7/29/2019 Plnt pos

1/15

Here it is explained how you can calculate the approximate position of theSunor aplanetas

seen from anotherplanet.Further downthe page it is explained how you can calculate the

approximate position of theMoonas seen from theEarth.

1. Positions of theSunand thePlanets

If you want to calculate the position of a celestial body in the sky or as seen from theSun

(heliocentric), then you need information about that body andperhapsabout theEarth. Which

information you need depends on what exactly you want to calculate and how accurate the

results should be. This page explains how you can calculate the position in the sky as seen from aparticular place onEarthor from theSunfor a celestial body that orbits theSunin a circle or

ellipse. For this, you need the following information:

the length a of thesemimajor axisof the orbit, which indicates the size of the orbit. ashould be measured inAstronomicalUnits (AU). Sometimes theperiheliondistance q is

given instead ofa. Then you can calculate a as follows:

(Eq. 1) a = q/(1 e)

theeccentricitye of the orbit, which indicates the shape of the orbit. the argument [omega] of the perihelion, which indicates how far the perihelion lies

beyond theascending nodeof the orbit. Warning: The symbol is also sometimes used

for the sum of "our" and . The symbol is also sometimes used for that sum, and that

symbol resembles . We use = + for that sum.

theeclipticlongitude [large omega] of the ascendingnodeof the orbit. theinclinationi of the orbit, which indicates how far the orbit is tilted compared to the

ecliptic.

themean anomalyM0 at a specific date d0, which indicates the position of theplanetin itsorbit at that date. Sometimes L0 = M0 + + is given instead ofM0, and sometimes a

date dperihelion is given instead when the object was in its perihelion (where M is equal to0). Here are the transformation formulas:

(Eq. 2) M0 = L0

(Eq. 3) M0 = n (d0 dperihelion)

In addition, you need the following information:

the obliquity [epsilon] of the ecliptic yourgeographicallongitude l (measured westward from Greenwich) andlatitude [phi]

(measured northward from theequator)

yourtime zonetz, measured as thenumberofhoursthat you have to add to your localclocktimet to get Universal Time (UTC)

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/reken/hemelpositie.html#maanhttp://aa.quae.nl/en/reken/hemelpositie.html#maanhttp://aa.quae.nl/en/reken/hemelpositie.html#maanhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Moonhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Moonhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Moonhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/en/reken/hemelpositie.htmlhttp://aa.quae.nl/en/reken/hemelpositie.htmlhttp://aa.quae.nl/en/reken/hemelpositie.htmlhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=astronomicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=astronomicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=astronomicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=astronomicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/reken/hemelpositie.htmlhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Moonhttp://aa.quae.nl/en/reken/hemelpositie.html#maanhttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sun

7/29/2019 Plnt pos

2/15

All in all, you need (besides the desired date and time) 16 separatenumbers: 6 to describe the

orbit and position of theplanetin space, 6 to describe the orbit and position of the Earth in space,

and 4 that describe the orientation of the Earth and your position on Earth. If you only want tocalculate the heliocentric position, then you need fewer numbers.

Theorbital elementsof theplanets(at noon UTC on 1 January 2000, at Julian Day 2451545) arelisted in the following table (with a measured in AU, and i, , , and M0 indegreesrelative to theequinoxof that date).

Table 1:Planets: Orbital Elementsa e i M0

Mercury 0.38710 0.20563 7.005 29.125 48.331 174.795

Venus 0.72333 0.00677 3.395 54.884 76.680 50.416

Earth 1.00000 0.01671 0.000 288.064 174.873 357.529

Mars 1.52368 0.09340 1.850 286.502 49.558 19.373

Jupiter 5.20260 0.04849 1.303 273.867 100.464 20.020Saturn 9.55491 0.05551 2.489 339.391 113.666 317.021

Uranus 19.21845 0.04630 0.773 98.999 74.006 141.050

Neptune30.11039 0.00899 1.770 276.340 131.784 256.225

Pluto 39.543 0.2490 17.140 113.768 110.307 14.882

For theSun,a is equal to 0, so also xSun = ySun = zSun = 0 so you can skip steps 1 through 4

below.

