True anomaly

<h2 id="formulas">Formulas</h2>
<h3>From state vectors</h3>
For elliptic orbits, the true anomaly ν can be calculated from <a href="/facts/Orbital_state_vectors/9PlwJYYP">orbital state vectors</a> as:

ν
        =
        arccos
        ⁡
        
          
            
              
                e
              
              ⋅
              
                r
              
            
            
              
                
                  |
                  e
                  |
                
              
              
                
                  |
                  r
                  |
                
              
            
          
        
      
    
    {\displaystyle \nu =\arccos {{\mathbf {e} \cdot \mathbf {r} } \over {\mathbf {\left|e\right|} \mathbf {\left|r\right|} }}}

(if r ⋅ v < 0 then replace ν by 2π − ν)
where: 

<ul><li>v is the <a href="/facts/Orbital_velocity_vector/9PlwJYYP">orbital velocity vector</a> of the orbiting body,</li>
<li>e is the <a href="/facts/Eccentricity_vector/uoemPe39">eccentricity vector</a>,</li>
<li>r is the <a href="/facts/Orbital_position_vector/9PlwJYYP">orbital position vector</a> (segment FP in the figure) of the orbiting body.</li></ul>
<h4>Circular orbit</h4>
For <a href="/facts/Circular_orbit/sSFMzmBv">circular orbits</a> the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the <a href="/facts/Argument_of_latitude/JqfNnhVD">argument of latitude</a> u is used:

u
        =
        arccos
        ⁡
        
          
            
              
                n
              
              ⋅
              
                r
              
            
            
              
                
                  |
                  n
                  |
                
              
              
                
                  |
                  r
                  |
                
              
            
          
        
      
    
    {\displaystyle u=\arccos {{\mathbf {n} \cdot \mathbf {r} } \over {\mathbf {\left|n\right|} \mathbf {\left|r\right|} }}}

(if rz < 0 then replace u by 2π − u)
where:

<ul><li>n is a vector pointing towards the ascending node (i.e. the z-component of n is zero).</li>
<li>rz is the z-component of the <a href="/facts/Orbital_position_vector/9PlwJYYP">orbital position vector</a> r</li></ul>
<h4>Circular orbit with zero inclination</h4>
For <a href="/facts/Circular_orbit/sSFMzmBv">circular orbits</a> with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses the <a href="/facts/True_longitude/fJjjLC5S">true longitude</a> instead:

l
        =
        arccos
        ⁡
        
          
            
              r
              
                x
              
            
            
              
                
                  |
                  r
                  |
                
              
            
          
        
      
    
    {\displaystyle l=\arccos {r_{x} \over {\mathbf {\left|r\right|} }}}

(if vx > 0 then replace l by 2π − l)
where:

<ul><li>rx is the x-component of the <a href="/facts/Orbital_position_vector/9PlwJYYP">orbital position vector</a> r</li>
<li>vx is the x-component of the <a href="/facts/Orbital_velocity_vector/9PlwJYYP">orbital velocity vector</a> v.</li></ul>
<h3>From the eccentric anomaly</h3>
The relation between the true anomaly ν and the <a href="/facts/Eccentric_anomaly/rmbJzkGl">eccentric anomaly</a> 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is:

cos
        ⁡
        
          ν
        
        =
        
          
            
              cos
              ⁡
              
                E
              
              −
              e
            
            
              1
              −
              e
              cos
              ⁡
              
                E
              
            
          
        
      
    
    {\displaystyle \cos {\nu }={{\cos {E}-e} \over {1-e\cos {E}}}}

or using the <a href="/facts/Sine/gQ6LXK8z">sine</a><a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> and <a href="/facts/Tangent/sF0jH3EI">tangent</a>:

sin
                ⁡
                
                  ν
                
              
              
                
                =
                
                  
                    
                      
                        
                          1
                          −
                          
                            e
                            
                              2
                            
                          
                          
                        
                      
                      sin
                      ⁡
                      
                        E
                      
                    
                    
                      1
                      −
                      e
                      cos
                      ⁡
                      
                        E
                      
                    
                  
                
              
            
            
