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- Overview
- Coordinates, angles and formulas
- Download or print the tables
- Using the tables
- Comparison to planisphere and accuracy

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This page describes the use of a set of tables from which you can find the transit, rise and set time and azimuth of the Sun and 5 major planets using only addition and subtraction for any place between 25 and 60 degrees latitude. You can also use the tables to find the transit, rise and set time and azimuth of any other object if you know the rough RA and declination of the object.

There are 4 separate tables...

**transit times**of the Sun and major planets for each week in the year - this table changes year on year, smallish changes for the Sun but huge changes for the major planets.**sidereal time**at midnight Local time for each day of the year. This table changes by a few minutes of sidereal time year on year depending where we are in the leap year cycle. This table is used to get RA from transit time for the Sun and planets, and the transit time from the RA from any other object you want to plan for.**'semi-dirunal arc'**for various observer latitudes and object declinations - this table does not change, and tells you the rise and set time for an object at your latitude if you know the transit time and declination.**'amplitude'**of rising or setting for various observer latitudes and object declinations - this table does not change from year to year.

Accuracy is usually better than 5 minutes of time and seldom worse than 10 minutes - better than a planisphere or graph and good enough for most planning purposes. You can also use the tables to find the rough RA of the Sun and any of the 5 major planets.

You can use the tables as they stand from a wide range of mid- Northern latitudes - tables are downloadable in Adobe Acrobat format to preserve the formatting. The tables are calculated using an MS Excel spreadsheet - you can download and customise the spreadsheet for different uses or latitude ranges.

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Imagine that you are looking due South at about 10:30 pm on a clear late December night in the Northern hemisphere, say around 52 degrees latitude. Just to your left, you notice the familiar shape of Orion, with Rigel as the bottom right foot of the Giant....

As time progresses, Rigel will cross your *meridian*,
where the star will reach its highest above the horizon, at about
11pm. Then the star will move westwards around the sky and set at
about 4:30 am the next morning. If you have a clear West horizon,
you will notice that the star sets about 13 degrees South of due
West (bearing 257 degrees).

The path that Rigel describes though the sky depends on the
* declination* of the star and your latitude on the Earth.
The time of transit depends on the * Right Ascension* of
the star and on the time of day told by the stars (the
*sidereal time*). The diagram below may clarify these
angles...

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Your horizon is shown in yellow. The plane of the page
(screen?) contains your *meridian*, the great circle
containing the North, and South points on your horizon and the
*Zenith* directly above your head. The North Celestial
Pole P is the pivot around which the wheel of the stars appears
to turn. When a star crosses your meridian, the Sidereal Time
will exactly equal the Right Ascension of the star, so the
**Hour Angle** (equal to ST - RA) of the star will
be zero. Before that time, the hour angle is negative (or
positive and larger than 12 hours) and after it is positive.

The **declination** of the star is the angle
between the star and the point where the great circle from the
Celestial Pole P through the star crosses the Celestial Equator.
The star in this diagram has a positive or North declination.
Rigel has a negative or South declination. It is the relationship
between the declination of a star and your latitude that defines
both the length of the arc above your horizon and the point on
your horizon at which the star will rise. The diagram below
defines two important angles...

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The **semi-diurnal arc** is half the arc that the
star is above the horizon - defined as the arc from T to 'sets'
in the diagram above. In the Northern hemisphere, the
semi-diurnal arc will be greater than 90 degrees or 6 hours if
the star has a positive declination, and less than 90 degrees or
6 hours if the star has a negative declination. These rules
reverse in the Southern hemisphere. A more elegant way of saying
this is to say that the semi-diurnal arc is greater than 90
degrees if the names of the declination and your latitude are the
same, and less if the names are contrary.
The semi-diurnal arc is usually listed in hours and minutes of
* sidereal* time, and this time can be added and
subtracted to the local *solar* time of transit to give
the rough times of rise and set for the star. Strictly speaking,
we should multiply the semi-diurnal arc period by 0.99727 to give
the period in UT hours, but the maximum error is 2 minutes in 12
hours, so I neglect this correction!

The **amplitude** of the star is the size of the angle
between the point on your (mathematical, fictional) horizon where
the star sets and due West. It is the same as the angle between
the point on your horizon where the star rises and due East. If
the declination of the star and the latitude have the same name,
then the amplitude subtracts from 90 and adds to 270 to give the
bearings of rising and setting. For contrary names, the amplitude
adds to 90 and subtracts from 270 degrees.

