Chapter 64

Examination of Parallax

Kepler can demonstrate the possible maximum parallax of Mars. He will reperform the calculations and comparisons of chapters 61 and 62 with parallax, and show that the results do not accord with reason.

What is diurnal parallax?
What determines the observed location of a heavenly body? Two things do: the location of the body, and the location of the observer. The location of the observer on the Earth changes over time, since the Earth both revolves around the Sun, and rotates about its axis. During the daily (“diurnal”) rotation, your location on the Earth moves. The effect of the motion of the observer on the body of the planet is diurnal parallax. See also chapter 11, where this was discussed earlier.

Chapter 61 Calculation

In chapter 61, Kepler found the nodes to be exactly opposite each other in zodiacal position: 16° 461/3′ Taurus and Scorpio. When he recalculates the location of the nodes, incorporating a slight parallax of 1′ to 2′ in his observations, he finds the nodes to be at 16¼° Taurus and 17¾° Scorpio. Thus, the use of a parallax correction causes the nodes not to be opposite each other, which is impossible of the plane of the Martian orbit intersects the body of the sun.

Chapter 62 Calculation

This concerns calculating the theoretical latitude of Mars, and comparing it with the observations. Compare the calculated latitudes in chapter 62 with the observed latitudes from chapter 15. Choosing years for which parallax was taken into account in formulating the table of chapter 15, Kepler reconsiders the observations, without the parallax correction he had previously made. (This explains the two values for latitude from chapter 15 in the table in chapter 62.) It is consistently true that not making a correction for parallax gives values that better agree with his theory:

Year
Calculated latitude
Observation w/o
parallax correction
Observation w/
parallax correction
1587
3°37′N
3°37′N
3°41′N
1589
1°5′N
1°7′N
1°12¾′N
1602
4°7½′N
4°8′N
4°10′N
1604
2°18½′N
2°21½′N
2°26′N

Conclusion

Thus, it is true that the parallax is very small, not to exceed 2 or 2½ minutes, which is within the limits of the Brahean observations. Using this parallax would require a change of half a minute to the inclination, making it 1°51′.

Afterword:

A rough calculation of the maximum parallax of Mars, using modern data for the ratio of the body of the earth to its orbit (unavailable to Kepler), gives a maximum possible parallax at opposition of around ½′. These are the numbers used:

  • Mean earth radius: 3963 miles
  • Earth-sun distance at aphelion: 94,500,000 miles
  • Mars-sun distance at perihelion: 127,400,000 miles
  • Closest Earth-Mars distance (if the lines of apsides were aligned this way): 32,900,000 miles.

The mean earth-sun distance is 92,960,000 miles, which puts the sun-earth distance as 92,960,000 / 3963 ~ 23,500 semidiameters of the earth. Compare this with Kepler’s estimate of 700-2000 semidiameters in chapter 11. Interestingly, with better astronomical equipment and improved accuracy of observations, the more exact determination of parallax allowed the calculation of astronomical distances. Can you think of how this was done?

Chapter 68