John Pazmino
  NYSkies Astronomy Inc
 2012 January 8 initial
 2013 July 3 current
    The NYSkies Astronomy Seminar of 6 January 2012 was the first of a 
two-part set on the Venus and Mars apparition of-2012. Mars was 
treated on the 6th and Venus will be the topic for the meeting on 
January 20th.
    During the Mars discussion the work of Kepler and Copernicus was 
compared with the Ptolemaeus model of the planets. We saw by sketches 
on the white blackboard that the geocentric and heliocentric schemes 
yield the same line of sight to a planet, to the first level of 
    The main errors in both Copernicus and Ptolemaeus are their use of 
only uniform circular motions which can not properly duplicate the 
real elliptical motion of the planets. The errors are greatest for the 
more excentric orbits of Mars and Mercury but are tempered for the 
more circular orbits of Venus, Jupiter, Saturn. 
    I here go thru one calculation with the geocentric model for Venus 
and Jupiter. By good chance Venus has both an almost circular orbit 
and appearance in the night sky now. Mars, while in the night sky now, 
has too elliptical an orbit for good results.
The exercise 
    Besides Jupiter also having an almost circular orbit like Venus, 
there happens to be on 2012 March 14 a mutual conjunction of Jupiter 
and Venus. The two planets stand on the same ecliptic longitude, with 
Venus a few degrees north of Jupiter in latitude. Since we ignore 
latitude here, we allow the two planets to coincide on that date. 
     Conjunctions were one way that early astronomers cross-checked 
their work. They calculated such conjunctions and looked if in fact 
the two planets lined up in the sky. If one or the other computation 
was wrong, because of math or procedure error, the planets would not 
be in the proper places and the conjunction occurs early or late. 
    With the simplified uniform circular model here there will be a 
discrepancy of a couple degrees in our computation here. In spite of 
this, the exercise brings out some curious and important features of 
planet motion. 
Geocentric model
    In the simplified scheme of pure circular paths and pure uniform 
circular motion a planet moves in a small circle, the epicycle, that 
in turn moves around the Earth in a larger circle, the deferent. There 
is no real body at the center of the epicycle, just an empty point. 
    Since to the ancient astronomers the planets had no material 
substance like mass or bulk, there was no problem with them moving in 
the opicycle with no apparent cause. This factor also maintained the 
premise that real motion may be only around Earth as a real center. 
    The speed of the epicycle around Earth is the mean angular speed 
of the planet and its position along the deferent is the mean 
longitude of the planet. 
    Ptolemaeus and his school knew nothing of orbits as such but did 
understand that there were two periods associated with each planet. 
One was the sidereal period which we know as the planet's own year. 
    The other is the synodic period, the time to complete one round of 
the epicycle relative to the Earth. It is also the period for a 
complete apparition of the planet relative to the Sun in the sky. 
    The interplay between these two period, when judiciously adjusted, 
does described closely the planets as they course thru the stars. Such 
was the strength of the Ptolemaeus system. In the era of naked eye 
observations and crude mathematical tools, the system did work. 
    The Prolemaeus system treated each planet independently of the 
others. There was no 'system' of 'unified organism' of planets. He 
notes in his book 'Almagestum' that there is no way to know the 
distances to the planets to place their deferents in rank above the 
Earth. He used a traditional order of the planets based on angular 
speed thru the zodiac. The Moon moved fastest to earn the lowest 
deferent. Next up were Mercury, Venus, Sun, Mars, Jupiter. Saturn, the 
angularly slowest, occupied the highest deferent. 
    Since distance were not factored into the Ptolemaeus model, the 
deferent of each planet was set to unity radius for each planet. This 
amounts to superimposing all the deferents into one with the many 
epicycles coursing along it. Ptolemaeus did not suggest this. Future 
authors maintained ranked deferents to prevent collisions among the 
planets as they sweeped along in their epicycles. 
    The epicycle is sized relative to unity to make the planet move as 
it does in the sky. In adjusting the epicycle size Ptolemaeus ended up 
making the epicycle and deferent ratio equal to the ratio of size for 
the planet's and Earth's orbit around the Sun. This was beyond him and 
his school to appreciate until Copernicus inverted the epicycle into 
an orbit around the Sun. 
