitudes (of more faint stars) seen in the same exposure time (but it needs a darker sky). However, such fast lenses (like f/1.4) have less quality, corner sharpness at widest aperture is notoriously poor. A 200 mm f/2.8 is 71.4 mm diameter, or 5.8 magnitudes more aperture than the 14mm f/2.8. Possibly something to consider, but you're surely talking about a tracking mount then. The 500 Rule of 500/200 = 2.5 seconds won't get it done.

### A Tracking Mount is a Better Solution about rotation blur

Do see the highlighted line at the bottom of this section before you leave. Might be interesting. The problem is that a fixed mount (like a camera tripod) is turned with the Earth as it rotates. This seriously blurs the stars (into long star trails) because exposure is necessarily more than a few seconds.

Andromedia galaxy, M31
Nikon D800, 70-200 lens at 200 mm
f/2.8 30 sec, ISO 3200
One single exposure
White Balance 4150K +3 Tint
This was Rule 6000, but it was tracked.

Using a tracking mount, this is M31 Andromeda with a full frame Nikon D800 and 70-200 mm camera lens, one single exposure at 200mm, f/2.8, 30 seconds at ISO 3200, November. That is EV -2 at ISO 3200, which is Light Value -7 EV (at ISO 100). This is heavily cropped, see a similar full frame image above, with same 200 mm lens, but not rotation was Not tracked then.

Andromeda is the closest and brightest spiral galaxy, 2.5 million light years distant, magnitude 3.4, and in a dark sky, its faint smudge of the small central core is the furthest thing seen with the naked eye. Or rather I should say detected by eye, it's a tiny vague smudge and takes a dark sky. It is actually a bit more than three degrees size (would appear six times larger in the sky than the full moon), but the faint size is not captured here in only 30 seconds. It seriously needs a longer exposure to have seen more. What we can perceive with naked eye is that fuzzy little bright center spot. Andromeda is on a collision course with our Milky Way galaxy (at 70 miles per second), expected in about 4.5 billion years. The frame also shows dwarf galaxies M32 (close right, mag 8) and M110 (left, mag 9).

The entire frame of this picture is about the same as the Andromeda frame shown above, but it is greatly cropped here to about 40% of FX frame height, and resampled to 33% size. It is just one single image (not stacked), but Long Exposure Noise Reduction (normal) was On. It needs much more exposure, but it was my first try with a tracking mount. The general idea today is that it is best to stack several 30 second exposures, instead of trying a longer single exposure.

The editing I did (surely considered very mild by some standards) in Adobe Camera Raw (ACR) was to crop it tighter, and set white balance by eye to find the small pleasant spot between Temperature too blue or too yellow, and Tint between too green or too magenta (this white balance was 4150K° and +3 Tint, but nearby city lights can have effect). It used the ACR lens profile to correct vignetting (but none of the frame corners are in the view of this crop), and added the Curve's Strong Contrast. Exposure +1, Blacks -80 to darken the sky, Clarity +35 to bring out the galaxy a bit, and Saturation +50 to bring out the colors. There are both some red and some blue stars. And then resampled the crop to 33% size for the web.

An "Equatorial mount" lets the telescope or camera rotate on an axis aligned to be parallel to the Earth's axis. This axis is pointed to the pole near Polaris, the North Star, so it rotates same as Earth on its axis. This lets one simple rotation motion stay locked on a star (instead of a complicated X and Y motion that requires a computer to perform it precisely, like a regular camera tripod which is called an alt-azimuth mount). On this equatorial axis, a motor can keep turning the camera back at near the same slow rate the Earth turns forward (actually at the Sidereal rate for the stars), so that the camera is locked onto the same spot in the sky with no relative motion.

For this purpose, there are a few motorized star tracking camera mounts available (ballpark cost around \$500 US, from iOptron, Vixen, etc, at Amazon, B&H, etc). The iOptron SkyGuider Pro I used has a small fixed telescope literally built into the hollow polar axis to align the mount on Polaris, easy to use, with an illuminated scale marking the offset amount and direction from Polaris. Then tonight's current direction to set it too is obtained from a smart phone app showing current month/day/hour location to properly align the mount (see page 15 of the user manual). This is extremely handy. Using the South Pole is a similar situation. This "Full package" system includes all but include the tripod or the ball mount to mount and aim the camera.

