»  Solutions to puzzles in my National Review Online Diary

  September 2010


In my September diary I posed the following brain-teaser.

Around this time of year we get Derbhenge. See, I rise early and walk my dog right after breakfast. Derb Manor is at the eastern end of my street, which is straight in this stretch. Walking back towards the house my compass heading is a tad south of east — to be precise, it is 90° 37′ 27.3″ from true north. Given that my house is at latitude 40° 51′ 36.1″ N, on what fall date shall I experience Derbhenge, i.e. the sun rising precisely at the end of my street (and right into my eyes, dammit)?

(To forestall the more punctilious kind of reader, I'll add that my house is at a height of 240 feet above sea level. By all means factor that into your calculation. I didn't because I can't believe it makes one day's difference to the date. If you can prove me wrong, I'll send you a free copy of We Are Doomed.)



My first shot at this was a quick lookup on Table 5 in Section 7 here. That, with some linear interpolation, got me October 4.

Unfortunately that lookup was way too quick, as readers were brutal at pointing out when I tried it on them. I'd skipped a row — which is to say, a week — somehow. Excuse: It's an unwieldy table. Working with the correct rows, I get September 25, which is more like my actual experience (though with contours & treeline, it's hard to be sure).

Several readers went way beyond the call of duty here. Reader A:

I used JPL's Horizon solar system ephemeris generator (setting longitude to 73 West).

I asked for solutions for apparent azimuth and elevation. I zeroed in to one minute temporal resolution around sunrise 6:45 ET on September 24, though it wasn't a particularly good match this year, high by 0.037. (Horizons only offers one minute resolution, but linear interpolation between the minutes bracketing sunrise should be pretty accurate.) Of course, it varies depending on the year, but the closest day is September 24.

When I reran the case for 10:44 UT (6:44 ET) for the same 11 years, with an elevation 1240 feet, neither the azimuth nor the elevation changed, with 0.0001 degree resolution, so as you assumed, the results are quite insensitive to elevation.

However, the results are not insensitive to longitude. The apparent azimuth of the sun at sunrise on this date at this latitude changes by about 0.000976 degrees per degree of longitude. (And not coincidently by about 0.001952 degrees per day).

Of course, if you moved Derbyhenge to West Longitude: 129° 44′ 35.2″, the solution for 2010 would more or less be exact.

Table of sun at sunrise for Derbyshire located at 73° W longitude for various years. Notice it decreases by about 0.1248 degrees per year until leap year when it jumps up by 0.3904 (just a but more than 0.1248 × 3, the Gregorian calender 3/4 century rule probably accounts for most of this.)

  Year  DoW      Azimuth   Error
  2006 Wednesday 90.66122  0.037023
  2007 Monday    90.53617 -0.08803
  2008 Wednesday 90.92531  0.301114
  2009 Thursday  90.80598  0.181782
  2010 Friday    90.67959  0.055393
  2011 Saturday  90.55324 -0.07096
  2012 Monday    90.94486  0.32066
  2013 Tuesday   90.81786  0.193656
  2014 Wednesday 90.69471  0.070507
  2015 Thursday  90.56829 -0.05591
  2016 Saturday  90.95504  0.330838

[The table I originally used] is a general kind of table valid for dates near the present and longitudes near Greenwich and certainly not offering better than 0.1 degree precision.

JPL Horizons offers very high precision, but would probably give about the same date most of the time. But for someone on the other side of the world, at 107 degrees East longitude, the results might indeed be different by a day, since the sun will move about half a degree along the ecliptic during 12 hours between sunrise at the two locations. An extreme case would be two observers a foot apart but straddling the international date line. They would observe exactly the same sunrise and each would assign it a different date.

As a further check, I created a DerbyHenge (location 40° 51′ 36.1″ N 73° 26′ 2.4″ W) in Starry Night Pro 6.0 and came up with the following solar azimuth at sunrise for these dates in 2010, so Starry Night would have picked September 25, but I trust JPL Horizons more.

(Note: a respectable astronomer would place the units indicator, ′, before the decimal point: 89° 57′.743 . Or would use minutes and seconds. Of course the Chaldeans did not stop at seconds of arc; and fractional parts of a second of arc would be expressed as thirds of arc with symbol ′′′, representing 1/216,000 degrees.)

  9/24  89° 57.743′
  9/25  90° 38.445′
  9/26  90° 59.510′
  10/4  95°  6.036′

Phew! Reader B came up with September 25 via some antique well-tested software:

As the attached screenshot shows, on 25 Sep 2010 the Sun will (did, actually) rise at your house at 5:44am GMT, 6:44 EST, at an azimuth of 90 degrees, plus. Note that for calculation purposes I assumed a nearby longitude, which is not critical.

This is from a neat little 3½-inch floppy DOS program that runs nicely on an ancient 8086 HP palmtop which would now be called a pda.

P.S. [quoting me] "Walking back towards the house my compass heading is a tad south of east — to be precise, it is 90° 37′ 27.3″ from true north. Given that my house is at latitude 40° 51′ 36.1″ N …"  My, what precision; were you using a theodolite, or more likely, taking it off your house survey plat?

No Sir, I was working from the USGS map of my neighborhood. That'll get you a tenth of a minute (about ten feet).

These solutions reflect the general opinion: September 25 or 24, split around 2:1. Practical astronomy is hard.