In low-latitude locations such as Singapore (1.3521° N) or anywhere that Polaris is not visible, polar alignment can be difficult and time-consuming to achieve. One method for doing initial alignment in altitude is to use the mount's latitude scale, or a digital level of some sort.
This method is not sufficiently accurate because the mount's latitude scale normally only has 2° increments, and digital levels (in spite of their supposed high accuracy) actually have a tiny pendulum inside which is insensitive to small angle changes and really only has a resolution of 0.5° which is 30 arc-minutes and insufficient for a good polar alignment.
Here we can see an EQ mount set to zero latitude (as indicated by the bubble level) but the digital inclinometer is claiming 0.50° angle (which can be zero'ed out, but illustrates the inaccuracy of digital inclinometers).
However, we can take advantage of a bubble level that has a 45° vial (or our untrustworthy inclinometer) to achieve more accurate angle measures.
The Starrett bubble level in the photo above has an accuracy of 1mm in 1m, or about 0.045° or 2.7 arc-minutes, hence we can be reasonably sure (within 5.4 arc-minutes or 1/11 of a degree) that the mount is indeed at zero latitude.
We then take note of the position of the altitude adjustment knob:
The next step is to keep rotating the altitude knob (counting rotations as we go) until the mount is at 45° latitude (as confirmed by our bubble level or digital inclinometer):
Again we have an uncertainty of 5.4 arc-minutes in this measurement; adding to the uncertainty in the zero measurement, yields a total potential error of around 11 arc-minutes or 0.2°. On the other hand, if we used the digital inclinometer to zero the mount and also measure the 45° angle, our total uncertainty is 1°.
It is obvious that the bubble level provides lower error than a digital inclinometer.
In the case of the mount pictured, it took 39.60 - 39.75 turns of the knob to reach 45° +/- 0.1° latitude; this means that every turn of the knob yields 1.1295 - 1.1338° degrees (67.77 - 68.028 arc-minutes, or an average of 67.89 arc-minutes) of altitude.
We can see that the mount's latitude scale indicates roughly 45° as well, however since we don't know if the mount and tripod are level, and the latitude scale has no vernier and only 2° increments, the latitude scale alone is insufficient for setting the latitude.
If we had used the digital inclinometer, with its systematic error of 0.5° (total 1° including uncertainty about the zero point) then over 45° +/- 0.5° every turn of the knob yields 1.119 - 1.149° (67.14 - 68.94 arc-minutes or an average of 68.04 arc-minutes) of altitude.
Most EQ mounts use a sort of tangent arm assembly to adjust the altitude; as the angle gets higher, there is an increasing error. At small angles, θ ≈ sin(θ) but as θ gets larger, the error also increases. For example, at 45° (0.785 radians), sin(45°) = 0.707, a difference of 11%.
Hence, we need to reduce our calculated arc-minutes per turn by around 11% - so 68.04 arc-minutes per turn of the knob is reduced to 61 arc-minutes per turn.
The Astro-Physics Mach1 has a very accurate altitude adjustment, and is spec'ed for 62 arc-minutes per rotation with 16 ridges on the altitude knob (3.875 arc-minutes per ridge); this is in very close agreement with our calculated 61 arc-minutes per turn.
It is obvious that by using the 0° and 45° points as reference, we can significantly reduce the effects of systematic error in the bubble level or digital inclinometer.
After resetting the mount to zero latitude (again confirming with the bubble level), we can set it to 1.3521° by rotating the knob (1.3521° * 60 / 3.875 arc-minutes) = 21 knob ridges. If we had used our derived value of 61 arc-minutes per rotation (3.8125 arc-minutes per turn), (1.3521° * 60 / 3.8125) = 21 knob ridges which is the same as the (known for the Mach1) setting.
On the other hand, if we need to set the latitude to a higher value, say 30°, we would use the 45° angle as the reference. In the case of 30°, we would zero the mount at 45°, then lower it by 15° (which is 900 arc-minutes) which would require 14.5 turns of the knob (14 turns and an additional 8 knob ridges).
This method would allow accurate altitude alignment to within (5.4 arc-minutes error of the level) + (8 arc-minutes from the knob uncertainty) of about 13 arc-minutes (0.22°) worst-case; actual error may be half that. Such a result is good but not great, a drift alignment can achieve better accuracy. With this amount of altitude error, there is approximately 3.5 arc-seconds of drift per minute, hence an unguided exposure of 1 minute with a typical DSLR will still show round stars.