Theinclinationof the orbit of theEarthrelative to theequinoxof the date is zero by definition,

so that orbit has nonodesand the value of theeclipticlongitudeof theascending nodeisimmaterial. The value of that I provide for theEarthin the table is the average of the valuesthat you find a fewyearsbefore and after 1 January 2000, when the inclination (compared to the

equinoxof J2000.0) is different from zero.

If you multiply the inclination iby 1, add 180degreesto , and suitably adjust the argument of theperihelion, then you still have the same orbit (but with the roles of ascending and

descending nodesexchanged). For the otherplanetsthan the Earth, the inclination is very

different from zero, so you can always choose it to be positive (by applying the factor of 1

when needed). Only for the orbit of the Earth can be inclination be positive or negative, and onesource can choose to take the future inclination to be positive while another source can take the

past inclination to be positive.

When the inclination is very small, then the length of the ascendingnodeis not determined very

accurately, so the values that different sources give for the length of the ascending node of the

Earth can differ by many degrees (even if you take a possible sign difference for the inclinationinto account) but this does not make much difference to the position of the Earth that you

calculate using that value, because theeclipticlatitudecannot be greater (in absolute sense) than

the inclination.

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbershttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/en/boom/planeet.html#planetshttp://aa.quae.nl/en/boom/planeet.html#planetshttp://aa.quae.nl/en/boom/planeet.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/tijd.html#yearshttp://aa.quae.nl/en/antwoorden/tijd.html#yearshttp://aa.quae.nl/en/antwoorden/tijd.html#yearshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=descending%20nodeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=descending%20nodeshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodehttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=descending%20nodeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equinoxhttp://aa.quae.nl/en/antwoorden/tijd.html#yearshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ascending%20nodehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=nodeshttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=inclinationhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/en/boom/planeet.html#planetshttp://aa.quae.nl/en/antwoorden/zonpositie.html#equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbers

7/29/2019 Plnt pos

3/15

Here is a table with some more handynumbersfor eachplanetthat depend only on theorbital

elements.n is measured in degrees per day, a(1e2) inAU, and in degrees. These numbers areexplained later.

Table 2:Planets: Other Orbital Numbersn

a(1

e2

)

Mercury4.092317 0.37073 77.456

Venus 1.602136 0.72330 131.564

Earth 0.985608 0.99972 102.937

Mars 0.524039 1.51039 336.060

Jupiter 0.083056 5.19037 14.331

Saturn 0.033371 9.52547 93.057

Uranus 0.011698 19.17725 173.005

Neptune0.005965 30.10796 48.124

Pluto 0.003964 37.09129 224.075

The steps you must follow to calculate the position of aplanetas seen fromEarthat date darethen:

1. Calculate themean anomalyMfrom d, M0, d0 and a.2. Calculate thetrue anomaly [nu] from Mand e.3. Calculate the distance r to theSun, from a, e and .4. Calculate the rectangular (= cartesian)heliocentriceclipticcoordinates(xplanet, yplanet,

zplanet) from , r, , and i.

5. Calculate the rectangulargeocentriceclipticcoordinates (x, y, z) from (xplanet,yplanet, zplanet) and (xearth, yearth, zearth).

6. Calculate the geocentric eclipticlongitude [lambda] andlatitude [beta] from (x, y,z).

7. Calculate theright ascension [alpha] anddeclination [delta] from , and theobliquity [epsilon] of the ecliptic.

8. Calculate thesidereal time [theta] from t, yourgeographicallongitude l and yourtimezonetz.

9. Calculate thehour angleHfrom and .10.Calculate the height h and theazimuthAfrom H, and your geographical latitude [phi].11.You can calculate theelongation [psi] of the planet from the geocentric ecliptic

coordinates (cartesian or polar) of theSunand the planet.

If you want to calculate the heliocentric position of a planet, then you should take (xplanet,

yplanet, zplanet) for(x, y, z) in step 5.

