Re: [Phys-L] moonrise celestial mechanics

[John Denker <jsd@av8n.com>]

On 03/03/2014 09:30 PM, John Mallinckrodt wondered:

> if I'm making another mistake because my calculations from Bob's film
> seemed to work out so well under the assumption that the moon's orbit
> was essentially equatorial.

Here's how I think about it:  To lowest order, the only thing
that matters is that the celestial sphere rotates.  Every object
on the sphere goes around once per sidereal day.  This gives 
the object an angular velocity proportional to the cosine of 
its declination.

There are corrections for things like the sun, moon, and planets
that move across the celestial sphere.  This is approximately 
a 3.7% correction in the case of the moon.  I assume everybody 
has already taken this term into account.

To lowest order, the moon is always near the celestial equator.
Apparently the calculation was done using this approximation.
So let's see how big are the next-order correction terms.
That will tell us how good the approximation is.

The biggest correction term says that the moon is always near
the ecliptic, and the ecliptic is inclined 23.4 degrees to
the equator.  This means the declination is somewhere between 
minus 23.4 and plus 23.4 degrees, with a peculiar distribution.

The next correction term says that the moon's orbit is
inclined 5.1 degrees relative to the ecliptic.  This means
the declination is somewhere between 28.5 degrees south and 
28.5 degrees north, with an even more peculiar distribution.
Circles within circles.

These two terms (and any other declination-related terms)
contribute an error of the form (1-cos(declination)). Looking 
at the distribution, I find that the error ranges from 0% to 
12.1% slow, with an average (i.e. mean) of 4.3% slow.

For more about the distribution, see note below.

Another correction comes from atmospheric lensing.  As a
rough estimate, if the moon pops straight up out of the
horizon, as could happen when observing from the equator,
this could slow things down by an additional 10%, depending
on details such as elevation and weather.

  In contrast, at high latitudes, the moon rises on a more
  nearly horizontal trajectory, and the lensing effect is
  less pronounced.

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

Note on peculiar probabilities:  There is a deplorable
tendency for people to assume that every probability 
distribution is Gaussian.

In the case of the moonrise rate error, the distribution 
of errors is highly non-Gaussian.  Because of the cosine, 
the probability density in the vicinity of zero error is 
very high, indeed singular.
 -- You would never know this by looking at the mean, standard
  deviation, min, max, or anything like that.
 -- You wouldn't necessarily know this by looking at an 
  ordinary histogram.

Therefore I have invented a thing called a /diaspogram/ which
shows the histogram /and/ a scatter plot of the raw data.  In 
cases where the histogram does not do justice to the data, this
gives you a chance to notice the problem.

I cobbled up some Monte Carlo data for the moonrise problem.
Notice that there are waaaay too many points hugging the 
zero-error edge of the plot:
  http://www.av8n.com/physics/img48/moon-dec-error-density.png

For more about this technique, see
 http://www.av8n.com/physics/probability-intro.htm#sec-diaspogram

Another good way of seeing what is going on is to plot the
/cumulative/ probability, not just the probability density:
  http://www.av8n.com/physics/img48/moon-dec-error-cume.png
A vertical rise in the cumulative probability corresponds to
an infinite probability density.

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

There is a lot of trickery involved in getting a spreadsheet
app to make nice plots like this.
 -- sorting is non-obvious (and needed for plotting the
  cumulative probability)
 -- the nice way to make a stair-step plot is non-obvious
 -- counting the number of points in each bin of the
  histogram is non-obvious

If I had to figure out all these tricks from scratch, it 
would be unreasonably laborious, especially for a problem 
that is not all that important to me.  Fortunately, I have
long since figured out how to do it, and I kept notes
  http://www.av8n.com/physics/spreadsheet-tips.htm
so I can barf out plots like this in less time than it takes 
to tell about it.