    Note carefully that for an inferior planet the epicycle maps to 
the planet's heliocentric orbit and the deferent maps to Earth's 
  orbit. The epicycle radius is (planet orbit radius) as a fraction of 
the Earth orbit radius. 
    For a superior planet the epicycle maps to Earth's orbit while the 
deferent maps to the planet's orbit. The epicycle radius is 1/(planet 
orbit radius). In both cases the deferent has a radius of unity. We 
have for the two planets 
    parameter       | Venus       | Jupiter         | Sun 
    synodic period  | 583.9 day  | 398.9 day        | --- 
    symodic arc     | 575.51 deg | 393.17 deg       | --- 
    sidereal period | 224.6 day  | 4332.6  day      | 365.25 day 
    deferent radius | 1.000      | 1.000            | 1.000 
    epicycle radius | 0.723      |  1/5.202 = 0.192 | ---
    prev apogee     | 2011-08-16 | 2011-04-06       | 2011-03-20 
    The 'apogee' for the Sun is his vernal equinox. We do not use the 
real apogee of the Sun in his excentric deferent. 
    These are the fundamental parameters describing the planets. As we 
progress thru the exercise we compute others based on these. The 
values are from modern solar system parameters. Note well that the SUn 
has no proper synodic period or epicycle. 
    The parameters are discussed in detail further in this piece. 
Mercury and Venus 
    These two planets posed major problems for the early astronomers. 
While the Moon, Sun, Mars, Jupiter, Saturn nitida mente fell into rank 
above Earth, Mercury and Venus showed no obvious proper place. Authors 
placed both below the Sun, both above, one above and one below. 
    No author that I know of suggested that Mercury or Venus were so 
close to the Sun that its epicycle overlapped the Sun's deferent. There 
could be times when the planet was under the Sun, closer to Earth, and 
others when over the Sun, farther from Earth. 
    The usual scheme has Mercury, then Venus, then Sun. in order above 
Earth. This is the order employed for naming the days of the week, 
making it impossible now to promote a revised order 
Superior conjunction 
    The concept of superior, and inferior, conjunction implies that 
the planet actually passes above or below the Sun. We are so used to 
these terms that we may forget that in the pure Ptolemaeus system the 
ranking of planet deferents was indeterminate. 
    The more historicly correct statement is that Venus comes to 
apogee, farthest from Earth, regardless of where the Sun is. The 
inferior conjunction is the perigee, again with no regard to the real 
location of the Sun. 
    That Venus reaches apogee when she is lined up with the Sun is 
treated as good luck, there being nothing in the model of either Sun 
or Venus to require the two line up at Venus's apogee. They just do. 
    In the usually adapted order of planets, the interval between 
deferents allows that their epicycles do not interfere. Venus never 
ascends higher then the Sun. At the highest farthest point on her 
epicycle Venus is still lower than the Sun. Both of her conjunctions, 
at the apogee and perigee of the epicycle, are inferior to the Sun. 
    By great fortune we can keep the terms inferior and superior 
conjunction for the same aspect of the planets relative to the Sun! 
When Venus is at superior conjunction she is in fact beyond the Sun in 
her orbit. For inferior conjunction she is under the Sun. In 2012 she 
actually crosses his face. 
    The angular distance in the sky of a planet from the Sun is the 
elongation of the planet. It is measured downrange along the ecliptic, 
being (planet's longitude) minus (Sun's). Due regard for crossing 360 
deg/0 deg is taken at the vernal equinox.
    By this sense of the difference, elongations east of the Sun are 
positive to appeal to the planet's presence in the evening sky for 
convenient viewing. West elongations, with the planet in the morning 
sky, are negative. 
    Elongation may be stipulated around the full 360 degrees entirely 
as a positive value or as east (positive) and west (negative) value up 
to 180 degrees. The latter is the more common practice altho it 
requires rectifying the longitude difference into this range. Whole 
revolutions around the ecliptic are generally discarded to leave the 
elongation within 360 degrees. 