              
                tan
                ⁡
                
                  ν
                
                =
                
                  
                    
                      sin
                      ⁡
                      
                        ν
                      
                    
                    
                      cos
                      ⁡
                      
                        ν
                      
                    
                  
                
              
              
                
                =
                
                  
                    
                      
                        
                          1
                          −
                          
                            e
                            
                              2
                            
                          
                          
                        
                      
                      sin
                      ⁡
                      
                        E
                      
                    
                    
                      cos
                      ⁡
                      
                        E
                      
                      −
                      e
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\sin {\nu }&={{{\sqrt {1-e^{2}\,}}\sin {E}} \over {1-e\cos {E}}}\\[4pt]\tan {\nu }={{\sin {\nu }} \over {\cos {\nu }}}&={{{\sqrt {1-e^{2}\,}}\sin {E}} \over {\cos {E}-e}}\end{aligned}}}

or equivalently:

tan
        ⁡
        
          
            ν
            2
          
        
        =
        
          
            
              
                1
                +
                e
                
              
              
                1
                −
                e
                
              
            
          
        
        tan
        ⁡
        
          
            E
            2
          
        
      
    
    {\displaystyle \tan {\nu  \over 2}={\sqrt {{1+e\,} \over {1-e\,}}}\tan {E \over 2}}

so

ν
        =
        2
        
        arctan
        ⁡
        
          (
          
            
            
              
                
                  
                    1
                    +
                    e
                    
                  
                  
                    1
                    −
                    e
                    
                  
                
              
            
            tan
            ⁡
            
              
                E
                2
              
            
            
          
          )
        
      
    
    {\displaystyle \nu =2\,\operatorname {arctan} \left(\,{\sqrt {{1+e\,} \over {1-e\,}}}\tan {E \over 2}\,\right)}

Alternatively, a form of this equation was derived by <a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> that avoids numerical issues when the arguments are near 
 
 
 
 ±
 π
 
 
 {\displaystyle \pm \pi }
 
, as the two tangents become infinite. Additionally, since 
 
 
 
 
 
 E
 2
 
 
 
 
 {\displaystyle {\frac {E}{2}}}
 
 and 
 
 
 
 
 
 ν
 2
 
 
 
 
 {\displaystyle {\frac {\nu }{2}}}
 
 are always in the same quadrant, there will not be any sign problems.

tan
        ⁡
        
          
            
              1
              2
            
          
          (
          ν
          −
          E
          )
        
        =
        
          
            
              β
              sin
              ⁡
              
                E
              
            
            
              1
              −
              β
              cos
              ⁡
              
                E
              
            
          
        
      
    
    {\displaystyle \tan {{\frac {1}{2}}(\nu -E)}={\frac {\beta \sin {E}}{1-\beta \cos {E}}}}
  
 where 
  
    
      
        β
        =
        
          
            e
            
              1
              +
              
                
                  1
                  −
                  
                    e
                    
                      2
                    
                  
                
              
            
          
        
      
    
    {\displaystyle \beta ={\frac {e}{1+{\sqrt {1-e^{2}}}}}}

so

ν
        =
        E
        +
        2
        arctan
        ⁡
        
          (
          
            
            
              
                
                  β
                  sin
                  ⁡
                  
                    E
                  
                
                
                  1
                  −
                  β
                  cos
                  ⁡
                  
                    E
                  
                
              
            
            
          
          )
        
      
    
    {\displaystyle \nu =E+2\operatorname {arctan} \left(\,{\frac {\beta \sin {E}}{1-\beta \cos {E}}}\,\right)}

<h3>From the mean anomaly</h3>
The true anomaly can be calculated directly from the <a href="/facts/Mean_anomaly/dzs98drn">mean anomaly</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 via a <a href="/facts/Fourier_expansion/s3fsxPAT">Fourier expansion</a>:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

ν
        =
        M
        +
        2
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            1
            k
          
        
        
          [
          
            
              ∑
              
                n
                =
                −
                ∞
              
              
                ∞
              
            
            
              J
              
                n
              
            
            (
            −
            k
            e
            )
            
              β
              
                
                  |
                
                k
                +
                n
                
                  |
                
              
            
