go west, young man!

I like precision. If my watch is more than 2 seconds off the real time, I feel the need to rectify it (if only Aus had an atomic radio signal like most of the civilised world). Not only do I like to know precisely what time it is, I like to know precisely where I am and where I’m supposed to be going. So when people head east or west, north or south, I don’t want them heading off a few degrees either side of these headings!

So it is pretty clear that I wouldn’t fit in the ancient world with this kind of attitude. They didn’t have watches at all, and the old wrist sundial was notoriously imprecise. Furthermore, it seems probable that people had no idea where they were going (well, not by my standards of precision). And it is that consideration which has provided the opportunity to introduce a little mathematical fun to my blogging today.

But before we get to the maths (or “math” for those of you in the US), let’s recap what we know about directions. In a previous post on the location of Eden I referred to the possible significance of the easterly migration through Gen 3–11 and the subsequent instruction by God to Abram, “Go west, young(ish) man!” Now it’s well known that, in ancient Israel, the primary direction (the direction you’d put at the top of the map) was east, where the sun rose.

Actually, we here in Aus know that the ancient Egyptians had it right, aligning the top of their map with the source of the Nile. This link will show you why this correctly reveals the true arrangement of a world map.

Anyway, a while back this got me wondering about just how imprecise using the sunrise as the basis for orientation could be, so I set out estimating just how far the sun ranges from sunrise in mid winter to sunrise in mid summer around Jerusalem. For those of you who’d rather not wade through the maths, skip to the end. For the rest of you, here goes (and if you spot any glaring mathematical errors, please let me know).

Before we begin, some housekeeping. I’m using standard spherical polar coordinates here, as described at MathWorld.

For simplicity, we shall make a couple of assumptions about our system, which are:

  • The earth is a perfect sphere. In fact, the earth is better described as an oblate spheroid, but this somewhat complicates the calculations and would only minimally alter the results (refer to the discussion at the end of this section), and
  • The sun is infinitely distant from the earth. (This actually assumes that the ratio of the earth’s radius over the distance to the sun is close to zero, a reasonable consideration. Furthermore, any error introduced into these calculations by this would be made redundant by the fact that the sun’s radius exceeds the earth’s and so the earth-sun vector below would still point at the sun, even if not precisely at its centre.)

Given these constraints, we may proceed to calculate the extreme points of sunrise. To do so we require the following data:

  • Angle between axis of earth and normal vector to orbital plane: 23.45°.
  • Latitude of Jerusalem: 31° 46′ North of the equator.

Jerusalem is thus defined as the point P on the globe, and since we are assuming that the sun is infinitely distant, we may (for simplicity) simply define this as a unit vector:

P = (1, θ, 90°–31° 46′)

Where θ ranges from 0° to 360° as the earth rotates during the day. For our purposes we shall determine what value of θ is appropriate for sunrise in mid-summer and mid-winter.

The position of the sun will remain constant. For mid-summer in Jerusalem, we shall position the sun at:

s = (1, 0°, 90°–23.45°)

For mid-winter the position of the sun will be:
s = (1, 0°, 90°+23.45°)

The local coordinate system at Jerusalem may thus be defined by the spanning vectors:

spanning vectors

(Here k is the vector (0, 0, 1) in the main coordinate system.) This system defines the local directions up = iJ, east = jJ, and north = kJ. Consequently, the local vector to the sun may be defined as:

(siJ, sjJ, skJ)

At last we are in a position to define sunrise (or sunset) as occuring when the direction to the sun has no vertical component in the local coordinate system—that is when

siJ = 0

Converting these values from spherical to rectangular coordinates gives the following:

iJ = ( sin(58°14′) cos θ, sin(58°14′) sin θ, cos(58°14′) )
s = ( sin(66.55°), 0, cos(66.55°) )

Substituting into the previous equation and solving for θ gives:

Calculating the result gives: θ = ±105° 34′ 52″

Given this value, we may now determine the position on the horizon where the sun will rise mid-summer in Jerusalem by converting the vector s to local coordinates. We now know that at sunrise, the position of Jerusalem is defined as:

P = (1, –105° 34′ 52″, 58° 14′)

This allows us to determine the entire set of local coordinates (according to the set of equations set out above) and gives the rectangular values:

iJ = ( –0.2284, –0.8190, 0.5265 )
jJ = ( 0.9633, –0.2686, 0 )
kJ = ( 0.1414, 0.5071, 0.8502 )

Converting s to the local coordinate system gives the following result:

sJ = ( 0, 0.8837, 0.4681 )

Consequently, the angle from true East to the position of sunrise in mid-summer here works out as:

That is, in mid-summer, the sun rises 27° 54′ 32″ north of true east. Now to save going through this all again, I’ll just note that the winter calculation reveals sunrise occurs at a similar angle south of true east, so the entire angular range through which sunrise travels throughout the year is 56°.

Now recall that we did make a couple of assumptions so as to keep the calculations above simple, so this is a rough value. But it does clearly show that using sunrise is not a precise means to define a direction.

Does this actually tell us anything useful? No, not really. Just a curiosity.

Perhaps my next post will contain something which proves to be more than an opportunity to play with some maths. We shall have to wait and see!

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