    Ecliptic latitude is ignored for elongation. The planet's actual 
angular separation from the Sun is the diagonal of the planet's 
latitude and elongation. It is very nearly just the elongation except 
when the planet is angularly very close to the Sun. 
    Aspects of the planet relative to the Sun are certain standard 
elongations. Of interest in this piece are the conjunctions, when the 
elongation is zero. Note well that the planet may be up to several 
degrees of latitude north or south of the Sun. In this exercise it 
happens that Venus at her inferior conjunction with the Sun in 2012 is 
exacta mente on top of (under, in front of) the Sun in a transit with 
almost zero latitude. 
Calendar maths 
    We need a way to do calendar maths. Some calculettes, mainly those 
for finance and business, have calendar functions. Sci/tech models 
generally lack them. Here is a method of maths that is quick simple 
and accurate consistently to within one day over many centuries. 
    Write the two dates as 'year-month-day'. We do this step by step 
for the interval from Venus's apogee to the Venus-Jupiter conjunction 
      2012  03  14  Ven-Jup conj 
    -(2011  08  16) previous apogee 
    Subtract EACH FIELD SEPARATELY! Do NOT borrow or carry! This is an 
algebraic operation with the signum specificly written. I put the 
subtrahend date in parens to remind that the minus applies to all of 
its fields and not just to the year. 
      2012  03  14   Ven-Jup conj 
    -(2011  08  16)  prior apogee 
      ---- --- ---
        +1 -05 -02 
    Do NOT miss out the signa! Leave one out or mistakenly assume it 
will cause no end of grief later in your calculations. 
    We now convert all fields into days. Each year is 365.25 day, This 
is the Julian year, close enough to the tropical or sidereal year of 
Earth. Each month is 30.44 day, 1/12 of the year. Each day is itself. 
Hours within a day are (hour)/24 decimal of a day. 
    Add together the converted parts of the difference between dates, 
minding very well the signum of each: 
    +01 year  = +365.25 day 
    -05 month = -152.20 day 
    -02 day   =  -02.00 day 
                +211.05 day 
    Venus moved from her prior apogee to conjunction with Jupiter in 
211.05 days. 
    This method is surprisingly good for just about all home astronomy 
purposes, being consistently within one day of the precise result. 
    Do not use this maths for critical work, like occultations. For 
such applications you have to go thru a more accurate computation or 
avail of a fin/biz calculette. An other way is to convert all calendar 
dates to Julian Day Numbers and convert the result to calendar date. 
Synodic period 
    The planet moves in the zodiac  at a speed different from the 
Sun's. It must at some moment line up with the Sun and be lost from 
view in daylight or strong twilight. After a few weeks the planet 
emerges into weaker twilight to begin an apparition. It stays in view, 
moving thru the zodiac until eventually it is lost from view again at 
the opposite twilight. A few weeks later it passes the Sun again. 
    The interval between Sun passes, conjunctions, is the synodic 
period. It is also the length of each apparition or complete round of 
elongations from the Sun, the apogee on the epicycle.
    There is an other sense of synodic period is often lost in modern 
home astronomy. If Mars is 40 degrees east of the Sun now, he'll again 
stand 40 degrees east of the Sun one synodic period later. He was 
there one synodic period earlier. It is well to say that the synodic 
period is the interval between recurrences of the same place in the 
planet's epicycle. 
    The conjunction of the planet with the Sun in traditional 
astronomy is not visible in the sky. It occurs against the Sun in 
daylight. In modern times we can observe the planet at conjunction in 
pictures captured of the Sun by the SOHO satellite. It orbits the Sun 
at the Sun-Earth L1 Lagrange point, well beyond the daylight caused by 
our atmosphere. 
    Its LASCO-3 camera cover a 7-degree radius around the Sun to image 
the outer corona. The pictures show background stars that float by the 
Sun in successive pictures. By examining pictures taken within a few 
days of conjunction the planet slides past the Sun. 