In contrast, had we set the mount's latitude directly from the digital inclinometer, we could have been in error by 1° which would lead to a 15 arc-second drift in 1 minute, thus limiting maximum exposure times to around 20 seconds before star elongation would be visible.
This method is not sufficiently accurate because the mount's latitude scale normally only has 2° increments, and digital levels (in spite of their supposed high accuracy) actually have a tiny pendulum inside which is insensitive to small angle changes and really only has a resolution of 0.5° which is 30 arc-minutes and insufficient for a good polar alignment.
Here we can see an EQ mount set to zero latitude (as indicated by the bubble level) but the digital inclinometer is claiming 0.50° angle (which can be zero'ed out, but illustrates the inaccuracy of digital inclinometers).
However, we can take advantage of a bubble level that has a 45° vial (or our untrustworthy inclinometer) to achieve more accurate angle measures.
The Starrett bubble level in the photo above has an accuracy of 1mm in 1m, or about 0.045° or 2.7 arc-minutes, hence we can be reasonably sure (within 5.4 arc-minutes or 1/11 of a degree) that the mount is indeed at zero latitude.
We then take note of the position of the altitude adjustment knob:
The next step is to keep rotating the altitude knob (counting rotations as we go) until the mount is at 45° latitude (as confirmed by our bubble level or digital inclinometer):
Again we have an uncertainty of 5.4 arc-minutes in this measurement; adding to the uncertainty in the zero measurement, yields a total potential error of around 11 arc-minutes or 0.2°. On the other hand, if we used the digital inclinometer to zero the mount and also measure the 45° angle, our total uncertainty is 1°.
It is obvious that the bubble level provides lower error than a digital inclinometer.
In the case of the mount pictured, it took 39.60 - 39.75 turns of the knob to reach 45° +/- 0.1° latitude; this means that every turn of the knob yields 1.1295 - 1.1338° degrees (67.77 - 68.028 arc-minutes, or an average of 67.89 arc-minutes) of altitude.
We can see that the mount's latitude scale indicates roughly 45° as well, however since we don't know if the mount and tripod are level, and the latitude scale has no vernier and only 2° increments, the latitude scale alone is insufficient for setting the latitude.
If we had used the digital inclinometer, with its systematic error of 0.5° (total 1° including uncertainty about the zero point) then over 45° +/- 0.5° every turn of the knob yields 1.119 - 1.149° (67.14 - 68.94 arc-minutes or an average of 68.04 arc-minutes) of altitude.
Most EQ mounts use a sort of tangent arm assembly to adjust the altitude; as the angle gets higher, there is an increasing error. At small angles, θ ≈ sin(θ) but as θ gets larger, the error also increases. For example, at 45° (0.785 radians), sin(45°) = 0.707, a difference of 11%.
Hence, we need to reduce our calculated arc-minutes per turn by around 11% - so 68.04 arc-minutes per turn of the knob is reduced to 61 arc-minutes per turn.
The Astro-Physics Mach1 has a very accurate altitude adjustment, and is spec'ed for 62 arc-minutes per rotation with 16 ridges on the altitude knob (3.875 arc-minutes per ridge); this is in very close agreement with our calculated 61 arc-minutes per turn.
It is obvious that by using the 0° and 45° points as reference, we can significantly reduce the effects of systematic error in the bubble level or digital inclinometer.
After resetting the mount to zero latitude (again confirming with the bubble level), we can set it to 1.3521° by rotating the knob (1.3521° * 60 / 3.875 arc-minutes) = 21 knob ridges. If we had used our derived value of 61 arc-minutes per rotation (3.8125 arc-minutes per turn), (1.3521° * 60 / 3.8125) = 21 knob ridges which is the same as the (known for the Mach1) setting.
On the other hand, if we need to set the latitude to a higher value, say 30°, we would use the 45° angle as the reference. In the case of 30°, we would zero the mount at 45°, then lower it by 15° (which is 900 arc-minutes) which would require 14.5 turns of the knob (14 turns and an additional 8 knob ridges).
This method would allow accurate altitude alignment to within (5.4 arc-minutes error of the level) + (8 arc-minutes from the knob uncertainty) of about 13 arc-minutes (0.22°) worst-case; actual error may be half that. Such a result is good but not great, a drift alignment can achieve better accuracy. With this amount of altitude error, there is approximately 3.5 arc-seconds of drift per minute, hence an unguided exposure of 1 minute with a typical DSLR will still show round stars.
In contrast, had we set the mount's latitude directly from the digital inclinometer, we could have been in error by 1° which would lead to a 15 arc-second drift in 1 minute, thus limiting maximum exposure times to around 20 seconds before star elongation would be visible.
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