          
          ]
        
        sin
        ⁡
        
          k
          M
        
      
    
    {\displaystyle \nu =M+2\sum _{k=1}^{\infty }{\frac {1}{k}}\left[\sum _{n=-\infty }^{\infty }J_{n}(-ke)\beta ^{|k+n|}\right]\sin {kM}}

with <a href="/facts/Bessel_functions/hSyvFPqC">Bessel functions</a> 
 
 
 
 
 J
 
 n
 
 
 
 
 {\displaystyle J_{n}}
 
 and parameter 
 
 
 
 β
 =
 
 
 
 1
 −
 
 
 1
 −
 
 e
 
 2
 
 
 
 
 
 e
 
 
 
 
 {\displaystyle \beta ={\frac {1-{\sqrt {1-e^{2}}}}{e}}}
 
.
Omitting all terms of order 
 
 
 
 
 e
 
 4
 
 
 
 
 {\displaystyle e^{4}}
 
 or higher (indicated by 
 
 
 
 
 
 O
 
 
 ⁡
 
 (
 
 e
 
 4
 
 
 )
 
 
 
 {\displaystyle \operatorname {\mathcal {O}} \left(e^{4}\right)}
 
), it can be written as<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

ν
        =
        M
        +
        
          (
          
            2
            e
            −
            
              
                1
                4
              
            
            
              e
              
                3
              
            
          
          )
        
        sin
        ⁡
        
          M
        
        +
        
          
            5
            4
          
        
        
          e
          
            2
          
        
        sin
        ⁡
        
          2
          M
        
        +
        
          
            13
            12
          
        
        
          e
          
            3
          
        
        sin
        ⁡
        
          3
          M
        
        +
        
          
            O
          
        
        ⁡
        
          (
          
            e
            
              4
            
          
          )
        
        .
      
    
    {\displaystyle \nu =M+\left(2e-{\frac {1}{4}}e^{3}\right)\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}+{\frac {13}{12}}e^{3}\sin {3M}+\operatorname {\mathcal {O}} \left(e^{4}\right).}

Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity 
 
 
 
 e
 
 
 {\displaystyle e}
 
 is small.
The expression 
 
 
 
 ν
 −
 M
 
 
 {\displaystyle \nu -M}
 
 is known as the <a href="/facts/Equation_of_the_center/UqGx4w4c">equation of the center</a>, where more details about the expansion are given.

<h3>Radius from true anomaly</h3>
The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by the formula

r
        (
        t
        )
        =
        a
        
        
          
            
              1
              −
              
                e
                
                  2
                
              
            
            
              1
              +
              e
              cos
              ⁡
              ν
              (
              t
              )
            
          
        
        
        
      
    
    {\displaystyle r(t)=a\,{1-e^{2} \over 1+e\cos \nu (t)}\,\!}

where a is the orbit's <a href="/facts/Semi-major_axis/l7JES2wo">semi-major axis</a>.
In <a href="/facts/Celestial_mechanics/FA3uR78J">celestial mechanics</a>, Projective anomaly is an angular <a href="/facts/Parameter/zFvVFMv9">parameter</a> that defines the position of a body moving along a <a href="/facts/Keplerian_orbit/XvdtxmKl">Keplerian orbit</a>. It is the angle between the direction of <a href="/facts/Periapsis/VaL6ow9w">periapsis</a> and the current position of the body in the projective space.
The projective anomaly is usually denoted by the 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 and is usually restricted to the range 0 - 360 degree (0 - 2 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 radian).
The projective anomaly 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is one of four angular parameters (anomalies) that defines a position along an orbit, the other two being the <a href="/facts/Eccentric_anomaly/rmbJzkGl">eccentric anomaly</a>, true anomaly and the <a href="/facts/Mean_anomaly/dzs98drn">mean anomaly</a>.
In the projective geometry, circle, ellipse, parabolla, hyperbolla are treated as a same kind of quadratic curves.