The amplitude and the semi-diurnal arc depend only on the
values of you latitude and the declination of the star - so the
tables of these two angles will be valid for *any year*.
In fact, I found tables of amplitudes and the semi-diurnal arc in
a set of 8 figure tables dating from 1924.

**Limits: ** I can't see Canopus rise from 52
North. Canopus has a declination of 52 degrees 42 minutes south.
The most southern star I can see peep over the horizon has a
declination of 37 degrees 30 minutes south. By the same token,
Deneb in the Swan is always above the horizon up here (although
masked by the Sun in Winter). The semi-diurnal arcs of Canopus
and Deneb will be 'off the table' at my latitude. The
*colatitude* or 90 - latitude gives the limiting
declination.

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A star rises and sets when the altitude of the star is zero (we neglect refraction in these tables). Using the formulas for converting equatorial coordinates to horizon coordinates and setting altitude equal to zero, we can derive expressions for the semi-diurnal arc and amplitude...

a = altitude, A = Azimuth dec = declination, RA = Right Ascension, H = Hour angle phi = latitude of observer LST = local sidereal time at observer's longitude H = LST - RA hour angle standard conversion formula gives Sin(a) = sin(dec) * sin(phi) + cos(dec)*cos(phi)*cos(H) a = 0 at rising or setting implies 0 = sin(dec) * sin(phi) + cos(dec)*cos(phi)*cos(H) so Cos(H) = - sin(dec) * sin(phi) --------------------- cos(dec) * cos(phi) = -tan(dec)*tan(phi)

The formula above gives the hour angle at setting or rising. Remember that the Hour angle at transit is zero by definition, so H is also the value of the semi-diurnal arc.

The tables of semi-diurnal arc simply tabulate acos(-tan(dec) *tan(phi)) for various combinations of latitude and declination.

The standard conversion formula for Azimuth gives Cos(A) = sin(dec) - sin(phi) * sin(a) ---------------------------- cos(phi) * cos(a) setting a = 0 at a rising or setting event gives Cos(A) = sin(dec) - 0 = sin(dec) ------------- -------- cos(phi) * 1 cos(phi) The tables of amplitude tabulate the values of Ampl = 90 - acos ( sin(dec) / cos(phi) for various values of declination and latitude.

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I use an MS Excel spreadsheet with VBA user defined functions to calculate the tables. Methods are approximate - mean elements for the planet positions, and a simple 'low precision formula' for the Sun. The tables are then saved as Adobe Acrobat (.PDF) documents to preserve the formatting. Sets of tables for the next few years can be downloaded below, and you can also download the Excel spreadsheet to customise the tables to your liking.**Tables available **
| 2002
| 2003
| MS Excel Spreadsheet (70 Kb .ZIP)

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I think the best way to explain these tables is through examples.

Suppose we want to know when Castor in Gemini transits during the day 1st January 2002. Castor has an RA of 7h 35m and a declination of 32 deg North. We need to use the sidereal time table to find the sidereal time on the day in question.

Sidereal time at midnight Jan 1st 2002 local time is 6h 41m, so the hour angle of Castor at midnight is LST - RA = 6h 41m - 7h 35m = -1h + 6m = -54 min. Negative hour angle means that Castor has yet to transit, so transit will occur 54 minutes after midnight, or 0h 54m. ICE gives transit time of 0h 52m 42 sec TDT.

Now we look at Castor nearer June, say June 14th.

Sidereal time at midnight June 14th 2002 local time is 17h 28m, so the hour angle of Castor at midnight is H = LST - RA = 17h 28m - 7h 35m = 10h - 7m = 9h 53m min. Positive hour angle means that Castor has already transited, so transit will occur at 24h - 9h 53min = 14h 7min on June 14th. As this lies between 9h and 15 h, we can subtract 2 minutes of time to reflect the shorter sidereal day, so I'd take the transit time as 14h 5min ICE gives transit time of 14h 03m 55.8 sec TDT. Alternatively, take Sidereal time at midnight the next day (June 15th) as 17h 32m. Then we get H = 17h 32 - 7h 35 = 9h 57m Transit = 24 - 9h 57 = 14h 03m

The transit times for the Sun and the 5 major planets are given directly by the tables. The declination is also given to the nearest degree - good enough for the purpose of these tables. To find the transit time of Jupiter on the 14th Feb 2002, just look up 14th Feb in the 2002 tables - the transit time in local time is 20h 51m and Jupiter's declination is 23 degrees North.

The Sun for that day has transit time of 12h 14min and a declination of 13 degrees South. Jupiter will be well visible in the evening sky, being 8h 37m 'behind' the Sun.

