Extra-credit Opportunity: Calculate the true orbital periods and radius of the
planets orbits from their configurations.
Use the planetary configurations listed below to calculate the true orbital periods of the planets and their true
orbital radii assuming that the orbits are circular and that the planets move a constant speeds on their orbits.
(This assumption is not strictly true, but allows for a simple solution to the two problems that is reasonably
accurate.) 5 points will be awarded for each correct calculation of the orbital period or orbital radius of Jupiter,
Mars and Venus for a possible total of 30 points. Mercury (an inferior planet) and Saturn (a superior planet) are
used as example calculations below. Check your own calculations for Jupiter Mars and Venus with published
values in the appendices of the text or online sources.
Planetary Configurations
Planet
Saturn
Jupiter
Mars
Venus
Mercury
Configuration
Opposition
Quadrature
Opposition
Opposition
Quadrature
Opposition
Opposition
Quadrature
Opposition
Inferior Conjunction
Inferior Conjunction
Maximum Elongation
Inferior Conjunction
Inferior Conjunction
Maximum Elongation
Date
2010/03/21
2010/06/19
2011/04/03
2010/09/21
2010/12/16
2011/10/28
2010/01/29
2010/05/04
2012/03/03
2010/10/29
2012/06/05
Average over many cycles
2010/01/04
2010/04/28
Average over many cycles
Julian Date
2455277.50
2455367.55
2455655.48
2455460.98
2455546.98
2455863.56
2455226.29
2455321.29
2455990.34
2455498.88
2456084.58
46.1
2455201.18
2455315.29
22.2
Calculating the true orbital period of an inferior planet (Example: Mercury)
1st: Sketch out the picture that corresponds to two adjacent inferior conjunctions. Inferior planets must orbit inside the
Earth’s orbit sine they can never be seen at opposition.
Configuration
Date
Julian Date
Inferior Conjunction #1
2010/01/04 2455201.18
Mercury
Inferior Conjunction #2
2010/04/28 2455315.29
The time between the two conjunctions equals the difference between the Julian dates: 114.11 days which is about
1/3 of a year.
#1

#1

Angle 
#2
 #2
2nd: Calculate the angle  that the Earth moved in its orbit between the two inferior conjunctions using a simple
proportion.

360

114.11 days 365.241 days
  112.47 degrees
The Earth moved 112.47 around its orbit in the 114.11 days between inferior conjunctions of Mercury.
3rd: Calculate the angle the inferior planet moved in its orbit between the two inferior conjunctions using simple
reasoning.
It is reasonable to assume that Mercury moved once around the Sun plus the 112.47 of angle  for a total angle
of 472.47 in the 114.11 days between inferior conjunctions. Note: this assumption may be different for other
inferior planets.
4th: Calculate the time required for the inferior planet to complete one orbit using a simple proportion.
472.47 
360 

114.11 days T days
T  86.9 days
Mercury requires about 86.9 days to complete one 360 orbit around the Sun. This is the true orbital period of
Mercury (approximately).
Calculating the true orbital period of a superior planet
Calculating the true orbital period of an inferior planet (Example:Saturn)
1st: Sketch out the picture that corresponds to two adjacent inferior conjunctions. Superior planets must orbit outside the
Earth’s orbit sine they are seen at opposition and conjunction.
Configuration
Date
Julian Date
Opposition
2010/03/21 2455277.50
Saturn
Opposition
2011/04/03 2455655.48
The time between the two oppositions equals the difference between the Julian dates: 377.98 days which is just
over 1 year.
#1
#1


#2

#2
Angle  is the angle
the Earth moved
around the Sun in
377.98 days, which
will equal 360 plus
the angle Saturn
moved in that time
period between
oppositions.
2nd: Calculate the angle  that the Earth moved in its orbit between the two inferior conjunctions using a simple
proportion.

360

377.98 days 365.241 days
  372.56 degrees
The Earth moved 372.56 around its orbit in the 377.98 days between oppositions of Saturn. The Earth completed
slightly more than one orbit.
3rd: Calculate the angle the superior planet moved in its orbit between the two oppositions using simple reasoning.
It is reasonable to assume that Saturn moved only slightly around the Sun; 372.56 minus the 360 of angle  for
a total angle of 12.56 in the 377.98 days between oppositions. Note: this assumption may be different for other
superior planets.
4th: Calculate the time required for the inferior planet to complete one orbit using a simple proportion.
12.56
360

377.98 days T days
T  10,834 days  29.7 years
Saturn requires about 29.7 years to complete one 360 orbit around the Sun. This is the true orbital period of
Saturn (approximately).
Calculating the true orbital radius of an inferior planet
1st: Sketch out the picture that corresponds to maximum elongation. Inferior planets must orbit inside the Earth’s orbit
sine they can never be seen at opposition.
Configuration
Mercury
Date
Average
over many
cycles
Maximum Elongation
Julian Date
22.2
Mercury
True Orbital
Radius
Maximum
Elongation Angle
Earth


1 AU
#2
 #2
2nd: Construct the right triangle with a hypotenuse of 1 AU and an interior angle equal to the maximum elongation
angle. See the figure above.
3rd: Calculate the true orbital radius using the trigonometric properties of right triangles.
opposite
hypotenus
opposite  hypothenus  sin(max. elongation )
sin( max. elongation ) 
 1 AU  sin(22.2  )
 0.378 AU
Mercury orbits the Sun about 0.378 AU from it.. This is the true orbital radius of Mercury (approximately).
Calculating the true orbital radius of a superior planet
This is the most difficult of the calculations, because you need to find two angles before you can calculate the distance.
Planet
Saturn
Configuration
Opposition #1
Quadrature #2
Date
2010/03/21
2010/06/19
Julian Date
2455277.50
2455367.55
1st: Sketch out the picture that corresponds to
opposition and the following quadrature. Superior
planets must orbit outside the Earth’s orbit sine they
are seen at opposition and conjunction. The time
between opposition and quadrature for Saturn is the
difference between the Julian dates 90.05 days
#1

#1

1 AU
#2
#2
2nd: Construct the right triangle with a hypotenuse
from the Sun to the superior planet at quadrature.
See the figure above.
Angle  is the angle
the Earth moved
around the Sun in
377.98 days, which
will equal 360 plus
the angle Saturn
moved in that time
period between
oppositions.
3rd: Calculate the angles that Saturn and the Earth moved through during the time between opposition and
quadrature. These are the two blue arrows in the figure above.
 Earth
360
90.05 days 365.241 days
  88.76 degrees

 Saturn
360
90.05 days 10,834 days
  2.99 degrees

4th: Calculate the interior angle to the red triangle (i.e. the red angle). Its value is just the difference between the angle
the Earth moved and the angle Saturn moved: 88.76-2.99 = 85.77
5th: Calculate the true orbital radius using the trigonometric properties of right triangles.
adjacent
hypotenus
adjacent
hypotenus 
cos( red interior angle)
1 AU

cos(85.77  )
 13.6 AU
cos( red interior angle) 
Saturn orbits the Sun about 13.6AU from it. This is the true orbital radius of Saturn (approximately). Note: this
orbital radius is about 30% different from the current accepted value. That is due to lack of precision in our
dates for the quadrature. The red interior angle should be 83.9 . The difference is a 2 day error in identifying
quadrature.