Voyager Earth to Jupiter journey time

Question

Voyager 1 was launched on Sept 5, 1977 and encountered Jupiter March 5, 1979, a journey of less than 2 years. When I try to calculate the trajectory of the probe I assume a Hohmann transfer from Earth to Jupiter even if I'm not 100% about that (the Internet doesn't have a lot of information on that point).

The thing is, when I compute the theoretical time of flight with the following formula:

$$time = \pi*\sqrt{\frac{a^3}{\mu_{sun}}}$$

with $a$ the semi-major axis and $\mu_{sun}$ the standard gravitational parameter

$$a=(149598023+778298361)/2 $$

$$\mu_{sun} = 1.32712428e11$$

I find $time=2.7$ years

But in reality Voyager managed to reach Jupiter in less than 2 years so how is that possible? I thought about the velocity gained by the probe from the Titan 3E rocket which certainly played a role. But how can I integrate this velocity in the time formula?

Answer 1

It wasn't a Hohmann transfer. The rocket simply boosted it to a higher velocity from Earth in order to enable a Jupiter flyby that would send the spacecraft to Saturn.

You can get the approximate orbit elements from the JPL HORIZONS Web-Interface, picking a day in the middle of the Earth to Jupiter transit (which by the way was 1.5 years):

giving:

$$SOE
2443666.500000000 = A.D. 1978-Jun-07 00:00:00.0000 TDB 
 EC= 7.969091448846140E-01 QR= 1.003858168243335E+00 IN= 1.036132511480259E+00
 OM= 3.430028790053406E+02 W = 3.591112970337812E+02 Tp=  2443392.027035175357
 N = 8.974748184533540E-02 MA= 2.463325742763491E+01 TA= 1.257125684251016E+02
 A = 4.942901873513668E+00 AD= 8.881945578784002E+00 PR= 4.011254606791074E+03
$$EOE

where:

  Symbol meaning [1 au= 149597870.700 km, 1 day= 86400.0 s]:

    JDTDB    Julian Day Number, Barycentric Dynamical Time
      EC     Eccentricity, e                                                   
      QR     Periapsis distance, q (au)                                        
      IN     Inclination w.r.t XY-plane, i (degrees)                           
      OM     Longitude of Ascending Node, OMEGA, (degrees)                     
      W      Argument of Perifocus, w (degrees)                                
      Tp     Time of periapsis (Julian Day Number)                             
      N      Mean motion, n (degrees/day)                                      
      MA     Mean anomaly, M (degrees)                                         
      TA     True anomaly, nu (degrees)                                        
      A      Semi-major axis, a (au)                                           
      AD     Apoapsis distance (au)                                            
      PR     Sidereal orbit period (day)                                       

There you can see that $a$ is almost $5\,\mathrm{AU}$ — much more than the $\approx 3\,\mathrm{AU}$ for a Hohmann transfer. If it hadn't flown by Jupiter, the orbit would have had an apoapsis of almost $9\,\mathrm{AU}$.

Answer 2

edit: @MarkAdler's answer's new edit is conclusive. Here is some additional perspective.

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In short - the orbits were elliptical, but already had a semi-major axis much larger than a Hohmann ellipse to Jupiter. The orbits were carefully aimed nearly at Jupiter but slightly behind it as it moved in its orbit.

The close flybys would boost their speeds even more and give them more range. $\Delta v$ from Voyager 1's flyby of Saturn (after Jupiter's flyby) to tilt it's orbital plane far out of the ecliptic. That allowed it to take valuable data at Jupiter first, then to explore the solar system environment out of the ecliptic plane past Jupiter as well. Voyager 2 used four flyby Maneuvers, at Jupiter, Saturn, Uranus, and Neptune, and used the last one to also move out of the ecliptic plane.

You can see a little 3D map-GIF of their orbits below - originally from here. The farthest orbit is that of Pluto, you can see on this scale that Jupiter is relatively close to the Sun.

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below: data for the Sun, planets, Pluto, Voyager 1 and Voyager 2, from January 1, 1969 (a good year to start things) until now. Dots are now. Data is from NASA JPL Horizons. Reposted from this question, where additional details are given.

below: A diagram of the trajectories that enabled NASA's twin Voyager spacecraft to tour the four gas giant planets and achieve velocity to escape our solar system. From here.