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Mike Hannibal asked-

>I have looked through the archives but have been
>unable to find the way to calculate predicted
>inter-stellar, interplanetary,
>stellar/planetary/lunar/solar etc distances.
>
>If someone could point me in the right direction I
>would be grateful. I assume that one is simply solving
>a spherical triangle but when i try to sort it out I
>get confused.
>
>I'd also appreciate guidance on correcting for
>refraction and limb/HP/SD in this regard.
>
>I intend to try to write a SciLab script to do the
>calcs.
>================

To clear up possible confusion, the distances Mike refers to are ANGULAR
"distances", the angle-in-the-sky subtended between two bodies, and not
true distances measured in miles or light-years

Bill has offered some code to do the job, but perhaps Mike wishes to
understand the problem and how to solve it, rather than paste in another's
code.

My first word of advice, to Mike and to many other listmembers, is to
acquire a copy of "Astronomical Algorithms", by Jean Meeus.

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

Take the simplest case first, the angle between two stars. They are so far
away, there's no parallax. And they are points of light, so no
semidiameter. Just a refraction correction.

There are different ways to do the job, but the following method is simple
to grasp; it assumes that Mike knows where he is on the Earth's surface,
roughly if not precisely, and the GMT.

First, get predicted values from the almanac of the position in the sky for
the two stars at the moment of the observation, in terms of dec and GHA
(from SHA and Aries).

Second,  knowing his lat and long, convert those two positions to altitude
and azimuth as seen by him in the sky from where he stands at that moment,
if there were no refraction. I think Mike is already familiar with how to
do this, but for others it's in Meeus, equations 13.5 and 13.6.

Where lat and dec are negative if southerly-

alt = arcsin (sin lat sin dec + cos lat cos dec cos(local hour angle))
az = arctan ( sin(local hour angle) / (cos(local hour angle) sin lat - tan
dec cos lat)

Then  (after converting az from radians to degrees if necessary) put az
into the right sector by the following rules, applied in this order-
If az is negative, add 180 to az
If hour angle is between 0 and 180, add another 180 degrees to az.

You can short-cut this procedure for az by using, if available,
In some calculators, POL( ((tan dec cos lat - cos HA sin lat), -sin HA),
when variable Y will contain az in the right quadrant,
or apply the ATAN2 function to ( -sin HA, (tan dec cos lat - cos HA sin lat))

az now corresponds to the navigator's convention of measuring clockwise
from North, rather than Meeus' "astronomers's convention" of measuring
azimuth from the South (though that distinction is unimportant for this
exercise).

Now we have alt1 and az1, alt2 and az2, for the two stars respectively.

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

Refraction in the atmosphere makes objects appear to be higher in the sky
than their predicted altitude, so now we must add the refraction, which
varies with altitude. Note that this is OPPOSITE to what we do when
correcting a sextant observation for refraction, when we subtract.
Refraction doesn't alter the azimuth, so that remains unchanged.

To get refraction, use the table for "stars and planets" in the almanac, or
calculate fit rom the Bennett formula..

Now calculate the corrected angle between the stars, as we would measure it
when observed through the atmosphere. For this we can adapt Meeus' formula
17.1. We will need the difference between the two azimuths, az1-az2.

Then the corrected angle, subtended between the stars, is-
arccos (sin alt1 sin alt2 + cos alt1 cos alt2 cos (az1-az2) ), which is
what's needed.

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

For planets, Sun, and Moon, the calculation can be done in a similar way,
starting from almanac predictions of dec and GHA, deriving alt and az, and
correcting alt for refraction as before. But for the Sun, don't use the
special Sun refraction tables in the almanac, because these are combined
with semidiameter and parallax corrections; a complication we do not need
for our present purpose, which is concerned with the Sun's centre, not its
limbs. For ALL bodies, then, the refraction table in the almanac, headed
"stars and planets", should be used (or the Bennett formula). Using this
table, the altitude should be corrected by adding refraction, just as before.

But as well as refraction, parallax should also be taken into account,
especially for the Moon. Parallax makes objects appear to be lower in the
sky than their predicted value (which was calculated as though they were
beeing seen from the Earth's centre). So for those bodies, parallax should
be subtracted from the predicted altitude. Again, this is OPPOSITE to what
one would do when correcting a sextant reading. And again, parallax doesn't
alter the azimuth of the body.

How much should this parallax correction be? It depends on the altitude,
varying as cos (alt). It's maximum, then, at zero altitude, in the
horizontal direction, which is why the value at zero altitude is known as
Horizontal Parallax, or HP. For the Moon, this is so enormous (around a
degree), and varies so greatly, that it's tabulated at 3-hour intervals; so
multiply the tabulated HP value by cos alt and subtract it from the
calculated alt. For the Sun, the HP varies little, over the year, from
0.15', and can be taken as constant over time. For planets, parallax is
negligible for Jupiter and Saturn, but for Venus and  Mars it can become
significant at times when they approach the Earth, and the "additional
correction", already allowing for the (cos alt) factor, alongside the
refraction table for stars and planets, can be applied.

So make those corrections, refraction and parallax, to the altitudes of the
two bodies, get the difference between azimuths, and get the subtended
angle just as for two stars.

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

We have left out a final step. The calculation has been of the angle
between the centres of the bodies. That's OK for stars, and for planets if
we treat them as points of light. But for Sun and Moon, we must, for
accuracy, measure from one limb or another rather than from the centre. In
the case of the Sun, we can choose the near limb or the far one, as
convenient.  For the Moon, we must choose the lit limb, the one nearest the
Sun. Then we add or subtract a semidiameter appropriately, from the angular
distance already corrected.

For the Sun, we can take the semidiameter to be a constant 16', or if
striving for greater accuracy, allow for a seasonal variation between New
Year, when it's nearly 0.3' greater, or early July, 0.3' less.

For the Moon, the semidiameter varies much more, and we can take it as 0.27
times the Moon's HP, as taken from the daily almanac tables.

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

The calculation method shown above is only possible if the observer can
state, at least roughly, where in the World he is. I have explained it that
way because it seems to me the easiest method to comprehend, although not
the shortest to calculate. There are other ways to do the job which avoid
calculating those altitudes and azimuths.

Perhaps Mike is asking to solve the same problem that faces a navigator
using "lunars", when he doesn't have that knowledge of his position. In
that case, his problem becomes the traditional "clearing" method, in which
the altitudes of the two bodies are actually observed with a sextant, and
the included angle between the bodies is then corrected on that basis,
without prior knowledge of position. If that's what he needs, he should ask
again, making that clear.

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

There are many fine-points that I have omitted. One is that the almanac
predictions of planets allow for phase effects of the illumination, so show
the centre-of-light, rather than the geometrical centre.


I do hope that I've got it all right....

George.











===============================================================
Contact George at george@huxtable.u-net.com ,or by phone +44 1865 820222,
or from within UK 01865 820222.
Or by post- George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13
5HX, UK.

   

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