Always orient a sundial towards polar north?

Question

I had a horizontal sundial made specially for my latitude (approx. 38° N). The default correct way to orient a sundial (in the northern hemisphere) is with the gnomon pointing towards polar north. The caveat is that my latitude (approx. 122° W) causes "clock noon" (the time when a clock on my wall indicates noon) is approximately 8 minutes behind solar noon (the time when a sundial pointing polar north indicates noon), hence the time indicated by the sundial will never match clock time, not even on days when the Equation of Time (EoT) is zero.

If I were to instead orient the sundial such that the gnomon pointed at noon when it's noon according to my clock (which means the sundial would be rotated a few degrees counter-clockwise from polar north) on an EoT zero day, would that be "better" overall or make discrepancies worse overall? Or should I just orient the gnomon to polar north?

Answer 1

A sundial needs to be aligned with the planet's axis of rotation, with the gnomon pointed in towards the nearest pole, for it to work correctly. The point of the shadow will then give you your Local Time (LT).

As you've discovered, LT only coincides with "clock time" (i.e. your time zone when there's no daylight saving) if your longitude exactly corresponds to the UTC offset for that time zone. This is somewhat rare. Each degree east or west of that exact UTC-offset longitude creates a 4-minute discrepancy between local time and clock time.

The problem is that if you align your gnomon so that local noon equals clock noon, it will no longer be aligned with the Earth's axis and the "mean time" curve imprinted on your sundial may no longer match the diurnal path of the gnomon's shadow. In effect, you've distorted the conic section. You can picture this distortion more easily if you imagine exaggerating the adjustment: say, 90 degrees (eg gnomon pointing due west).

A 1.5 degree distortion is probably trivial, but some would say that a 6-minute discrepancy between LT and clock time is equally trivial (especially when the EoT discrepancy can be much more than that).

A much more accurate solution is possible. If your sundial is correctly oriented, the point of the gnomon's shadow will closely follow the mean time line on the sundial's horizontal plane on "EoT zero day", with the discrepancy from "clock time" remaining more or less consistent throughout the day. Since the problem is not the device but the time labels, the solution is to change the labels. You could use a permanent marker to incorporate the six-minute delay, so that (for example) either the label for the noon mark on the sundial reads 11:54am, or a new mark labelled 12:00pm is placed at the 12:06pm point on the sundial.

Answer 2

There's another possible solution: tilt the sundial's dial plate so that it aligns with the earth surface at the "nominal longitude" for your timezone, while the gnomon still is aligned with the axis of rotation.

Why does that work?

If you use your sundial at a place where local time and clock time match (center meridian of a timezone), it shows the correct time. Of course, this does not magically know where you are on the globe, but just depends on the geometry of the dial plate and the gnomon relative to the sun and the axis of rotation.

So, wherever on Earth you mount your sundial, if you orient the gnomon and the dial plate exactly the way you'd do at the "nominal" location, it will show the same "sundial time". As the earth isn't flat, when installing the sundial in a different location, you'd need to compensate for the curvature of the surface.

How to find the correct orientation of the dial plate and gnomon?

You can calculate the necessary tilt, but that involves quite some trigonometry (see EDIT below).

Or you can calibrate it on a sunny day:

• As you're living a bit west of your timezone's center meridian, by 2 degrees, lift the west end of the plate by 1.5 degrees. E.g. use wedges to temporarily hold that position. Then rotate the dial plate until it shows the correct clock time.
• Check the accuracy at different times of day and adjust lift and rotation until you're satisfied.
• The gnomon's edge (not the 12 o'clock line of the dial plate) should always point to the north and have an upward angle corresponding to your latitude of 38° N, if you can manage to measure that.
• When you're satisfied, mount the sundial permanently.

Don't expect too much: the sun's diameter always causes a blurry shadow instead of a sharp edge. That means that you can't reliably read out times better than maybe two minutes.

EDIT (Here's the math as per OP's comment):

You want to rotate the sundial aound the gnomon's axis by the angle $\lambda$ that you have to compensate (i.e. 2° counterclockwise for your 2° west of the center meridian).

So, you can move the west end of the dial plate up and south, in a direction perpendicular to the gnomon's axis, by an amount so that the overall rotation is exactly what you want to compensate.

• Let $\lambda_{rel}$ be the longitude offset (west = positive).
• Let $\phi$ be your latitude (north = positive).
• Use a fixed point at the east end of the dial plate.
• Find its west counterpart. Let $L$ be the distance between these east and west points.
• Lift the west point up by $L \cdot \sin \lambda_{rel} \cdot \cos \phi$.
• Move the west point southwards by $L \cdot \sin \lambda_{rel} \cdot \sin \phi$ (i.e. rotate the dial plate)
• Make sure the north-south axis of the dial plate is still horizontal (both ends at the same height). That is not universally correct, but for deviations of a few degrees it's a very good match.

If you're east of the nominal meridian, you'll get negative values for the west-end movement, implying you should move the west end down and north. But you can of course instead hold the west end, and move the east end up and south.

If you're on the southern hemisphere, you'll get negative move-south values, meaning to move north instead.