    An outer planet like Jupiter has only one conjunction with the Sun 
per apparition. It is, in the Copernicus model, on the far side of the 
Sun in superior conjunction. 'Superior' is commonly omitted because 
there is only this one lineup with the Sun. 
    Venus and Mercury, inferior planets, have two conjunctions with 
the Sun. Some authors say there are two apparitions, one between each 
pair of conjunctions. The more usual treatment is to consider the full 
cycle of Venus as one apparition with two conjunctions and two 
intervals of viewing. 
Synodic arc 
    While the planet is cycling around its epicycle, the epicycle is 
carried around the Earth on the deferent. After one synodic revolution 
in the epicycle, the planet has displaced a fixed distance downrange 
in the zodiac. Staying with the start of the cycle at apogee, superior 
conjunction, the epicycle center is lined up with the planet, so the 
displacement between apogees is also that for the very planet in the 
zodiac. Wanderings of the planet, the retrograde loops and stations, 
cancel out along the way. 
    By counting the motion from apogee we also place the Sun against 
the planet. The angular length of the synodic arc is the distance run 
by the Sun during one synodic period of the planet. The synodic period 
times the daily motion of the Sun yields this angular displacement. 
    We state the length of the arc as the distance run by the Sun in 
the zodiac between apogees. This is NOT necessarily the amount the 
epicycle, carried by the deferent, moved! Jupiter and Saturn do not 
whiz around the zodiac in a year and then a few more degrees. 
    Jupiter's synodic period is about 393 days and its synodic arc is 
the distance the Sun runs in those days, or about 388 degrees. The 
very planet move only those extra 28 degrees, over the one lap of 360 
degrees. This leads to the rule that Jupiter spends one year per 
zodiac sign as he orbits the Sun. 
Sun's arc 
    The Sun's model has no epicycle and no synodic period. To track 
the movement of the Sun we bank him off of the vernal equinox. When 
the Sun stands at this point in the ecliptic his longitude is zero 
degrees. For spans of a few decades we can neglect the effects of 
precession. Precession causes a westward drift of the vernal equinox 
along the ecliptic of quite one degree in 72 years. 
    The motion of the Sun is smooth enough to fully describe with a 
slightly excentric deferent. There is an apogee in early July and a 
perigee in early January. The Sun advances in the ecliptic a bit 
faster or slower but for our purposes here we assume a uniform speed 
thruout the year. 
    This speed is the full 360 degrees circuit of the zodiac divided 
by the 365.25 days it takes to complete it, or (360 deg)/(365.25 day) 
=  0.9856 deg/day. For casual calculations, we commonly use 1 deg/day, 
like for assessing the position of constellations for a given date. 
Sidereal period 
    A second period associated with the planet is the sidereal period. 
This is not recta mente observable but was well known to the ancient 
astronomers. They needed both periods to manipulate their model of 
planet motion. They knew utterly nothing about orbits as such. 
    The sidereal period was found by watching the planet go thru its 
synodic cycle and applying the formula 
    (planet sid cyc) = (Earth sid cyc) - (planet syn cyc) 
    In this formula we count CYCLES, ROUNDS, LAPS and NOT periods, 
spans,intervals. This is important!! We also count the laps of an 
inferior planet as a NEGATIVE number and a superior planet as a 
POSITIVE number. 
    For the two planets we have from almanacs covering 2008 thru 2012 
the following superior conjunctions 
    Venus       | Jupiter 
    2008 Jun  9 | 2009 Jan 24 
    2010 Jan 11 | 2010 Feb 28 
    2011 Aug 16 | 2011 Apr  6 
    2013 Mar 18 | 2012 Nay 13 
    For Venus we have 3 synodic cycles completed in 1743.93 days or 
4.7746 Earth sidereal cycles. Recall that we define the year by the 
rounds of Earth in her orbit or of Sun in his deferent. 