Polaris: Setup out in the field first carefully aims the polar axis of the mount at the North pole near Polaris (RA: 2h 31m 48.7s, dec: +89° 15' 51"). Polaris is actually about 3/4 degree from the pole, perhaps 44 minutes. This separation is near about 1.5 times the full moons diameter (so in a telescope, it's further than you might imagine). The direction of Polaris around the North pole rotates during the night, and over the course of a year. If the mount’s rotation axis is aligned on that correct spot, then the mount’s rotation around the pole will track accurately for long periods of time. If misaligned a bit, then it will skew a bit during that time.

Using this tracking mount, the small image is a 127x78 pixel 100% crop from this same image (greater enlargement, slightly left of galaxy center) showing round stars without trails (but without tracking, 200 mm at 30 seconds would otherwise compute a 6000 Rule, and 55 pixels of trail at declination 41° degrees).

But there is another very popular and inexpensive DIY idea for a suitable tracker for a camera, and if serious about it, you may want to investigate an easy-to-build barn door hinge tracker for the camera. Do look deep into that Google list, because there are many versions shown. The hinge axis is aimed at Polaris, the North Star, so the stars revolve around it. The accuracy of that aim will affect the tracking precision. The distance from hinge to drive screw is a precise calculation to match the Earth's rotation to the screw thread pitch turning at one RPM (360° per minute). It can get fancy, some versions add a small One RPM motor and gears, but if no motor, this drive screw can be manually turned, for example as 1/4 turn every 15 seconds, which would then compute motion blur as 15 seconds (still suitable for wide lenses, but allowing a much longer overall exposure). Don't shake the camera doing the adjusting. The straight line screw travel won't be adequate tracking accuracy over many minutes, but it is very popular and very suitable for a few minutes, if polar aligned fairly well.

### Stars and Rules

If anyone is interested, here is how changes to Rule or Crop Factor affect these values.

These properties are about the rotation with time, and the length of the star trail, in degrees, mm, and pixels. Pixel values shown are for 24 megapixels (the 6000 pixel width), which will vary with your own megapixels. These are Sidereal numbers.

When all else is unchanged, Crop Factor changes only Pixels and X CoC. The mm path length stays the same, but the sensor is smaller, so it appears larger in the image.

A Rule adjusted for Crop Factor leaves those two results unchanged, but changes time and all the others. The short time may not be usable.

Changing only megapixels in same sensor width changes trail in pixels, but not degrees or mm, so the image xCoC looks the same (half the pixels of width, in half of the width, looks the same).

Changes are shown as the columns progress downward.

Focal
Length
Pixels
Width
RuleCrop
factor
Time
seconds
Trail
degrees
Trail
mm
Trail
pixels
Trail
X CoC
500 Rule for 1x adjusts time for focal length
15 mm6000500133.330.1396.081.221x
20 mm"""250.104"""
30 mm"""16.670.0696"""
40 mm"""12.50.0522"""
60 mm"""8.330.0348"""
500 Rule Not adjusted for crop factor (not a good idea)
20 mm60005001250.10450.036466.0771.22
"""2"""12.152.43x
"""4"""24.314.86x
"""8"""48.619.72x
Rule adjusted for crop factor is less seconds, but same blur result
20 mm60005001250.10450.036466.0771.22
""333.31.516.6650.069630.02443""
""312.51.615.6250.065280.02279""
""250212.50.052230.01823""
""12546.250.02610.00912""
Half the pixel dimension is same trail in a half smaller image
20 mm30005001250.10450.036463.0381.22x
"""2"""6.0772.43x
"""4"""12.154.86x
"""8"""24.319.72x
2x focal length at half the exposure is same pixels and X CoC
40 mm6000500112.50.052230.036466.0771.22x
"""2"""12.152.43x
"""4"""24.314.86x
"""8"""48.619.72x
412 Rule gives 1 X CoC if adjusted for Crop
20 mm6000412120.60.086070.030045.0071x
""274.71.513.7350.057390.02003""
""257.51.612.8750.053790.01878""
""206210.30.043030.01502""
""10345.150.021520.007511""
Focal
Length
Pixels
Width
RuleCrop
factor
Time
seconds
Trail
degrees
Trail
mm
Trail
pixels
Trail
X CoC