## Wednesday, October 25, 2017

### Select Questions-Answers From All Experts Astronomy Forum (3. Angles of the Sun at Sunrise in Winter etc.)

Question -
I live in a house with windows on two sides, South and East. When sitting
up in bed, on the longest day, the sun appears at about 4 am through one
window facing approximately due East. On the shortest day when sitting in
the same position the sun appears through a window approximately south to
south-east. The horizon between the two views is  at the same level to
within a few metres. the horizon is about 3-4 Kms away

My question is what is the relationship between the angle of the sun
rising on the longest/shortest day and my position on the globe -that is
my latitude (Salisbury UK)

First, let's set out the reference frame for the angle- which we call
'azimuth'.

We can begin by noting the azimuth angle positions for the four cardinal
points of the compass:

North: A  =   0 degrees

East:  A  =  90 degrees

South:  A  = 180 degrees

West"  A =  270 degrees

Note that there may be some deviations for differing systems, but in every
astronomy course I've taught, this is how azimuth has been defined. Thus,
I call the angle more exactly "astronomical azimuth".

Now, for rising times at the equinoxes (approximately on March 21, and
Sept. 23) the rising angle will always be at 90 degrees. Thus, due east.

By the same token, the setting angle will be:

90 + 180  =  270 degrees or due west

Thus at those two dates only will you observe the Sun rise at *due east*

Now, what about on the shortest and longest days of the year?

We can employ a simple trig relationship to obtain the azimuth angle on
these dates, and make appropriate conclusions.

That relationship is expressed:

cos (A) = sin(d)/ cos (L)

where A is the azimuth of the Sun

d is the Sun's declination (which may be obtainable from a table - though
we know it right off from the equinox and solstice positions: +/- 23.5
degrees at solstices, 0 degrees at equinoxes).

L is the latitude for which I am using 51.5 degrees N, for London.
(Salisbury is only 10-12 miles further south on its own latitude line, so
the difference will be negligible for these purposes).

Now, for the December Solstice the Sun is over the Tropic of Capricorn
(lat. 23.5 S) so its declination is  -23.5 degrees.

We have for the Sun's azimuth at sunrise on Dec. 21 near your locale:

cos (A) = sin (-23.5)/ cos (51.5)

which gives approximately, 130 degrees./

Where is this on our observing circle reference?

We know that 180 degrees is due South so that this must be:

40 degrees SOUTH of due East. (90 + 40 = 130)

Now, on the longest day of the year (~ June 21) the Sun is over the Tropic
of Cancer at 23.5 N latitude, so the Sun's declination is + 23.5 degrees.
Then the azimuth for that date is:

cos (A) = sin (23.5)/ cos (51.5)

And A = 50 degrees,

This would put the Sun's rising position North of due East, specifically
40 degrees North of due East.

Based on the preceding results, between the shortest and longest day of
the year you should be seeing the Sun move from south of due east to north
of due east, by the amount of degrees difference indicated.

This also discloses that on the longest day  you cannot be observing
the Sun at true due East, but rather forty degrees North of that position
at rise time. On the shortest day, you'll be seeing the Sun 40 degrees
south of East, and you  correctly noted "approximately south to
south-east".

Re: the difference between the two views and being meters apart, and "the
horizon 3-4 km away", these linear measures are all meaningless in terms
of Sun positions. Mainly because they are subjective and variable,
dependent on the observer and his unique domain, landscape etc..

This is why angular (e.g. azimuth, hour angle) are the measures we use
since they dispense with the idiosyncrasies and peculiarities of different
linear distances. In other words, given the right instruments - e.g.
alt-azimuth measuring device- they apply to ALL observers irrespective of
particular location or house.