<h2 id="projective-parameters-and-projective-anomaly">projective parameters and projective anomaly</h2>
An orbit type is classified by two project parameters 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 as follows, 

<ul><li>circular orbit 
 
 
 
 β
 =
 0
 
 
 {\displaystyle \beta =0}
 
</li></ul>
<ul><li>elliptic orbit 
 
 
 
 α
 β
 <
 1
 
 
 {\displaystyle \alpha \beta <1}
 
</li></ul>
<ul><li>parabolic orbit 
 
 
 
 α
 β
 =
 1
 
 
 {\displaystyle \alpha \beta =1}
 
</li></ul>
<ul><li>hyperbolic orbit 
 
 
 
 α
 β
 >
 1
 
 
 {\displaystyle \alpha \beta >1}
 
</li></ul>
<ul><li>linear orbit 
 
 
 
 α
 =
 β
 
 
 {\displaystyle \alpha =\beta }
 
</li></ul>
<ul><li>imaginary orbit 
 
 
 
 α
 <
 β
 
 
 {\displaystyle \alpha <\beta }
 
</li></ul>
where

 
 
 
 α
 =
 
 
 
 (
 1
 +
 e
 )
 (
 q
 −
 p
 )
 +
 
 
 (
 1
 +
 e
 
 )
 
 2
 
 
 (
 q
 +
 p
 
 )
 
 2
 
 
 +
 4
 
 e
 
 2
 
 
 
 
 
 2
 
 
 
 
 {\displaystyle \alpha ={\frac {(1+e)(q-p)+{\sqrt {(1+e)^{2}(q+p)^{2}+4e^{2}}}}{2}}}

β
 =
 
 
 
 2
 e
 
 
 (
 1
 +
 e
 )
 (
 q
 +
 p
 )
 +
 
 
 (
 1
 +
 e
 
 )
 
 2
 
 
 (
 q
 +
 p
 
 )
 
 2
 
 
 +
 4
 
 e
 
 2
 
 
 
 
 
 
 
 
 
 {\displaystyle \beta ={\frac {2e}{(1+e)(q+p)+{\sqrt {(1+e)^{2}(q+p)^{2}+4e^{2}}}}}}

q
 =
 (
 1
 −
 e
 )
 a
 
 
 {\displaystyle q=(1-e)a}

p
 =
 
 
 1
 Q
 
 
 =
 
 
 1
 
 (
 1
 +
 e
 )
 a
 
 
 
 
 
 {\displaystyle p={\frac {1}{Q}}={\frac {1}{(1+e)a}}}

where 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is semi major axis,
 
 
 
 e
 
 
 {\displaystyle e}
 
 is <a href="/facts/Eccentricity_(mathematics)/SzgpW5R7">eccentricity</a>, 
 
 
 
 q
 
 
 {\displaystyle q}
 
 is <a href="/facts/Perihelion_distance/VaL6ow9w">perihelion distance</a>、
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 is <a href="/facts/Aphelion_distance/VaL6ow9w">aphelion distance</a>.
Position and heliocentric distance of the planet 
 
 
 
 x
 
 
 {\displaystyle x}
 
, 
 
 
 
 y
 
 
 {\displaystyle y}
 
 and 
 
 
 
 r
 
 
 {\displaystyle r}
 
 can be calculated as functions of the projective anomaly 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 :

 
 
 
 x
 =
 
 
 
 −
 β
 +
 α
 cos
 ⁡
 θ
 
 
 1
 +
 α
 β
 cos
 ⁡
 θ
 
 
 
 
 
 {\displaystyle x={\frac {-\beta +\alpha \cos \theta }{1+\alpha \beta \cos \theta }}}

y
 =
 
 
 
 
 
 
 α
 
 2
 
 
 −
 
 β
 
 2
 
 
 
 
 sin
 ⁡
 θ
 
 
 1
 +
 α
 β
 cos
 ⁡
 θ
 
 
 
 
 
 {\displaystyle y={\frac {{\sqrt {\alpha ^{2}-\beta ^{2}}}\sin \theta }{1+\alpha \beta \cos \theta }}}

r
 =
 
 
 
 α
 −
 β
 cos
 ⁡
 θ
 
 
 1
 +
 α
 β
 cos
 ⁡
 θ
 
 
 
 
 
 {\displaystyle r={\frac {\alpha -\beta \cos \theta }{1+\alpha \beta \cos \theta }}}

