The "500 Rule" was a rough guide (for 35 mm film) about maximum allowable exposure time of star photographs due to trails when camera is on a fixed tripod. It says the maximum star exposure time (without excessive rotation trail due to a fixed mount) is the "Rule" number / focal length (for example, 500 Rule / 20 mm lens = 25 seconds maximum time). It does ignore some factors, like sensor size or pixel size.
This calculator offers other more specific goal-oriented methods to examine the trailing problem, and offering visibility of the situation. We can compute the time that limits the star trail length to a specific length in digital pixels, and for film or digital, 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, 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 (on a fixed mount).
The 500 Rule was for 35 mm film, and it does not account for other sensor sizes. 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 for crop factor by dividing the Rule instead of multiplying the focal length.
The "(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 (500):333, then it's a 333 Rule that was adjusted for your selected cropped sensor (500 / 1.5 crop = 333), or 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 shown. If you specified a Rule, then that is what you get, but both versions are shown, trying to be clear.
If using "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 implemented, which is the precise rate of the stars, but it's only 0.27% different than Solar clock rate. This difference is important over many minutes, but it will have little effect on short fixed mount exposures like 30 seconds. Solar days are 86400 seconds for 360 degrees. Sidereal days are 86164.1 seconds for 360 degrees. Sidereal rate is the precise number, but it won't make a noticeable difference in 30 seconds. Solar time is appropriate for the Sun, and Lunar for the Moon. Lunar rate is slower, about 2.4% less than sidereal, losing about 1/2 degree per hour against the stars (13 degrees per day, 27.3 days per sidereal revolution). However the moon is lighted by direct sunlight, and does not need long exposures like even 1 second.
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 technically zero 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, from either pole. Directly above our head is celestial declination equal to our Earth latitude. Technically, star rates right at the horizon are also affected by angular refraction, which varies.
Results. Seconds is Exposure time, Degrees is rotation in that time, at selected rate. Trail mm computes Pixels and X CoC, both 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 computed seconds vary.
Arcseconds per pixel and pixels per degree and can aid stellar angular measurements from the photo. But note the method is limited to spans of "small angles", not more than about 10 degrees (but even 15° is still pretty 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. 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 compute Small angles on any images that are not resampled, if you can enter the sensor and focal length properties used then.
For future images too. Wikipedia says the Sun's angular size is 31′31″ – 32′33″ (about 0.53°) and the Moon's size is 29′20″ – 34′6″ (also about the same). The distance to the elliptical orbits varies slightly with position, but both diameter sizes are close to 0.5° (a very small angle). So this pixels per degree number will tell you what size of those to expect on your sensor and lens. The one half degree will fill half of that pixels per degree number of pixels. Exposure depends on phase, the Full moon is about usual Sunny 16 exposure. A quarter phase moon (side lighted) will need +2 or +3 EV more exposure, but won't be as much as a second or two of exposure. The Moon's reflectivity (albedo) is 12%, so it should look middle gray dark, not white.
Circular polar star trails: If using an interval timer with one second spacing, the calculator can compute the one second gap size in pixels between shots (or just use wide angle). Specify the declination of your widest view, because of course, 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 interval needs to be 33 seconds, so that it does not skip shots.
But this calculator is NOT about intended 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.
This calculator is NOT about proper 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 20 or 30 seconds (you may like the Milky Way to be longer). Since the earth rotates, some trail is of course inevitable without a tracking mount, and this will be a conflict.
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.
Focal length is the actual focal length (actual, NOT any Equivalent Focal Length).
Sensor Width is the largest dimension in pixels of the picture the camera took (if 6000x4000, then 6000 is the width here, even if you hold it vertical).
Options 1-3 are three ways to specify camera sensor size.
See more at Determine Crop Factor.
Film sizes should be accurate in the list, but the "1/x inch" sensor numbers are Not meaningful. The "1/x inch" description is NOT the sensor dimension, not even related to the sensor. It's a false specification, it compares the sensor to the picture size of an old glass vidicon tube (1950s, before CCD). But the "1/x inch" dimension was that outer glass tube diameter, and there is absolutely nothing about the digital sensor that is that dimension (its diagonal is probably no more than 70% of that dimension). It is an inflated false number. Instead, we need to know the sensors actual real W x H size in mm. Or an accurate Crop factor can compute it here. The list here tries to provide some known sensor sizes for the 1/x numbers, but there are usually a few different sensor sizes claiming the same 1/x number, so it can be wrong. The correct calculation really needs the correct sensor size, WxH in mm.
Measuring the blur trail:
This 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 on fixed tripod. 4000 may be a bit too much. :) This trail length is 9.7x CoC mm. On a fixed mount, 20 mm would blur less than 200 mm.
This image is a 100% crop from it. Calculating for 20 seconds (35.9 mm sensor width, 7360 pixels width, 1x crop factor), the result is 59.4 pixels of trail (if 99.6% length at 5.4° declination). Computing zero degrees would seem a better general case for the entire field of view which often includes zero declination.
When this image is rotated 27° (to be horizontal), and enlarged to about 800% to see the pixels, I count a trail to be about 69x10 pixels (a tight marquee crop shows the count, or you can actually crop it and examine resulting size). But don't count the entire trail. Star blobs are round, so if the width of the line is 10 pixels, the additional blur trail of a round star is 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 three Bayer pixels for every image RGB pixel. There's always a few more neighboring pixels, but the addition due to rotation is what is calculated.
