# The Last Days of Cassini: the Grand Finale

### A.  Introduction

A central premise of Physics from Planet Earth is that students will be more motivated to study physics if their coursework addresses important and newsworthy scientific events.  A good example is the Cassini-Huygens mission to Saturn, which ended dramatically on September 15, 2017 when the spacecraft dipped into the Saturnian atmosphere and self-destructed — its grand finale.  Cassini‘s fiery demise was carefully planned by NASA seven years ago (Ref. 1), when the spacecraft’s fuel supply began to run low, limiting opportunities for further exploration of Saturn and its surroundings.  Flight engineers chose to end the mission by crashing into Saturn to eliminate any possibility of the spacecraft contaminating Enceladus or Titan, two moons of Saturn that potentially harbor life.

Cassini was launched from Cape Canaveral 20 years ago.  After 4 gravity assists (2 from Venus, 1 from Earth, and 1 from Jupiter), it entered orbit about Saturn in July 2004, the first spacecraft ever to do so.  The ESA’s (European Space Agency) Huygens spaceprobe was dispatched from Cassini on Christmas day, 2004, and soft-landed on Titan three weeks later, the most distant landing ever achieved by a spacecraft from Earth.  Over the next 13 years, Cassini discovered 7 new moons of Saturn (bringing the total to 62) and conducted close flybys of 13 moons.  To visit them, it employed gravity assists, or “targeted flybys,” to alter its flight trajectory with minimal expenditure of fuel.  Most of these flybys — 127 in all — involved Titan, Saturn’s largest moon and the only one with sufficient mass to alter Cassini‘s trajectory significantly.  By taking advantage of gravity assists, NASA flight engineers were able to extend the mission by 9 years.  Their efforts were amply rewarded.  Together, Cassini and Huygens revolutionized our understanding of the Saturnian system.  In addition to providing stunning close-up images of the planet, its rings and moons, the two spacecraft discovered subsurface water oceans on Titan, Enceladus and Dione, surface oceans of methane (CH4) on Titan, and gaseous plumes  containing H2O, CO2, CH4 and H2 emanating from Enceladus, making the latter the most promising candidate in our solar system for hosting extraterrestrial life.

Prior to its “grand finale” orbits, Cassini used a targeted flyby of Titan (designated T125) to enter a “ring-grazing” orbit, with periapse just outside Saturn’s F ring.  After completing 20 ring-grazing orbits, the spacecraft executed its final targeted flyby (T126) of Titan on April 22, 2017, and was inserted into a daring trajectory that — near its periapse — passed between the cloud tops of Saturn and its innermost ring, a gap only 3000 km wide.  Over the next 5 months, Cassini executed 22 grand finale orbits, until a high altitude flyby of Titan on September 12 caused the spacecraft to plunge into Saturn’s atmosphere three days later.

In this post, we focus on T126 to learn how Cassini was transferred from its last ring-grazing orbit to the first of its grand finale orbits.  This is done via a set of homework exercises and/or exam problems suitable for first year university physics students.  We gratefully acknowledge the generous help of Dr. Duane Roth, Cassini Navigation Team Chief, who kindly forwarded Reference 1 and provided detailed instructions for using the online software package WebGeoCalc (Ref. 2) to calculate the trajectory of Cassini relative to Saturn and Titan, both before and after T126.  We drew heavily WebGeoCalc’s calculations to write this post.  Thank you, Dr. Roth!

Cassini‘s grand finale was a headline-grabbing event, and perfectly timed for inclusion in the fall semester syllabus of an introductory mechanics course.  We hope your students will enjoy learning the physics behind the headlines.  Farewell, Cassini!  Right up to your final moments, you performed brilliantly!

### B.  The T126 flyby

In the exercises below, the T126 flyby should be treated as an elastic “collision” between Cassini and Titan.  The following information will be useful for solving the problems.