<h2 id="keplers-equation">Kepler's equation</h2>
The projective anomaly 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 can be calculated from the eccentric anomaly 
 
 
 
 u
 
 
 {\displaystyle u}
 
 as follows,

<ul><li>Case : 
 
 
 
 α
 β
 <
 1
 
 
 {\displaystyle \alpha \beta <1}
 
</li></ul>

 
 
 
 tan
 ⁡
 
 
 θ
 2
 
 
 =
 
 
 
 
 1
 +
 α
 β
 
 
 1
 −
 α
 β
 
 
 
 
 tan
 ⁡
 
 
 u
 2
 
 
 
 
 {\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {1+\alpha \beta }{1-\alpha \beta }}}\tan {\frac {u}{2}}}

u
 −
 e
 sin
 ⁡
 u
 =
 M
 =
 
 
 (
 
 
 
 1
 −
 
 α
 
 2
 
 
 
 β
 
 2
 
 
 
 
 α
 (
 1
 +
 
 β
 
 2
 
 
 )
 
 
 
 )
 
 
 3
 
 /
 
 2
 
 
 k
 (
 t
 −
 
 T
 
 0
 
 
 )
 
 
 {\displaystyle u-e\sin u=M=\left({\frac {1-\alpha ^{2}\beta ^{2}}{\alpha (1+\beta ^{2})}}\right)^{3/2}k(t-T_{0})}

<ul><li>case : 
 
 
 
 α
 β
 =
 1
 
 
 {\displaystyle \alpha \beta =1}
 
</li></ul>

 
 
 
 
 
 
 s
 
 3
 
 
 3
 
 
 +
 
 
 
 
 α
 
 2
 
 
 −
 1
 
 
 
 α
 
 2
 
 
 +
 1
 
 
 
 s
 =
 
 
 
 2
 k
 (
 t
 −
 
 T
 
 0
 
 
 )
 
 
 α
 (
 
 α
 
 2
 
 
 +
 1
 
 )
 
 3
 
 
 
 
 
 
 
 {\displaystyle {\frac {s^{3}}{3}}+{\frac {\alpha ^{2}-1}{\alpha ^{2}+1}}s={\frac {2k(t-T_{0})}{\sqrt {\alpha (\alpha ^{2}+1)^{3}}}}}

s
 =
 tan
 ⁡
 
 
 θ
 2
 
 
 
 
 {\displaystyle s=\tan {\frac {\theta }{2}}}

<ul><li>case : 
 
 
 
 α
 β
 >
 1
 
 
 {\displaystyle \alpha \beta >1}
 
</li></ul>

 
 
 
 tan
 ⁡
 
 
 θ
 2
 
 
 =
 
 
 
 
 α
 β
 +
 1
 
 
 α
 β
 −
 1
 
 
 
 
 tanh
 ⁡
 
 
 u
 2
 
 
 
 
 {\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {\alpha \beta +1}{\alpha \beta -1}}}\tanh {\frac {u}{2}}}

e
 sinh
 ⁡
 u
 −
 u
 =
 M
 =
 
 
 (
 
 
 
 
 α
 
 2
 
 
 
 β
 
 2
 
 
 −
 1
 
 
 α
 (
 1
 +
 
 β
 
 2
 
 
 )
 
 
 
 )
 
 
 3
 
 /
 
 2
 
 
 k
 (
 t
 −
 
 T
 
 0
 
 
 )
 
 
 {\displaystyle e\sinh u-u=M=\left({\frac {\alpha ^{2}\beta ^{2}-1}{\alpha (1+\beta ^{2})}}\right)^{3/2}k(t-T_{0})}

The above equations are called <a href="/facts/Kepler%2527s_equation/pww75aK5">Kepler's equation</a>.