These steps are explained in more detail below. As an example, we'll calculate the position of

Jupiterat 01:00hoursCentral European StandardTime(0:00 hoursUTC) on 1 January 2004from 52 north latitude and 5 east longitude (which is near Utrecht in the Netherlands).

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbershttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/en/antwoorden/tijd.html#time%20zonehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geographicalhttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=orbital%20elementshttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numbers

7/29/2019 Plnt pos

4/15

The section aboutapproximationsdetails approximations that provide less accurate results with

less calculations.

WARNING: These calculations are based on the assumption that theplanetsorbit around the

Sunin elliptical orbits that never change. In reality these orbits do change, but slowly and not by

much. These formulas are very suitable to determine whether a particular planet will be visible ata certain time, or whatstarit will be close to in the sky, but not (for example) to calculate

whether a particular planet willeclipsea particularstar. The accuracy of the results of these

formulas is discussedlater.

WARNING 2: Make sure that you use the right units for the trigonometric functions such assin

andarctan. In computer programs those functions commonly work with radians. Many

calculators can be configured to work withdegreesor radians. Calculate sin(1) with yourcalculator or computer program. If the result is approximately 0.8415, then the unit is the radian.

If the result is approximately 0.01745, then the unit is thedegree. To transform from degrees to

radians, multiply thenumberof degrees by /180 0.01745329.

1.1. The MeanAnomaly

First, calculate whichanglen theplanettraverses on average per day, as seen from theSun:

(Eq. 4) n = 0,9856076686/(aa)

and then themean anomalyM for date d (measured in days) from the start date d0, the mean

anomaly M0 on the start date, and n:

(Eq. 5) M = M0 + n (d d0)

Fordand d0 you could very well useJulian Day Numbers. The values ofn for theplanetsareshown in thesecondtable at the top of this page.

0:00UTCat 1 January 2004 is 1460.6 days after thetimefor which M0 from the table is valid, so

d d0 is equal to 1460.5. The mean anomaly ofJupiterat the desired time is equal to MJupiter=

20.020 + 0.083056 * 1460.5 = 141.324and for theEarthit is MEarth = 357.529 +

0.985608 * 1460.5 = 357.009 (after subtracting multiples of 360).

1.2. The TrueAnomaly

Thetrue anomaly [nu] is theanglebetween the line from the focus of the orbit (theSun) to theperihelionof the orbit and the line from the focus to theplanet. To calculate the true anomaly,

you need to solve the Equation ofKepler. See thePage about the Equation of Keplerabout this.

ForJupiter, the true anomaly that goes with MJupiter = 141.324 and eJupiter = 0.04849 is

equal to Jupiter = 144.637. For theEarth, the true anomaly that goes with MEarth = 357.009

and eEarth = 0.01671 is equal to earth = 356.907.

http://aa.quae.nl/en/reken/benaderingenhttp://aa.quae.nl/en/reken/benaderingenhttp://aa.quae.nl/en/reken/benaderingenhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/antwoorden/sterren.html#starhttp://aa.quae.nl/en/antwoorden/sterren.html#starhttp://aa.quae.nl/en/antwoorden/sterren.html#starhttp://aa.quae.nl/en/antwoorden/verduisteringen.html#eclipsehttp://aa.quae.nl/en/antwoorden/verduisteringen.html#eclipsehttp://aa.quae.nl/en/antwoorden/verduisteringen.html#eclipsehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starhttp://aa.quae.nl/en/reken/hemelpositie.html#nauwkeurigheidhttp://aa.quae.nl/en/reken/hemelpositie.html#nauwkeurigheidhttp://aa.quae.nl/en/reken/hemelpositie.html#nauwkeurigheidhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Julian%20Day%20Numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Julian%20Day%20Numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Julian%20Day%20Numbershttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Keplerhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Keplerhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Keplerhttp://aa.quae.nl/en/reken/kepler.htmlhttp://aa.quae.nl/en/reken/kepler.htmlhttp://aa.quae.nl/en/reken/kepler.htmlhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/reken/kepler.htmlhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Keplerhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=perihelionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=true%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Julian%20Day%20Numbershttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=mean%20anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anomalyhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=numberhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/hemelpositie.html#nauwkeurigheidhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starhttp://aa.quae.nl/en/antwoorden/verduisteringen.html#eclipsehttp://aa.quae.nl/en/antwoorden/sterren.html#starhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/reken/benaderingen

