Specific orbital energy

In the <a href="/facts/Gravitational_two-body_problem/vWg6qg9X">gravitational two-body problem</a>, the specific orbital energy 
 
 
 
 ε
 
 
 {\displaystyle \varepsilon }
 
 (or specific vis-viva energy) of two <a href="/facts/Orbiting_body/ecPWQ8p6">orbiting bodies</a> is the constant quotient of their <a href="/facts/Mechanical_energy/gI4oMzoY">mechanical energy</a> (the sum of their mutual <a href="/facts/Potential_energy/I03KLbmn">potential energy</a>, 
 
 
 
 
 ε
 
 p
 
 
 
 
 {\displaystyle \varepsilon _{p}}
 
, and their <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a>, 
 
 
 
 
 ε
 
 k
 
 
 
 
 {\displaystyle \varepsilon _{k}}
 
) to their <a href="/facts/Reduced_mass/GQgUNp8D">reduced mass</a>. 
According to the <a href="/facts/Vis-viva_equation/mA6SnCd9">orbital energy conservation equation</a> (also referred to as vis-viva equation), it does not vary with time:

ε
              
              
                
                =
                
                  ε
                  
                    k
                  
                
                +
                
                  ε
                  
                    p
                  
                
              
            
            
              
              
                
                =
                
                  
                    
                      v
                      
                        2
                      
                    
                    2
                  
                
                −
                
                  
                    μ
                    r
                  
                
                =
                −
                
                  
                    1
                    2
                  
                
                
                  
                    
                      μ
                      
                        2
                      
                    
                    
                      h
                      
                        2
                      
                    
                  
                
                
                  (
                  
                    1
                    −
                    
                      e
                      
                        2
                      
                    
                  
                  )
                
                =
                −
                
                  
                    μ
                    
                      2
                      a
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\varepsilon &=\varepsilon _{k}+\varepsilon _{p}\\&={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}

where

<ul><li>
 
 
 
 v
 
 
 {\displaystyle v}
 
 is the relative <a href="/facts/Orbital_speed/6yAYqeqC">orbital speed</a>;</li>
<li>
 
 
 
 r
 
 
 {\displaystyle r}
 
 is the <a href="/facts/Orbital_state_vectors/9PlwJYYP">orbital distance</a> between the bodies;</li>
<li>
 
 
 
 μ
 =
 
 G
 
 (
 
 m
 
 1
 
 
 +
 
 m
 
 2
 
 
 )
 
 
 {\displaystyle \mu ={G}(m_{1}+m_{2})}
 
 is the sum of the <a href="/facts/Standard_gravitational_parameter/pzGlwadL">standard gravitational parameters</a> of the bodies;</li>
<li>
 
 
 
 h
 
 
 {\displaystyle h}
 
 is the <a href="/facts/Specific_relative_angular_momentum/FeW3STxI">specific relative angular momentum</a> in the sense of <a href="/facts/Relative_angular_momentum/7TuSVFxq">relative angular momentum</a> divided by the reduced mass;</li>
<li>
 
 
 
 e
 
 
 {\displaystyle e}
 
 is the <a href="/facts/Eccentricity_(orbit)/WGlB4NBn">orbital eccentricity</a>;</li>
<li>
 
 
 
 a
 
 
 {\displaystyle a}
 
 is the <a href="/facts/Semi-major_axis/l7JES2wo">semi-major axis</a>.</li></ul>
It is a kind of <a href="/facts/Specific_energy/zVRhSxPh">specific energy</a>, typically expressed in units of 
 
 
 
 
 
 MJ
 kg
 
 
 
 
 {\displaystyle {\frac {\text{MJ}}{\text{kg}}}}
 
 (mega<a href="/facts/Joule/9ndq9jD2">joule</a> per kilogram) or 
 
 
 
 
 
 
 
 km
 
 
 2
 
 
 
 
 s
 
 
 2
 
 
 
 
 
 
 {\displaystyle {\frac {{\text{km}}^{2}}{{\text{s}}^{2}}}}
 
 (squared kilometer per squared second). For an <a href="/facts/Elliptic_orbit/SfuXuATN">elliptic orbit</a> the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to <a href="/facts/Escape_velocity/bdtme0tx">escape velocity</a> (<a href="/facts/Parabolic_trajectory/vDNtF83E">parabolic orbit</a>). For a <a href="/facts/Hyperbolic_trajectory/4wEAXaez">hyperbolic orbit</a>, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as <a href="/facts/Characteristic_energy/249clhTX">characteristic energy</a>.

Specific orbital energy open-in-new

Specific orbital energy