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The point at which the spacecraft should leave the (almost circular) catch-up orbit can be calculated using this formula:  
 
The point at which the spacecraft should leave the (almost circular) catch-up orbit can be calculated using this formula:  
  
<math>\Delta \varphi = 2 \pi \left [ 1 - ( 2 m - 1 ) \left ( \frac {a}{r_z} \right )^{\frac{3}{2}} \right ] </math>
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<math>\Delta \varphi = \pi \left [ 1 - ( 2 m - 1 ) \left ( \frac {a}{r_z} \right )^{\frac{3}{2}} \right ] </math>
  
 
where
 
where
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The variable <math>n</math> again describes, how many orbits the chasing craft should wait in the immediate orbit until it meets the target.   
 
The variable <math>n</math> again describes, how many orbits the chasing craft should wait in the immediate orbit until it meets the target.   
  
If the time to rendezvous is shorter than one [[orbital period]], its possible to ignore most orbital mechanics and instead assume both craft are in a gravity-free space. That concept gets used very often in various [[radar]] rendezvous guidance systems ([[Gemini]], [[Soyuz]]). Instead of making orbital maneuvers, which would need fast computers to calculate, the chasing spacecraft uses a [[Doppler effect|doppler radar]] to measure the angle, distance and rate of change of the distance. By keeping the angle between [[line of sight]] to the target (measured by the radar) and its own velocity vector constant ([[proportional guidance]]), the chasing spacecraft approaches the target very effectively.  
+
If the time to rendezvous is shorter than one [[orbital period]], its possible to ignore most orbital mechanics and instead assume both craft are in a gravity-free space. That concept gets used very often in various [[radar]] rendezvous guidance systems ([[Gemini]], [[Soyuz]]). Instead of making orbital maneuvers, which would need fast computers to calculate, the chasing spacecraft uses a [[Doppler effect|doppler radar]] to measure the angle, distance and rate of change of the distance. By keeping the angle between [[line of sight]] to the target (measured by the radar) and its own velocity vector constant ([[proportional guidance]]), the chasing spacecraft approaches the target very effectivly.  
  
 
When better computers are available, its also possible to solve [[Hills equations]], which describe the movement of an object relative to another in an orbit. This is more effective in terms of fuel and more accurate, but requires better onboard computers, as it requires solving differential equations.
 
When better computers are available, its also possible to solve [[Hills equations]], which describe the movement of an object relative to another in an orbit. This is more effective in terms of fuel and more accurate, but requires better onboard computers, as it requires solving differential equations.
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{{HasPrecis}}
 
{{HasPrecis}}
  
[[Category: Articles]]
 
 
[[Category:Celestial mechanics]]
 
[[Category:Celestial mechanics]]

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