<h2 id="generalized-anomaly">Generalized anomaly</h2>
For arbitrary constant 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
, the generalized anomaly 
 
 
 
 Θ
 
 
 {\displaystyle \Theta }
 
 is related as

 
 
 
 tan
 ⁡
 
 
 Θ
 2
 
 
 =
 λ
 tan
 ⁡
 
 
 u
 2
 
 
 
 
 {\displaystyle \tan {\frac {\Theta }{2}}=\lambda \tan {\frac {u}{2}}}

The eccentric anomaly, the true anomaly, and the projective anomaly are the cases of 
 
 
 
 λ
 =
 1
 
 
 {\displaystyle \lambda =1}
 
, 
 
 
 
 λ
 =
 
 
 
 
 1
 +
 e
 
 
 1
 −
 e
 
 
 
 
 
 
 {\displaystyle \lambda ={\sqrt {\frac {1+e}{1-e}}}}
 
, 
 
 
 
 λ
 =
 
 
 
 
 1
 +
 α
 β
 
 
 1
 −
 α
 β
 
 
 
 
 
 
 {\displaystyle \lambda ={\sqrt {\frac {1+\alpha \beta }{1-\alpha \beta }}}}
 
, respectively.

<ul><li>Sato, I., "A New Anomaly of Keplerian Motion", Astronomical Journal Vol.116, pp.2038-3039, (1997)</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Two_body_problem/YCG4geSH">Two body problem</a></li>
<li><a href="/facts/Mean_anomaly/dzs98drn">Mean anomaly</a></li>
<li><a href="/facts/Eccentric_anomaly/rmbJzkGl">Eccentric anomaly</a></li>
<li><a href="/facts/Kepler%2527s_equation/pww75aK5">Kepler's equation</a></li>
<li><a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a></li>
<li><a href="/facts/Kepler%2527s_laws_of_planetary_motion/jYQoIBjM">Kepler's laws of planetary motion</a></li>
<li>Projective anomaly</li>
<li><a href="/facts/Ellipse/1wUDnGxY">Ellipse</a></li>
<li><a href="/facts/Hyperbola/6R6ER0mY">Hyperbola</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Murray, C. D. & Dermott, S. F., 1999, Solar System Dynamics, Cambridge University Press, Cambridge. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-57597-4</li>
<li>Plummer, H. C., 1960, An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://www.worldcat.org/oclc/1311887">1311887</a> (Reprint of the 1918 Cambridge University Press edition.)</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.faa.gov/sites/faa.gov/files/about/office_org/headquarters_offices/avs/III.4.1.4_Describing_Orbits.pdf">Federal Aviation Administration - Describing Orbits</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Fundamentals of Astrodynamics and Applications by David A. Vallado <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Broucke, R.; Cefola, P. (1973). "A Note on the Relations between True and Eccentric Anomalies in the Two-Body Problem". Celestial Mechanics. 7 (3): 388–389. Bibcode:1973CeMec...7..388B. doi:10.1007/BF01227859. ISSN 0008-8714. S2CID 122878026. <a href="https://ui.adsabs.harvard.edu/abs/1973CeMec...7..388B/abstract" target="_blank">https://ui.adsabs.harvard.edu/abs/1973CeMec...7..388B/abstract</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series. American Institute of Aeronautics & Astronautics. p. 212 (Eq. (5.32)). ISBN 978-1-60086-026-3. Retrieved 2022-08-02. <a href="978-1-60086-026-3" target="_blank">978-1-60086-026-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series. American Institute of Aeronautics & Astronautics. p. 212 (Eq. (5.32)). ISBN 978-1-60086-026-3. Retrieved 2022-08-02. <a href="978-1-60086-026-3" target="_blank">978-1-60086-026-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Smart, W. M. (1977). Textbook on Spherical Astronomy (PDF). p. 120 (Eq. (87)). Bibcode:1977tsa..book.....S. <a href="https://wangsajaya.files.wordpress.com/2015/02/textbook-on-spherical-astronomy-smart-6ed-1977.pdf" target="_blank">https://wangsajaya.files.wordpress.com/2015/02/textbook-on-spherical-astronomy-smart-6ed-1977.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Roy, A.E. (2005). Orbital Motion (4 ed.). Bristol, UK; Philadelphia, PA: Institute of Physics (IoP). p. 78 (Eq. (4.65)). Bibcode:2005ormo.book.....R. ISBN 0750310154. Archived from the original on 2021-05-15. Retrieved 2020-08-29. <a href="0750310154" target="_blank">0750310154</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

True anomaly open-in-new

True anomaly