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I think it's worth quiestioning some dtails of Frank Reed's recent posting
on this topic.

He wrote-

>A big part of this error results from the fact that the ecliptic only looks
>like a sine curve. It's not a bad approximation ...

I fully agree with this, so far.

But then he continues-

>..., but the real curve has
>"broader shoulders" than a sine curve. It's easy to see this by imagining
>what the
>ecliptic would look like if the Earth's axis were tilted, say, 80 degrees
>instead of 23.45.

=================

Not that easy to see, Frank.

We are, I think we agree, presuming the Earth's orbit to be circular, so
the intervals between the solstices and the equinoxes are identical,
exactly a quarter of a year. And we assume that the Earth is precisely in
the plane of the ecliptic, which is almost exactly true by definition, so
the Sun's ecliptic latitude is zero.

Meeus, "Astronomical algorithms", in equation 13.4, gives an equation which
we can now simplify into-

sin dec = sin obl * sin long       giving "real" declination.

in which obl. is the obliquity of the Earth's equator to the ecliptic, of
something like 23.4deg.

Long is the Sun's celestial longitude, measured in terms of angle from the
vernal equinox. Under our crude assumption of no eccentricity, long will
increase steadily as time passes, to be 90deg at the moment of summer
solstice, and so on.

We are considering how good an approximation to the "real" solution, above, is-

dec = obl * sin long     "approx"; our sine-curve approximation

For small obliquity, dec and obl will both be small angles, and the sine of
a small angle is accurately proportional to the angle itself, so our sine
curve will be a good approximation. But less so, as obliquity increases,
and 23.4deg is not a small angle in those terms. Let's plot out some values
of real and approx declinations, for obl = 23.4, as long increases from
0deg to 90 deg.

long  real  approx
 0   0.000   0.000
15   5.900   6.056
30  11.453  11.700
45  16.309  16.546
60  20.111  20.265
75  22.558  22.603
90  23.4    23.4

It seems clear, from this, that the approximation, by a simple sine curve,
has (to use Frank's descriptive term) "broader-shoulders" than the true
declination curve; not the other way round as Frank suggested.

You can see that the approximation is good to about a quarter of a degree.
Another error of similar magnitude results from our neglect of the
eccentricity of the Earth's orbit.

George.

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