kb/data/en.wikipedia.org/wiki/Astronomia_nova-7.md

title	chunk	source	category	tags	date_saved	instance
Astronomia nova	8/11	https://en.wikipedia.org/wiki/Astronomia_nova	reference	science, encyclopedia	2026-05-05T08:51:05.564785+00:00	kb-cron

In the diagram above,

    A
  

{\textstyle A}

is the sun and

    B
  

{\textstyle B}

is the center of the circle;

    G
  

{\textstyle G}

is aphelion and

    D
  

{\textstyle D}

is the planet. The area of the sector

    G
    A
    D
  

{\textstyle GAD}

is the area swept by the line drawn from the planet to the sun. From the area law, this is proportional to the time that the planet has traversed the segment

    G
    D
  

{\textstyle GD}

in its orbit, and therefore also the mean anomaly. Thus the area

    G
    A
    D
  

{\textstyle GAD}

gives the mean anomaly. The eccentric anomaly is defined by the angle

    ∠
    G
    B
    D
  

{\textstyle \angle GBD}

. Since the angle of a sector, centered on a circle, is always proportional to its area, we can also express this by the area

    G
    B
    D
  

{\textstyle GBD}

. The relation between these two areas gives the relation between the mean anomaly (and therefore time) and eccentric anomaly. From the diagram, it is clear that the mean anomaly is simply the eccentric anomaly plus the area of the triangle

    D
    A
    B
  

{\textstyle DAB}

. The base of this triangle

          A
          B
        
        ¯
      
    
  

{\textstyle {\overline {AB}}}

is the eccentricity of the circle, and the height of the triangle is proportional to sine of the eccentric anomaly. This is the Kepler equation. If we write

    M
  

{\textstyle M}

for the mean anomaly,

    E
  

{\textstyle E}

for the eccentric anomaly, and

    e
  

{\textstyle e}

for the eccentricity, then this can be written as:

    M
    =
    E
    +
    e
    sin
    ⁡
    (
    E
    )
  

{\displaystyle M=E+e\sin(E)}

Kepler further shows that the true anomaly is given by the eccentric anomaly plus the angle

    ∠
    D
    B
    A
  

{\textstyle \angle DBA}

. Kepler refers to the angle

    ∠
    D
    B
    A
  

{\textstyle \angle DBA}

as the optical equation. For low eccentricities, this angle is approximately twice the area of the triangle

    D
    B
    A
  

{\textstyle DBA}

. If we write the true anomaly as

    ϑ
  

{\textstyle \vartheta }

, this gives the formula:

    ϑ
    ≈
    E
    +
    2
    e
    sin
    ⁡
    (
    E
    )
  

{\displaystyle \vartheta \approx E+2e\sin(E)}