    Jupiter goes thru 3 synodic cycles in 1205.51 days or 3.3032 Earth 
laps. Then we get 
    item          | Venus    | Jupiter 
    Earth sid cyc |  4.7746  | 3.3032 
    plan syn cyc  | -3       | +3 
    diff cycles   |  7.7746  |  0.3032 
    Earth/diff    |  0.6141  | 10.8946 
    plan sid per  | 224.31   | 3979.25 day 
    Venus sidereal period is close to the true value but Jupiter is 
too short. The maths are correct but the premise of uniform circular 
motion is off. In the 2008-2012 span Jupiter happens to be near his 
perihelion and moves faster in his orbit. This faster motion carried 
thruout the orbit by the calculation makes for a quicker completion of 
the lap. 
    The lesson here is that we don't need hundreds or thousands of 
years of record as some authors assert. Observations for only a few 
decades or a century are enough. This span smooths out the 
irregularities of the planet motion, like the faster pace near 
Jupiter's perihelion, to yield surprisingly accurate sidereal period. 
This method was in history the one employed by babylonian astronomers, 
who compiled the first good tables of planet motions. 
General procedure 
    We obtain the location of the planet and its elongation from Sun 
as follows. First, for each planet, count the days from its previous 
apogee to the conjunction. This gives us the fraction of both the 
synodic period and synodic arc for that planet. 
    We then count the days from the previous vernal equinox to the 
conjunction and also from the previous apogee. We need both intervals. 
The latter places the Sun at the planet, he being in conjunction with 
the planet. This is the longitude of the planet from which it steps 
forward to the Jupiter-Venus conjunction.
    The former gives the longitude of the Sun at conjunction, by which 
we obtain the elongation of the planetary encounter in the sky. 
    We finally get the deviation of the planet from the epicycle 
center. This is generated by the planet's movement in the epicycle and 
may be east or west of the epicycle center. This is the 'wandering' 
portion of the otherwise smooth direct movement of the planet. 
    The longitude of the planet is the longitude of the Sun at the 
last apogee, plus the fractional synodic arc to conjunction, plus the 
deviation around the epicycle. 
    The elongation of the planets, both in conjunction, is their 
longitude minus that of the Sun. If all goes well we should find the 
planets have almost equal longitudes and are situated in the early 
evening sky. 
Sun's longitude 
    We find the ecliptic longitude of the Sun by counting forward from 
the previous vernal equinox, when the longitude was zero degrees. The 
prior equinox before 14 March 2012 is 2011 March 20. We can also bank 
off of the next equinox of 2012 March 20 by backing off the Sun's 
longitude. We do both to cross-check our work. 
    item          | prev V E   | next V E 
    Ven-Jup conj | 2012 Mar 14 | 2012 Mar 14 
    vern equinox | 2011 Mar 20 | 2012 Mar 20 
    elapsed days | +359.25 day  | -6 days 
    Sun's long   | +354.09 deg  | -5.91 deg 
    tossed laps  | +354.09 deg  | +354.09 deg 
Apogee longitude 
    When the planet is at the apogee, at the farthest highest point in 
the epicycle, it is also in conjunction with the Sun. The longitude of 
the apogee at this moment is the longitude of the Sun. 
    It is found by pacing the Sun from the vernal equinox to the 
apogee. We could have banked off of the 2012 vernal equinox and 
probably should do so as a double-check on our work. I did this, not 
shown here, and all looks copasetic. 
    item        | Venus        | Jupiter 
    apogee date | 2011 Aug  16 | 2011 Apr   6 
    prev ver eq | 2011 Mar  20 | 2011 Mar  20 
    raw elapsed |    0  +5  -4 |    0  +1 -14  
    elapsed per |  148.20 day  |  16.44 day 
    elapsed arc |  146.07 deg  |  16.20 deg 
    apogee long |  146.07 deg  |  16.20 deg 
    Do understand well that is is the movement of the epicycle center, 
NOT of the very planet, from its previous apogee to the vernal 
equinox. Because the vernal equinox is zero degrees, this movement 
puts the apogee at its longitude. 