You need the transit times to work out the rise and set times for the planets or any other object.

Once you have the transit time and declination for an object,
you can use the **semi-diurnal arc table** to
estimate the rise and set times for the object at your location.
The procedure is

- Look up the value corresponding to the declination of the object and your latitude in the semi-diurnal arc table
- If the 'names' of the declination and latitude are the same (ie both South or both North) then the table gives the semi- diurnal arc directly.
- If the 'names' are contrary (one North, the other South) then
the table gives the semi-
*nocturnal*arc, ie half the time the object is*below*the horizon. To find the semi- diurnal arc just subtract the table figure from 12 hours. - Having found the semi-diurnal arc, just add and subtract this time to the time of transit for the object and you have an estimate of the rising and setting times.
- Times above 24h just mean the setting occurs on the next day - just subtract 24 hours from the time and adjust the date.

Jupiter transits at 20h 51m on Feb 14th, so we get rise and set times of 12h 22m and 29h 20m (ie 05h 20m on the 15th Feb) respectively.

The Sun on 14th Jan 2002 has a declination of -13 or 13
degrees South. The table gives the semi-*nocturnal* arc as
7h 17m for 55 degrees North, so the semi-diurnal arc is 12 - 7h
17m = 4h 43m. The Sun transits at 12h 14m, and so rises at
roughly 7h 31m and sets at about 16h 57m on this day.

ICE gives Jupiter rising at 12h 10m, transiting at 20h 47m and setting at 05h 24m the next day, and the Sun rising at 7h 25m, transits at 12h 14m and sets at 17h 04.

The estimated times from these tables do not take refraction into account, and so errors of 5 to 15 minutes might be expected.

To find the azimuth of rising and setting of an object, you need

- the
**declination**of the object - your
**latitude**

You then use the amplitude table to find the amplitude of the
object. If the declination of the object and your latitude have
the same name, then the amplitude is *added* to 270
degrees to estimate the azimuth of setting, and *subtracted
* from 90 degrees to estimate the azimuth of rising. I
remember this by visualising the object as **higher than
the equator on the Celestial sphere** and thus rising and
setting North of the East-West line and so getting **closer
to being circumpolar**.

If the declination and your latitude have opposite names (for
instance in the case of Rigel), then you *subtract* the
amplitude from 270 and *add* to 90 to estimate setting and
rising azimuth. Again, I remember this by visualising the object
as **lower on the celestial sphere** thus rising
South of the East-West line, and so getting **closer to
never appearing**.

As a practical example, suppose we want to know the Sun's azimuth of rising on 2002 May 28th, and my latitude is 52.5 degrees North. The Sun has a declination of 21 degrees North on that date. Entering the amplitude table at latitude 52.5 North and Declination 21 gives an amplitude of 36 degrees.

As the declination and latitude have the same name, I can say the Sun sets at an azimuth of 306 degrees and rises at an azimuth of 054 degrees. ICE gives azimuth 308 for setting and 052 for rising on this day (and the declination of the Sun is given as just under 21.5 degrees - so we might interpolate the amplitude as 37 degrees).

Suppose we estimate the azimuth of rising and setting for 2002 Dec 21st (traditionally taken as the shortest day up here at 52.5). The calculation is sumarised below:

- Sun's declination 23 South from tables
- At Latitude 52.5 North, we get amplitude 40 deg
- Declination and latitude have opposite names so subtract from 270 and add to 090
- Azimuth of setting is 230 and azimuth of rising is 130

The RA of a body at transit is equal to the sidereal time at transit. The Sun transits at 11:57 on 2002 May 28th local date, and the sidereal time at 0h on that day was 16:21. 16:21 + 11:57 = 28:18 or 4:18 h. ICE gives the RA of the Sun as 4:20:37 for that instant of TDT. In practice, the RA of the Sun changes little over a day or week, and so this value of 4:18 will stand for some days around the 28th.

To estimate the RA of a faster moving planet at times other than the transit time, just use the formula H = LST - RA. Suppose we want to know the RA of Venus on 2002 May 28th at 20:00 local time. Venus transits at 14:19 on that day.

- The HA of Venus is approximately 20:00 - 14:19 = 5:41h
- The LST at 20:00 is approximately 20:00 + 16:21 = 36:21 = 12:21 h
- LST - H = RA so 12:21 - 5:41 = 6:40 hrs
- ICE gives 6:43:36 hrs on that day

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The main corrections relate to your local time, and the use of sidereal time as if the hours are the same length as local time hours. The corrections are summarised below.

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Last Modified: 26th May 2002

Keith Burnett