7/29/2019 Plnt pos

5/15

1.3. The Distance to theSun

Theeccentricitye indicates the shape of the orbit and cannot be negative. Ife is equal to 0, then

the orbit is a circle. Ife is greater than 0 but less than 1, then the orbit is an ellipse. Ife is equal

to 1, then the orbit is a parabola or a straight line. Ife is greater than 1, then the orbit is a

hyperbola.

The size a of thesemimajor axisof the orbit indicates the size of the orbit. For a circular orbit, a

is equal to the distance between theplanetand theSun. For an elliptic orbit, a is equal to half of

the length of the longest line segment that fits within the ellipse.

The distance rof theplanetfrom theSunis:

(Eq. 6) r = a(1 e2)/(1 + ecos)

The values ofa(1e2) for theplanetsare shown in thesecondtable at the top of this page.

The distance betweenJupiterand theSun, measured inAU, is then rJupiter= 5.19037 / (1 +

0.04849 * cos(144.637)) = 5.40406and for theEarthrEarth = 0.99972 / (1 + 0.01671

* cos(356.907)) = 0.98331 .

1.4. The RectangularHeliocentricEclipticalCoordinates

We can now calculate the rectangular heliocentriceclipticcoordinates, all measured inAU. Of

those, xplanetis the distance from theSunas measured along the line from theSunto thevernal

equinox,yplanetthe distance to the line from theSunto thevernal equinox, measured in the plane

of theecliptic, and zplanetthe distance above the plane of the ecliptic.

(Eq. 7) xplanet = r (cos cos( + ) sin cos i sin( + ))

(Eq. 8) yplanet = r (sin cos( + ) + cos cos i sin( + ))

(Eq. 9) zplanet = r sin i sin( + )

ForJupiterwe find xJupiter= 5.40406 * (cos(100.464) * cos(273.867 + 144.637)

sin(100.464) * cos(1.303) * sin(273.867 + 144.637)) = 5.04289, yJupiter =5.40406 * (sin(100.464) * cos(273.867 + 144.637) + cos(100.464) *

cos(1.303) * sin(273.867 + 144.637)) = 1.93965, and zJupiter = 5.40406 *

sin(1.303) * sin(273.867 + 144.637) = 0.10478and for theEarthxEarth = 0.98331

* (cos(174.873) * cos(288.064 + 356.907) sin(174.873) * cos(0) *sin(288.064 + 356.907)) = 0.16811 , yEarth = 0.98331 * (sin(174.873) *cos(288.064 + 356.907) + cos(174.873) * cos(0) * sin(288.064 +

356.907)) = 0.96884, and zEarth = 0.98331 * sin(0) * sin(288.064 + 356.907) =0.00000

1.5. The RectangularGeocentricEclipticalCoordinates

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=vernal%20equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=vernal%20equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=vernal%20equinoxhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=semimajor%20axishttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eccentricityhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sun

7/29/2019 Plnt pos

6/15

If you have done steps 1 through 4 for theplanetand also for theEarth, then you have calculated

xplanet, yplanet,zplanet, xEarth, yEarth, zEarth. Then we can calculate the coordinates of theplanet relative to the Earth, measured inAU:

(Eq. 10) x = xplanet xEarth

(Eq. 11) y = yplanet yEarth

(Eq. 12) z = zplanet zEarth

For the rectangular geocentric coordinates ofJupiter, measured in AU, we find x= 5.04289

(0.16811) = 4.87477 , y= 1.93965 0.96884 = 0.97081 , and z= 0.10478 0.00000 = 0.10478

1.6. TheGeocentricEclipticalLongitudeandLatitude

Theeclipticlatitude [beta] indicates how far the celestial body is from theecliptic. The ecliptic

longitude

[lambda] shows how far the celestial body is from thevernal equinox, measuredalong the ecliptic. For the distance [large delta] of theplanetfrom theEarthwe find

(Eq. 13) = (x + y + z)

and then

(Eq. 14) x = cos cos

(Eq. 15) y = sin cos

(Eq. 16) z = sin

It follows that

(Eq. 17) = arctan(y, x)

(Eq. 18) = arcsin(z/)

Here, arctan(y, x) is the special arc tangent function with two arguments that calculates the

anglebetween the x axis and the line that goes from the point withcoordinates(0,0) to the point

with coordinates (x,y). This special arc tangent function is usually not available under that

name on electronic calculators, but is often available in programming languages (with a name

like "atan2"). Some calculators have a function that transforms rectangular coordinates to polar

ones (a distance and an angle), and you can use that to calculate the angle that goes with xand y,and that angle is . That function often has a name like "RP" or "Pol". To check: if you enterx

= 1 and y = 0, then it should yield 0 for the angle, if you enterx = 0 and y = 1 then the result

should be 90, and if you enterx= 1 and y= 1then the result should be 135 or 225(those differ by 360 so they indicate the same angle).

If that function is not on your calculator or in your programming language, then you can use the

ordinary arc tangent function, but then you have to do some extra work. In that case, calculate

http://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#arcsinhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/en/reken/notatie.html#arcsinhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=eclipticalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#planet

7/29/2019 Plnt pos

7/15

(Eq. 19) 0 = arctan(y/x)

Ifsin 0 has the same sign (plus or minus) as yand cos the same sign as x, then = 0.

Otherwise, = 0 + 180.

We find = ((4.87477)2

+ 0.970812

+ 0.104782

) = 4.97161, = arctan(0.97081,4.87477) = 168.737 , and = arcsin(0.104784.97161) = 1.208.

1.7. The Right Ascension andDeclination

The axis of rotation of theEarthis not at rightanglesto the plane of the orbit of theEarth, so the

equatorof the Earth makes ananglewith the plane of the orbit of the Earth. This angle [epsilon] is called the obliquity of theeclipticand its value at the beginning of 2000 was

(Eq. 20) = 23.4397

Theequatorialcoordinate systemin the sky is tied to the rotation axis of the Earth. Theequatorialcoordinatesare theright ascension [alpha] and the declination [delta]. Thedeclination shows how far the body is from thecelestial equatorand determines from which

parts of the Earth the object can be visible. The right ascension shows how far the body is from

thevernal equinox, as measured along the celestial equator, and determines (together with otherthings) when the object is visible. We have:

(Eq. 21)sincos = sin cos cos sin sin

(Eq. 22) cos cos = cos cos

(Eq. 23) = arcsin(sin cos + cos sin sin )

You can combine formulas21and22to:

(Eq. 24) = arctan(sin cos tan sin , cos )

where the special arc tangent function was used again that was described at step 6. The right

ascension is usually written as atime, withhours,minutes, andseconds. To transform it from a

time measured in hours to an angle measured indegrees(so you can use it in formulas withthings such as sine functions), you should multiply it by 15.

ForJupiterwe find = arctan(sin(168.737) * cos(23.4397) tan(1.208) *

sin(23.4397), cos(168.737)) = 170.120 = 11h20m29sand =arcsin(sin(1.208) * cos(23.4397) + cos(1.208) * sin(23.4397) *

sin(168.737)) = 5.567.

If you calculate the right ascension, declination, and distance for theSunand allplanetsfrom the

table, then you find

Table 3:PlanetPositions on 1-1-2004

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=angleshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=angleshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=angleshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorialhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorialhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinate%20systemhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinate%20systemhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinate%20systemhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/en/antwoorden/hemel.html#celestial%20equatorhttp://aa.quae.nl/en/antwoorden/hemel.html#celestial%20equatorhttp://aa.quae.nl/en/antwoorden/hemel.html#celestial%20equatorhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#arcsinhttp://aa.quae.nl/en/reken/hemelpositie.html#eq_21http://aa.quae.nl/en/reken/hemelpositie.html#eq_21http://aa.quae.nl/en/reken/hemelpositie.html#eq_21http://aa.quae.nl/en/reken/hemelpositie.html#eq_22http://aa.quae.nl/en/reken/hemelpositie.html#eq_22http://aa.quae.nl/en/reken/hemelpositie.html#eq_22http://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#tanhttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=minuteshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=minuteshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=minuteshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=minuteshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/reken/notatie.html#tanhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/hemelpositie.html#eq_22http://aa.quae.nl/en/reken/hemelpositie.html#eq_21http://aa.quae.nl/en/reken/notatie.html#arcsinhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/antwoorden/seizoenen.html#vernal%20equinoxhttp://aa.quae.nl/en/antwoorden/hemel.html#celestial%20equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinate%20systemhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorialhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=equatorhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=angleshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declination

7/29/2019 Plnt pos

8/15

Name right ascension declination distance

degrees time degrees AU

Sun 280.710 18h42m50s -23.074 0.98331

Mercury268.693 17h54m46s -20.296 0.70403

Venus 316.189 21h04m45s -18.614 1.3061Mars 8.335 0h33m20s +3.660 1.1115

Jupiter 170.120 11h20m29s +5.567 4.9716

Saturn 100.256 6h41m01s +22.420 8.0443

Uranus 333.148 22h12m36s -11.868 20.654

Neptune313.525 20h54m06s -17.459 30.973

Pluto 260.277 17h21m07s -14.497 31.700

1.8. TheSiderealTime

Where a celestial body appears to be in the sky for you depends on the orientation of theEarthatyour location compared to thestars. Thisangleis captured in thesidereal time (theta). Thesidereal timeat a certain moment is equal to theright ascensionthattransits(culminates, passes

through thecelestial meridian) at that moment. Ift is the local clock time and tz is how much

you have to add to the local time to get Universal Time (UTC), both measured inhours, then the

sidereal timemeasured indegreesis equal to

(Eq. 25) = MEarth+ Earth + 15 (t + tz) l

where

(Eq. 26) Earth = Earth + Earth

and Earth is shown in thesecondtable. The tzis equal to 1hourfor Dutch and Belgian

standard time(CET, Central European Time), and to 2 hours for Dutch and Belgiandaylight

savings time(CEDT, Central European Daylight Savings Time). It is usually close to l/15hours (reduced by one extra hour for daylight savings time).

The sidereal time is often written as a time rather than an angle, just like for the right ascension.

With the value ofEarth from the second table, formula25becomes = MEarth + 102.937 +

15 (t + tz) l. For01:00 hours Central European Standard Time at 5 eastlongitudewe

have t = 1, tz= 1 and l= 5. In step 1 we found that MEarth = 357.009. All in all, we

find = 357.009 + 102.937 + 15 (1 + (1)) (5) = 104.946 = 6 h59m47s

(after subtracting multiples of 360 or 24h).

1.9. TheHourAngle

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=siderealhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=siderealhttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#sidereal%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/en/antwoorden/tijd.html#standard%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#standard%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#daylight%20savings%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#daylight%20savings%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#daylight%20savings%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#daylight%20savings%20timehttp://aa.quae.nl/en/reken/hemelpositie.html#eq_25http://aa.quae.nl/en/reken/hemelpositie.html#eq_25http://aa.quae.nl/en/reken/hemelpositie.html#eq_25http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Forhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Forhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Forhttp://aa.quae.nl/en/reken/hemelpositie.html#eq_25http://aa.quae.nl/en/antwoorden/tijd.html#daylight%20savings%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#daylight%20savings%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#standard%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=secondhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=UTChttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/en/antwoorden/tijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=starshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=siderealhttp://aa.quae.nl/en/antwoorden/planeten.html#Plutohttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Neptunehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Uranushttp://aa.quae.nl/en/antwoorden/planeten.html#Saturnhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Marshttp://aa.quae.nl/en/antwoorden/planeten.html#Venushttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Mercuryhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=AUhttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=declinationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascension

7/29/2019 Plnt pos

9/15

Thehour angleH indicates how far the celestial body has passed beyond thecelestial meridian. Ifthe hour angle is zero, then the bodytransitsand is highest in the sky, on the celestialmeridian.The hour angle is often written as atimerather than an angle, just like for thesidereal timeand

theright ascension. That time is approximately equal to how long ago the body was highest in

the sky.

(Eq. 27) H =

ForJupiter, we find H= 104.946 170.120 = 65.174 = 6h59m47s 11h20m29s =

4h20m42ssoJupiterwill be highest in the sky about 4hourslater.

1.10. The Height andAzimuth

The position of a celestial body in the sky is specified by itsaltitudeh above thehorizon, and its

azimuth A. The altitude is 0 at thehorizon, +90 in thezenith(straight over your head), and90 in thenadir(straight down). The azimuth is the direction along the horizon, which we

measure from south to west. South has azimuth 0, west +90, north +180, and east +270 (or90, that's the same thing). The altitude and azimuth are thehorizontalcoordinates. If theEarthhad noatmosphere, then we'd have, for the horizontal coordinates:

(Eq. 28)sinAcosh = sin H cos

(Eq. 29) cos A cos h = cos H cos sin sin cos

(Eq. 30) sin h = sin sin + cos cos cos H

It follows that

(Eq. 31) A = arctan(sin H, cos H sin tan cos )(Eq. 32) h = arcsin(sin sin + cos cos cos H)

where the special arc tangent function is used which was described earlier.

The atmosphere of theEarthbendslightupward a bit so that it seems as if celestial bodies are a

bit higher in the sky than they would be without an atmosphere. This effect is on average about

0.57 at the horizon and decreases rapidly with height. Moreover, it varies significantly close tothe horizon, depending on the circumstances in the atmosphere. If you want to apply a correction

to h for this refraction, then you can use the following formula (if you measure h indegrees):

(Eq. 33) happarent = h + 0.017/tan(h + 10.26/(h + 5.10))

ForJupiterat 01:00hoursCentral European StandardTimeon 1 January 2004 as seen from 52

northlatitudeand 5 eastlongitudewe find h = arcsin(sin(52) sin(5.567) + cos(52)

cos(5.567) cos(65.174)) = 19.495 and A= arctan(sin(65.174),

cos(65.174) sin(52) tan(5.567) cos(52)) = 73.383.Jupiteris then about19.5 degrees above the horizon, about 16 degrees south of east. The correction for refraction is

about 0.05.

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20anglehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=meridianhttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=altitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=altitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=altitudehttp://aa.quae.nl/en/antwoorden/hemel.html#horizonhttp://aa.quae.nl/en/antwoorden/hemel.html#horizonhttp://aa.quae.nl/en/antwoorden/hemel.html#horizonhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizonhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizonhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizonhttp://aa.quae.nl/en/antwoorden/hemel.html#zenithhttp://aa.quae.nl/en/antwoorden/hemel.html#zenithhttp://aa.quae.nl/en/antwoorden/hemel.html#zenithhttp://aa.quae.nl/en/antwoorden/hemel.html#nadirhttp://aa.quae.nl/en/antwoorden/hemel.html#nadirhttp://aa.quae.nl/en/antwoorden/hemel.html#nadirhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizontalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizontalhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/atmosferen.html#atmospherehttp://aa.quae.nl/en/antwoorden/atmosferen.html#atmospherehttp://aa.quae.nl/en/antwoorden/atmosferen.html#atmospherehttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#tanhttp://aa.quae.nl/en/reken/notatie.html#arcsinhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/antwoorden/licht.html#lighthttp://aa.quae.nl/en/antwoorden/licht.html#lighthttp://aa.quae.nl/en/antwoorden/licht.html#lighthttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=degreeshttp://aa.quae.nl/en/antwoorden/licht.html#lighthttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/en/reken/notatie.html#arcsinhttp://aa.quae.nl/en/reken/notatie.html#tanhttp://aa.quae.nl/en/reken/notatie.html#arctanhttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#sinhttp://aa.quae.nl/en/antwoorden/atmosferen.html#atmospherehttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizontalhttp://aa.quae.nl/en/antwoorden/hemel.html#nadirhttp://aa.quae.nl/en/antwoorden/hemel.html#zenithhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=horizonhttp://aa.quae.nl/en/antwoorden/hemel.html#horizonhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=altitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=azimuthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hourshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Jupiterhttp://aa.quae.nl/en/antwoorden/planeten.html#Jupiterhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=right%20ascensionhttp://aa.quae.nl/en/reken/sterrentijd.html#sidereal%20timehttp://aa.quae.nl/en/antwoorden/tijd.html#timehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=transitshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=celestial%20meridianhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=hour%20angle

7/29/2019 Plnt pos

10/15

1.11. TheElongation

Theelongationof aplanetis a measure for the distance of thatplanetfrom theSunalong thesky. You can measure that distance (1) directly between theplanetand theSun, or (2) only along

theecliptic. These two methods in general give slightly different values, except forplanetsthat

are exactly on theecliptic.

If you know the eclipticlongitudeSunof theSunand also the ecliptic longitude andlatitudeof the planet, then you can calculate elongation (1) with

(Eq. 34) 1 =arccos(cos cos( Sun))

Elongation (2) is then equal to

(Eq. 35) 2 = Sun

It is customary to add or subtract multiples of 360 to or from 2 until the value lies between180 and 180. Elongation (1) can only be positive or zero, but elongation (2) can be negative,too. Positive means "east of the Sun" and negative "west of the Sun".

If you know the cartesianheliocentriceclipticcoordinates(xEarth, yEarth, zEarth) of Earth and

the cartesiangeocentricecliptic coordinates (x, y, z) of the planet, then you can calculate

elongation (1) also like this:

(Eq. 36) = (x2 + y2 + z2)

(Eq. 37) rEarth= (xEarth2 + yEarth

2 + zEarth2)

(Eq. 38) 1= arccos((xEarthx + yEarthy + zEarthz)/(rEarth))

We found earlier forJupiterthat = 168.737 and = 1.208. Similarly, we find for the Sun

that Sun = 279.844. It follows that 1= arccos(cos(1.208) * cos(168.737

279.844)) = 111.102and 2= 168.737 279.844 = 111.063. Earlier, we found

(xEarth, yEarth, zEarth) = (0.16811, 0.96884, 0.00000) and (x, y, z) = (4.87477,

0.97081, 0.10478)and, from that, = 4.97161 and rEarth = 0.98331, and with these we

now find 1= arccos((0.16811 * 4.87477 + 0.96884 * 0.97081 + 0 * 0.10478) /

(4.97161 * 0.98331)) = 111.102, just like before.

1.12. Accuracy

How accurate are the results that these formulas yield? I compared those results for every dayfrom 1 January 1980 through 1 January 2020 with the results that you get if you do the

calculations using a truncated version of theVSOPtheory according to the book"Astronomical

Algorithms" by J. Meeus, who includes the effects of the mutual gravitational attraction of theplanets, of abberation, nutation, the traveltimeoflight, and the difference betweenUTCand

TDT. The greatest differences are shown in the next table.

http://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=elongationhttp://aa.quae.nl/en/antwoorden/hemel.html#elongationhttp://aa.quae.nl/en/antwoorden/hemel.html#elongationhttp://aa.quae.nl/en/antwoorden/hemel.html#elongationhttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/antwoorden/planeten.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/en/boom/planeet.html#planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=planethttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/en/antwoorden/zon.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=ecliptichttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/planeten.html#planetshttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/en/antwoorden/hemel.html#ecliptichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=longitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/en/antwoorden/zonpositie.html#sunhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=latitudehttp://aa.quae.nl/en/reken/notatie.html#arccoshttp://aa.quae.nl/en/reken/notatie.html#arccoshttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/en/reken/notatie.html#coshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=heliocentrichttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=coordinateshttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/en/antwoorden/planeten.html#Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Earthhttp://aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=geocentrichttp://aa.quae.nl/cgi-b