Mean longitude 
    Mean longitude is the fraction of the synodic arc from the 
previous apogee plus the longitude of that apogee. We got the apogee 
longitude above because this is also the Sun's longitude on that date. 
    item        | Venus       | Jupiter 
    Ven-Jup con | 2012 Mar 14 | 2012 Mar 14 
    prev apogee | 2011 Aug 16 | 2011 Apr  6                          
    raw elapsed |   +1  -5 -2 |   +1  -1 +8 
    elapsed per | 211.05 day  | 342.81 day 
    synodic per | 583.9 day   | 398.9 day 
    symodic arc | 575.51 deg  | 393.17 deg 
    elapsed arc | 208.18 deg  | 367.45 
    apogee long | 146.07 deg  |  16.20 deg 
    mean long   | 354.25 deg  | 383.65 deg 
    tossed laps | 354.25 deg  |  23.65 deg 
    The apparent discrepancy between the longitude of Venus and 
Jupiter, hardly a conjunction situation, comes from the fact that both 
planets are off of their apogees, in an other pars of their epicycles. 
We must compute a correction based on the fractional revolution of the 
planet in the epicycle and this is the hard part of this exercise. You 
NEED trigonometry and a sci/tech calculette. 
The hard step 
    There is no real body at these mean longitudes, only the 
mathematical point at the center of the wpicylcles.
    The final step, which is the toughest, is to find the deviation of 
the planet, the angle from the epicycle center as seen from Earth. This 
involves some trigonometry to solve various triangles.
    At this point you BETTER make a sketch of the planets, which I 
have here for Venus. Due to ASCII limitations I leave out the actual 
circles of deferent and epicycle. 
                            +E                       +O 
    A is the apogee of the Venus epicycle. V is Venus somewhere more 
or less near her location on the epicycle. X is the center of the 
epicycle on the Venus deferent. E is Earth.
    We look from the north on to he planetary system. The epicycle is 
the circle around X, its center, passing thru A and V. The deferent is 
the circle around E passing thru X. 
    The angle from the vernal equinox O around Earth E to X is the 
mean longitude of Venus. Angle AXV is the synodic angle of Venus, the 
portion of the full 360 degree of the epicycle run by Venus since 
apogee. Angle XEV is the displacement of Venus throwing her off of the 
mean longitude location of line EX. 
    We got so far the mean longitudes of Jupiter and Venus and now 
must find the displacements from those positions.. 
Synodic angle 
    The angular movement of the planet around its epicycle is counted 
from the apogee and is in classical works called the anomaly. This 
angle is not an observable one, there being no such thing as an 
epicycle but is an intermediate quantity that goes into other calcs to 
yield the place and motion of the planet in the sky. 
    By convention, to keep the algebraic signa consistent, the synodic 
angle for an inferior planet is counted anticlockwise as a positive 
value and clockwise as a negative one for a superior planet. You can 
remember this by the direction the planet emerges from superior 
conjunction with the Sun. When Venus leaves apogee she heads into east 
elongation from the Sun, making her displacement a positive angle. 
Jupiter leaves his apogee into west, negative, elongation. 
    The angle is found by the portion of the synodic period elapsed 
since the previous apogee. This same portion was applied to the 
synodic arc to get the mean longitude since the apogee. 
    The synodic angle for Venus and Jupiter are 
    item          | Venus       | Jupiter 
    Ven-Jup conj  | 2012 Mar 14 | 2012 Mar 14 
    prev apogee   | 2011 Aug 16 | 2011 Apr  6 
    elapsed per   | 211.05 day  | 342.81 day 
    synodic per   | 583.9 day   | 398.9 day 
    synodic ratio |   0.3614    |   0.8594 
    synodic angle | 130.12 deg  | -309.38 deg 
Next step 
    Here's where your work can go horribly wrong without a diagram and 
clear understanding of how angles behave. The synodic angle AXV is the 
portion of 360 degrees traveled by the planets. 
    item         | Venus      | Jupiter     | in Venus diagram 
    deferent rad | 1.000      | 1.000       | line EX 
    epicycle rad | 0.723      | 0.192       | line XA 
    synodic ang  | 130.12 deg | -309.38 deg | angle AXV 
    interior ang |  49.88 deg |  129.38 deg | triangle EXV 
    planet dist  | 0.769      |  1.132      | line EV, Cosine Law 
    deviation    | 46.00 deg  |  16.33 deg  | angle XEV, Sine Law 
    I used the Law of Cosines and Law of Sines, wxplained below, 
to get the missing parts of triangle EXV, for Venus and a similar 
triangle for Jupiter. 
Planet longitude 
    We're almost done. The longitude of the planet is the mean 
longitude plus the deviation. Do watch for any rollover thru 360 
    item       | Venus      | Jupiter 
    mean long  | 354.25 deg |  23.65 deg 
    deviation  |  46.00 deg |  16.33 deg 
    true long  | 400.25 deg |  39.98 deg 
    tossed lap |  40.25 deg |  39.98 deg 
    sign       |  10 Taurus |   9 Taurus 
    The last line is merely the sign-angle notation, truncated to the 
degree, for the ecliptic longitude. The ecliptic is divided into 12 
signs of 30 degrees each. The 30-degree blocks of longitude are named 
after the 12 zodiacal signs. In astronomy and most astrology the 
degrees within a sign are numbered 0-29. Some astrologers use 1-30. 
    Where are the planets relative to the Sun and to the sky? This is 
shown by the elongation of the planets from the Sun, their longitude 
minus that of the Sun. Because we found the longitudes are the same, 
the planets being in mutual conjunction, we need only one calculation. 
        item       | planets        
        true long  |   40 deg, rounded 
        Sun's long |  355 deg, rounded 
        elongation | -315 deg 
        tossed lap |  +45 deg 
    The planets are about 45 degrees east of the Sun, placing them in 
the evening sky in twilight. They set in early night. Note that Venus 
is near her eastern greatest elongation, where she sat on March 27th.  
Sines and cosines
    I used the law of Sines and law of Cosines to get the deviation of 
the planet from the center of the epicycle. Because in astronomy these 
two trig tricks are so handy, yet so little known among home 
astronomers, I explain them here.
    First, I must remind that these are valid ONLY for plane, flat, 
triangles, NOT triangles on a sphere. You can not use these directly 
on the sky, for example. There are spherical versions of the laws 
which I sip here.
    The Law od Sines is the simpler of the two. In the triangle here 
the capital letters are the angles and small letters are the sides. 
Each side is opposite an angle of the same letter. The triangle is NOT 
a right triangle but we'll see what happens when it is. 
                 \                       / 
                   \                  / 
                     b             a 
                       \        /   
                         \   / 

        sin(A) / a = sin(B) / b = sin(C) / c
    This is easy! Yet if we know all three sides we can get all three 
angles. You do need that sci/tech calculette to get the angle itself 
that has the function value you found. 
    The Law of Cosines is a bit more fiddly and demands close 
attention to the singa of each item.
    c ^ 2 = a ^ 2 + b ^ 2 - (2 * a * b * cos(C))
A common mistake is to miss taking the square root of c^2 after doing 
the rest of the maths. Your result is all wrong, specially if side c 
goes into a downstream calculation. 
    For both formulae do not ever assume the signum of an item! Write 
it in and apply it in the maths. 
    What's with the right triangle? Angle C is 90 degrees. Cosine of 
90 degrees is zero. The whole term with cos(C) zeros out. We got 
    c ^ 2 = a ^ 2 + b ^ 2
which is our good friend the Pythagoras Law! 
    For the Law of Sines with anfgle C as the right angle, we note 
that c is the hyptonuse and sine(90) = 1. Then 
    sin(A) / a = sin(B) / b = 1 / c
For each term of this equation 
    sin(A) = a / c 
    sin(B) = b / c 
which are the very definition of the sine function in a right 
    On 14 March 2012 both planets are really in longitude 40 degrees, 
10 Taurus by modern computation methods. We ignored latitude effects. 
making the two planets essentially coincident on the ecliptic. They 
are really about 3 degree north of Jupiter. 
    The procedure here is definitely a first-order approximation and 
is intended to illustrate the way planet positions are calculated in 
the geocentric model of the solar system.