Turn VR Off on the tripod. VR (Vibration Reduction) is Nikon's image stabilization method. Not intended for tripods, and worse, in very dark conditions, VR can cause a visible red streak seen in the star image. This is Andromeda with a Nikon D800 and 70-200 mm VR lens at 200 mm f/2.8 and 30 seconds. It took me a little while to figure out what the red was.
Also note the vignetting (dark corners). That's going to happen, and it's worse at wide aperture, and with wide angle (this one of Andromeda is 200 mm, f/2.8, 20 seconds, fixed tripod). The Adobe 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.
Of course, camera settings for stars are:
Manual focus. Actually seeing stars in the viewfinder is difficult or impossible. 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. Be aware that the least touch later can change it, and zooming often changes focus, so check focus in your actual photo results often by zooming in greatly on the last LCD preview result. A piece of masking tape on the lens to hold the focus ring is helpful.
Live View Mode can help 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. When you can see the bright ones, 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.
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 of course, 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 of course 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 of course 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.
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)
Applying the "Rule" (for example Seconds = 500 / 18 mm) by using adjusted Rules for Crop factor (Rule / Crop factor) of course computes different maximum exposure times for different sensors, because the purpose of this Crop adjustment is to give comparable blur results. However, for the different exposure times computed for the different size cropped sensors of the same megapixels and same lens, the calculator computes the same blur trail in both mm and pixels and same CoC multiple (if the Rule is converted for crop factor). The Earth rotates at the same rate across the sensors, but the sensors are different sizes, and the Crop factor equalizes that aspect. But shorter exposure must be compensated with higher ISO or wider aperture. For a 1.5 crop sensor, the fewer seconds of time required by the Rule is 0.58 EV less exposure than a 1.0 crop sensor. This is just saying that for a smaller cropped sensor, the same blur trail of the Rule requires less seconds of time, seconds / 1.5, which is 0.58 EV less exposure.
There's a 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 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 of course 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 of course, 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 directly 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, (71.4/5) 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 fixed mount (like a camera tripod) is turned with the Earth as it rotates. This seriously blurs the stars because exposure is necessarily more than a few seconds.
This is Andromeda with a camera lens, 200 mm f/2.8, 30 seconds at ISO 3200 (very similar to the one above, but greatly cropped here, to about 40% of FX frame height, resampled to 33% size). It is one single image (not stacked), but Long Exposure Noise Reduction (normal) was On. It was my first try with a tracking mount. Andromeda is 2.5 million light years distant, and is actually three degrees size (six times larger than the full moon), but the faint size is not captured completely here in 30 seconds.
The editing I did (probably considered mild by some standards) in Adobe Camera Raw 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 was 4150K° and +3 Tint, but nearby city lights can have a big effect). It used the ACR lens profile to correct vignetting (probably not 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. And 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, but in reverse direction. 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 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 the same slow rate the Earth turns forward, 15 degrees per hour (but actually at the very similar Sidereal rate for the stars), so that the camera is locked onto the same spot in the sky with no relative motion.
Polaris: Setup 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"), which is about 3/4 degree from the pole, perhaps 44 minutes. This separation is about 1.47 times the full moons diameter (so it's further than you might imagine). Its direction rotates during the night, and over the course of a year. Then the mounts rotation around the pole will track accurately for longer periods of time. If misaligned a bit, then it will skew a bit.
For this purpose, there are a few motorized star tracking camera mounts available (ballpark cost around $400 US, from iOptron, Vixen, etc, at Amazon, B&H, etc). The iOptron SkyGuider Pro has a small fixed telescope literally built into the hollow axis to align the mount on Polaris, with an illuminated scale marking the offset amount and direction from Polaris. A smart phone app shows current month/day/hour location to properly align the mount (see page 15 of the user manual). The South Pole has a very similar situation. This "Full package" system does not include the tripod or the ball mount to aim the camera. Using this mount, here's a 127x78 pixel 100% crop from this same image showing round stars (200 mm at 30 seconds would otherwise compute a 6000 Rule and 68 pixels of trail at declination 41°).
But there is also a very popular and inexpensive DIY idea of this, and if serious about it, you may want to investigate an inexpensive and easy-to-build barn door hinge tracker for the camera. Look deep into that Google list, there are many versions shown. 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. Some versions add a small one RPM motor and gears, but this one RPM can be manually adjusted as 1/4 turn every 15 seconds, which would then compute blur as 15 seconds (still suitable for wide lenses, but allowing a much longer overall exposure). The straight line screw travel won't be adequate tracking accuracy over many minutes, but it is very suitable for a few minutes, if polar aligned fairly well. Don't shake the camera doing it.
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 of course 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 of course 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.
|Lens FL||Width||Rule||Crop||Seconds||Degrees||mm||Pixels||X CoC|
|Rule adjusted for crop factor is less seconds, but same blur result|
|Half the pixel dimension is same trail in a half smaller image|
|2x focal length at half the exposure is same pixels and X CoC|
|412 Rule gives 1 X CoC if adjusted for Crop|
|Lens FL||Width||Rule||Crop||Seconds||Degrees||mm||Pixels||X CoC|
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