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Gravity Assist for Interstellar Flight

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Gravity assist, also known as gravitational slingshot or swing-by is the use of gravity of a heavenly body, and the relative motion between the heavenly body and a spacecraft in-order to alter the path and speed of a spacecraft without requiring any propellant. 

Animation of Cassini Trajectory
Animation of Cassini Trajectory (Source: Phoenix 777)

Gravity assist is the most efficient method to successfully accomplish large distance interplanetary / interstellar missions, because it just requires the proper application of two basic laws of physics: momentum conservation and energy conservation. The phenomenon of gravitational slingshot can be considered to be the result of non-contact elastic collision between a heavenly body and a spacecraft.

The Soviet probe Luna 3 first used the gravity assisted maneuver in 1959 when it photographed the far side of Earth’s Moon. Since then, several notable slingshots have been done. Among them, Cassini mission to Saturn made the use of Slingshot effect four times with the space craft gaining the energy equivalent to 75 tons of rocket fuel. Launched on October 15, 1997, Cassini passed by Venus twice, then Earth, and finally Jupiter on its way to enter Saturn orbit on July 1, 2004.

Gravity assists in Cassini mission.
Gravity Assists in Cassini Mission (Source : NASA / Jet Propulsion Laboratory – Caltech)

Following section deals with the simple physical principles involved in gravity assist.

Momentum Conservation

Because of momentum conservation, the enormously massive heavenly body imparts acceleration (or deceleration) to the aircraft without the measurable change in its own. Let m and M be the masses of the spacecraft and the heavenly body respectively, and  (v_{i}, V_{i}) and  (v_{f}, V_{f}) be their respective velocities before and after the encounter. Then,

    \begin{equation*}      mv_i + MV_i = mv_f + MV_f  \end{equation*}

    \begin{equation*}      V_f - V_i = \frac{m}{M} (v_i - v_f)  \end{equation*}

Since m/M \approx 0

(1)   \begin{equation*}      V_f = V_i = V  \end{equation*}

Energy Conservation

Applying energy conservation in heavenly body frame, the spacecraft’s approach velocity  \Vec{u_i} = \Vec{v_i} - \Vec{V_i} relative to the body is deflected by the gravitational pull to relative departure velocity  \Vec{u_f} such that

(2)   \begin{equation*}     \mid \Vec{u_i} \mid = \mid \Vec{u_f} \mid \end{equation*}

Then in the space frame,

(3)   \begin{equation*}     \Vec{v_f} = \Vec{u_f} + \Vec{V} \end{equation*}

Accelerating Gravity Assist (Left) and Decelerating Gravity Assist (Right)

Consider the triangles in above figure (left), i.e. the case when the spacecraft passes behind the heavenly body. The constant velocity  \Vec{V} of the heavenly body is represented by the common base. In the heavenly body’s frame, the spacecraft’s approach velocity  \Vec{v_i} - \Vec{V_i} (dashed) is rotated as arrowed by the gravity to the departure velocity with unaltered magnitude since energy is conserved. Addition of  \Vec{V} gives  \Vec{v_f} .


 \mid \Vec{v_f} \mid > \mid \Vec{v_i} \mid

However, if the spacecraft is made to pass in front of the planet (figure to the right), we can clearly see that the speed is lost, i.e.  \mid \Vec{v_f} \mid < \Vec{v_i} \mid .

Interstellar Flight with Gravity Assist

For an interstellar travel we require a tremendous amount of  $v_f$ , and none of the heavenly bodies within our solar system can feasibly propel us enough. However in 1963, a legendary physicist Freeman Dyson proposed a theory for interstellar travel using the gravity assist from the binary system of white dwarf stars, which not only have huge mass and density but also spin much faster.

Gravity Assist with Binary System of White Dwarf Stars

For a binary star system with equal masses,

(4)   \begin{equation*}     V = \sqrt{\tfrac{GM}{4R}} \end{equation*}

If we assume white dwarfs of one solar mass and radius = 20,000 km each, we get  V = $1.3*10^6$ m/s. If the spacecraft could leave the system with 2V+u such that  u $\ll$ V , it will be moving almost at 1 % the speed of light. Since the nearest star Proxima Centauri B is at about 4.24 light years away, the speed will get us to it under 500 years. It is in-fact a small time (if we depended on Deep Space 1, it would take around 83,000 years!).

If we rather depend on the binary system of neutron stars instead of white dwarfs, then with their radii equal to 20 km and mass equal to one solar mass, the slingshot would provide spacecraft the speed of 0.27c. This speed can get us to the nearest star by 15.7 years!

Another Beauty : Time Dilation

15.7 years is measured by an observer on earth. At the speed of 0.27c, Lorentz factor

\gamma = \tfrac{1}{\sqrt{1-\tfrac{v^2}{c^2}}} = 1.04, and relativistic effect comes to play.
The person on the space craft would measure the contracted length of \tfrac{4.24}{\gamma} = 4.077 light years, and feels that he completed the mission in 15.1 years. That way, the person on flight would be 7.2 months younger than the person on earth!

Concluding Remarks

These binary systems are the theoretical gravitational machines which Dyson envisioned an advanced future civilization would be able to actually create and put them on space! For our current level of technological advancement, we can safely say that using gravity assist for interstellar travel is unfeasible.

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Sandesh Parajuli

Sandesh Parajuli

Sandesh is a founder of Cherrubics. He is an undergraduate aerospace major at Department of Mechanical and Aerospace Engineering, Pulchowk Campus, Nepal.

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