How long is the 85th parallel?

[Pleonast]

@Francis_Vaughan

Of course, this is the dope, we worry things to death.

Actually, the approximation I used isn’t correct either. Whilst the distance from the pole to the centre of the earth is less than the distance of centre to the equator, the radius of curvatures are the other way around - the earth curves more at the equator than at the pole. So the poles are flatter. The radius of curvature at the pole is larger than the radius of curvature at the equator, and thus the 85th parallel is actually longer than that on a perfect sphere, not less. At 85˚ it seems to be about 6398km.

So the answer is 3503.6 km or 2177 miles.

Still only 5 miles out from the back of the envelope calculation - which shows how useful such simple approximations are.

No, that’s not right. It’s the radius that matters, not the radius of curvature. The radius of curvature only matters when following a great circle. The 85th parallel does not.

To compute the length of the 85th parallel, we need the distance from the parallel to the Earth’s axis. That is, the radius of the circle that the parallel forms about the Earth’s axis. And then multiply that by 2 pi.

If we assume the Earth is spherical, the “mean” radius is 6371.0 km (3958.8 mi) (see factsheet). Multiply that by cos(85 deg) and we get the distance from the 85th parallel to the Earth’s axis: 555.27 km (345.03 mi). Mulitply by 2 pi and the length of the 85th parallel is 3488.9 km (2167.9 mi).

However, since we are so close to the pole, we can use the polar radius of 6356.8 km (3949.9 mi). Because the radius is smaller near the North pole, the distance from the 85th parallel to the axis is smaller: 554.03 km (344.26 mi). Even though the radius of curvature at the 85 parallel is larger than near the equator, because the radius of the 85th parallel about the Earth’s axis is smaller compared to a spherical model, it’s length is smaller: 3481.1 km (2163.0 mi).

Of course, better model is an oblate spheroid. With that model, the radius of the Earth at the 85th parallel is 6356.9 km (3950.0 mi). Yes, only a difference in the fifth significant digit. Using that radius, the 85th parallel is 3481.1 km (2163.1 mi). The difference is 62 m (204 ft), right at the edge of our precision.

There are more precise geoid models, but I can’t imagine they’ll make more than a few hundred of meters difference in this case. The bottom line is any method you use is going to be approximately correct. But be careful that any corrections you make are going towards more accurate, not less.

[ZenBeam]

I think Francis Vaughan is correct. The 85th parallel is defined by the angle of the surface, not by the angle of a line going through the center of the Earth. So the radius of curvature would be the correct radius to use.

See Wikipedia’s article on Latitude.

Geodetic latitude: the angle between the normal and the equatorial plane. The standard notation in English publications is φ. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid.