8.1 The Separation and PositionAngle Functions
Separation is useful for testing the degree of conjunctions, eclipses, and transits, but it has other applications as well.
Computing the apparant angular separation between objects.
You can use the function to find out the degree of an eclipse. If the separation of the Moon and the Sun is less than approximately one degree, then there is a partial solar eclipse visible from some point on the surface of the Earth.
Similarly, you can test for transits of Venus across the solar disk.
The planets Mars and Jupiter are 34 degrees apart in the sky on the given date.
In[4]:=Separation[Mars, Jupiter, {1993,11,17,3,20,0}]
Out[4]=
On this date, the Moon and the Sun are just 0.55 degrees apart relative to the center of the Earth. A partial solar eclipse, therefore, occurs on the given date.
In[5]:=Separation[Moon, Sun, {1994,11,4}]
Out[5]=
Venus and the Sun are separated by just 0.33 degrees; hence, Venus is nearly passing across the solar disk, which is 0.5 degrees wide.
In[6]:=Separation[Venus, Sun, {1882,12,7}]
Out[6]=
This shows that the MarsSunEarth angle is about 162 degrees.
In[7]:=Separation[Mars, Earth, {1993,11,17,3,20,0}, ViewPoint > Sun]
Out[7]=
Venus and the Sun are separated by only 0.17 degrees during the transit of 1769.
In[8]:=Separation[Venus, Sun, {1769,6,4,9,0,0}]
Out[8]=
In 1769 Captain Cook sailed to the Pacific to witness a transit of Venus across the solar disk. Transit of Venus happens only four times every 243 years. The 1769 transit occurred at about 09:00 on June 4 local time in Melbourne, but would have been about midday on June 3 in Tahiti (Captain Cook's location). You can compute the degree, or closeness, of the event using Separation.
In addition, the duration of the transit can be determined using the expressions EclipseBegin[Sun, Venus, Earth, {1769,1,1}] and EclipseEnd[Sun, Venus, Earth, {1769,1,1}].
The PositionAngle Function
A related function is PositionAngle. Separation gives the apparent angular separation of any two objects, but it does not give information about the orientation. Astronomers use a quantity called the position angle to represent the orientation of two objects. The position angle of one object relative to a second is the angle between the first object and the north celestial pole as measured relative to the second object.
Computing the apparent angle between three objects.
The constellation of Gemini consists of the two bright stars Castor and Pollux. These stars are separated by about 4.5 degrees, and the position angle from Castor out to Pollux can be computed using PositionAngle.
This shows that Pollux is rotated about 148 degrees around Castor. Remember that the usual definition of position angle is such that 0 degrees is in the direction of the north celestial pole, and counterclockwise is positive.
In[9]:=PositionAngle[Castor, Pollux]
Out[9]=
There are other uses of PositionAngle. Consider, for example, the following graphic, which uses the Horizon > True option to make the horizon line horizontal.
Castor and Pollux are setting into the northwest horizon. It appears that Pollux is almost directly above Castor.
In[10]:=RadialStarChart[Gemini, {1993,11,17,7,20,0}, Horizon > True, Ecliptic > False, StarLabels > True, Epilog > StarNames[Gemini]];
The zenithCastorPollux angle is about 354 degrees. That is, 354 degrees is needed to rotate counterclockwise from the zenith about Castor to Pollux. This is equivalent to 6 degrees clockwise.
In[11]:=PositionAngle[Zenith, Castor, Pollux, {1993,11,17,7,20,0}]
Out[11]=
The Elongation Function
Another related function is Elongation, which is useful for indirectly determining the rising and setting times of a planet relative to local sunrise and sunset.
Computing the angle along the ecliptic of an object to the Sun.
If the elongation of an object is sufficiently positive, the object is visible chiefly in the evening sky after dusk. If, however, the elongation is sufficiently negative, the object is mainly visible in the morning sky before dawn.
This shows that Mars is only 11 degrees east of the Sun on the given date. This is fairly close to the Sun, so Mars is hard to spot on this date.
In[12]:=Elongation[Mars, {1993,11,17,3,20,0}]
Out[12]=
By definition the elongation of the Sun is zero.
In[13]:=Elongation[Sun, {1993,11,17,3,20,0}]
Out[13]=
The Moon has a positive elongation on this date; hence, it appears mostly in the evening sky after dusk.
In[14]:=Elongation[Moon, {1993,11,17,3,20,0}]
Out[14]=
You can easily convert the elongation angle into a time by remembering that the Earth rotates 360 degrees in 24 hours, and that therefore each 15 degrees of positive elongation corresponds to 1 hour of time after sunset. Thus, an elongation of 45 degrees corresponds to 3 hours before sunrise.
Note that the definition of a new or full moon relates to the elongation of the Moon. The time returned by NewMoon is such that the ecliptic longitude of the Sun and the Moon are the same at that instant. That is, by definition, the value of Elongation[Moon, NewMoon[]] is zero.
Similarly, the time returned by FullMoon is such that the ecliptic longitude of the Sun and the Moon differs by 180 degrees. Therefore, by definition, the value of Elongation[Moon, FullMoon[]] is 180 degrees.
