kb/data/en.wikipedia.org/wiki/Astronomia_nova-2.md

title	chunk	source	category	tags	date_saved	instance
Astronomia nova	3/11	https://en.wikipedia.org/wiki/Astronomia_nova	reference	science, encyclopedia	2026-05-05T08:51:05.564785+00:00	kb-cron

The actual correction Kepler shows to be less than 1 minute of arc, which is smaller than the error in Tycho's observations, and for practical purposes can be ignored. So, opposition can be defined as the moment when the ecliptic longitude of the sun and Mars are

    180
    ∘
  

{\textstyle 180\circ }

apart. Although the ecliptic longitude of Mars is the same from the Earth and sun at this point, the same is not true for the ecliptic latitude. In the diagram above

    G
  

{\textstyle G}

is the sun,

    H
  

{\textstyle H}

is the Earth and

    I
  

{\textstyle I}

is Mars. The line

    G
    H
  

{\textstyle GH}

is in the plane of the ecliptic. The angle

    β
  

{\textstyle \beta }

is the angle that Mars appears above the ecliptic when viewed from Earth; this is the ecliptic latitude. The angle

    α
  

{\textstyle \alpha }

is the ecliptic latitude viewed from the sun, which is smaller than

    β
  

{\textstyle \beta }

. The relation between these angles tells us something about the ratio of the Earth-sun distance

    G
    H
  

{\textstyle GH}

and the Mars-Sun distance

    G
    I
  

{\textstyle GI}

. In chapter 11, Kepler attempts to determine the parallax of Mars. As Mars is close to the Earth, its position in the sky will appear to change slightly as the observer's position changes throughout the day, even if it is otherwise stationary. This effect, called parallax, would be greatest when Mars is at opposition, since at that time Mars is at its closest point to the Earth. The existing estimates of the distance to the sun, based largely on Aristarchus' method, suggested the parallax could be as high as 6 minutes of arc, but Kepler's own attempts to determine parallax gave values that were less than 2 arc minutes. In chapters 12-14, Kepler determines the longitude of Mars' ascending and descending nodes and the orbital inclination of Mars. To find the nodes, Kepler looks for observations of Mars where its ecliptic latitude is close to

    0
  

{\textstyle 0}

, then use interpolation to find the exact moment when it is zero and uses then use existing tables such as the Prutenic tables (which were based on Copernicus' theory) to compute the longitude of Mars at that time. Kepler located the ascending node at

      225
      
        ∘
      
    
    44
    
      
        
          1
          2
        
      
      ′
    
  

{\textstyle 225^{\circ }44{\frac {1}{2}}'}

and the descending node at

      46
      
        ∘
      
    
    
      
        
          1
          8
        
      
      ′
    
  

{\displaystyle 46^{\circ }{\frac {1}{8}}'}

. These values are not precisely

      180
      
        ∘
      
    
  

{\displaystyle 180^{\circ }}

apart. The longitudes in the Prutenic tables were measured from the mean sun. Kepler argues that if they were measured from the sun's actual position instead the values would be

      180
      
        ∘
      
    
  

{\displaystyle 180^{\circ }}

apart. The Prutenic tables also provided distances to the planets. This allows Kepler to solve the triangle in figure 1 above to compute heliocentric latitude from the observed geocentric latitude, from which he could deduce the orbital inclination of Mars by observing Mars when its latitude was greatest and computing its heliocentric latitude. He finds the orbital inclination to be

      1
      
        ∘
      
    
    
      50
      ′
    
  

{\displaystyle 1^{\circ }50'}

. This also allowed Kepler to demonstrate the important fact that plane of Mars orbit does not wobble in any way as many theories before him had suggested. Using observations from various points, he shows the orbital inclination is constant. In chapter 15, Kepler recomputes Tycho's 12 oppositions, so as to determine the precise moment Mars's ecliptic longitude is

      180
      
        ∘
      
    
  

{\textstyle 180^{\circ }}

from the sun. For each observation, he determines the ecliptic longitude and latitude of Mars as seen from the Earth, and the time when opposition occurs. In chapter 16, Kepler constructs his first model, the vicarious hypothesis, to account for the observations. This is a modification of the equant model of Ptolemy. In this model, the planet is assumed to move on a circular orbit, and the speed on the orbit varies in such a way that it appears uniform from some point called the equant. The line connecting the equant and the center of the circle is called the line of apsides, and it intersect the circle at two points, one where the planet moves at its fastest speed called the aphelion and the other where the planet moves at its slowest speed called the perihelion. For his model, Ptolemy had assumed that the center of the circle lies exactly halfway between the equant and the point from which opposition is measured (which for Kepler is the sun), this model is called bisected eccentricity. Kepler however considers the more general hypothesis that the center of the circle can be placed at any point along the line of apsides between the sun and the equant.