Titan is in a near-circular orbit (ε = 0.0288) about Saturn, well-aligned to the planet’s equatorial plane, with period $\boldsymbol{P_{T}=1.379\times 10^{6}}$ s (16.0 d).  Using Kepler’s 3rd law, its semi-major axis is found to be $\boldsymbol{a_{T}=1.222\times 10^{6}}$ km.  Prior to the T126 flyby, Cassini was in a highly eccentric orbit (ε = 0.795) about Saturn with inclination about 60° relative to the equatorial plane, period $\boldsymbol{P_{C}=6.205\times 10^{5}}$ s (7.2 d) and semi-major axis $\boldsymbol{a_{C}=7.178\times 10^{5}}$ km.  The flyby occurred at a distance $\boldsymbol{r=1.206\times 10^{6}}$ km from Saturn, not far from the spacecraft’s apoapse.  (See Figure 2.)  At the encounter point, the spacecraft crossed the plane of Titan’s orbit from below to above.  As a result of the flyby, Cassini entered an even more eccentric orbit (ε = 0.906) which, at periapsis, passed between the planet and its innermost ring.

Exercises for students:

1. Titan is in a near-circular orbit about Saturn with period $\boldsymbol{P_{T}=1.379\times 10^{6}}$ s.  The T126 flyby occurred when Titan was $\boldsymbol{1.206\times 10^{6}}$ km from Saturn.  Find its speed at this time.  Use $\boldsymbol{GM_{S}=3.793\times 10^{7}\:\mathrm{km^{3}/s^{2}}}$.  (Ans: $\boldsymbol{v_{T}=5.645\: \mathrm{km/s}}$)

2.  Just before the flyby, Cassini was $\boldsymbol{1.19\times 10^{6}\: \mathrm{km}}$ from Saturn and moving with speed $\boldsymbol{v_{C}=3.291\: \mathrm{km/s}}$.  The eccentricity of its orbit was 0.795.  Show that, at periapsis, the spacecraft was just outside Saturn’s F ring, which has a radius of about $\boldsymbol{1.40\times 10^{5}\: \mathrm{km}}$.

3.  Titan’s orbit is within 0.3° of the plane of Saturn’s rings, whereas Cassini‘s orbit is tilted by about 60°.  As shown in this video,  Entering Final Orbits: Cassini’s Grand Finale (Ref 3), Cassini passed in front of Titan during T126.  (The encounter occurs 20 s from the beginning of the video.)  Let $\boldsymbol{\vec{u}}$ and $\boldsymbol{\vec{{u}'}}$ be the velocities of the spacecraft relative to Titan before and after the flyby.  Make a rough sketch of the velocity vectors $\boldsymbol{\vec{v}_{T}}$$\boldsymbol{\vec{u}}$ and $\boldsymbol{\vec{{u}'}}$, and explain why the spacecraft’s speed decreased due to the flyby.  No calculations are necessary.

4.  In a reference frame whose x– axis is parallel to $\boldsymbol{\vec{v}_{T}}$ , and whose z-axis is perpendicular to the moon’s orbit, the velocity of Titan just before the flyby may be expressed as $\boldsymbol{\vec{v}_{T}=5.645\, \hat{i}\: \mathrm{km/s}}$, and the velocity of Cassini as $\boldsymbol{\vec{v}_{C}=1.161\, \hat{i}-0.701\, \hat{j}+2.998\, \hat{k}\: \mathrm{km/s}}$.   Calculate the angle between the two vectors and also find the relative velocity vector $\boldsymbol{\vec{u}=\vec{v}_{C}-\vec{v}_{T}}$.  (Ans: 66°)

5.  As Cassini approached Titan, the angle between the relative velocity vector $\boldsymbol{\vec{u}}$ and $\boldsymbol{\vec{v}_{T}}$ was 145.5°.  The flyby deflected the direction of the relative velocity by 9.1°.  Show that this reduced the spacecraft’s speed relative to Saturn by about 850 m/s.

6.  True or false:  the change in the speed of the spacecraft is equal to $\boldsymbol{\left | {\vec{v}}'_{C}-\vec{v}_{C} \right |}$, i.e., the absolute value of the change in the velocity of the spacecraft.

7.  At its closest approach to Titan, Cassini was only 980 km above the surface of the moon, or 3554 km from its center.  What was its speed relative to Titan at this time?  (Ans:  $\boldsymbol{u_{max}= 5.880 \: \mathrm{km/s}}$)

8.  The unfueled mass of Cassini was 2125 kg, and the exhaust speed of its MMH/NTO fuel mixture is 2.8 km/s.  What is the minimum mass of fuel that would have been needed to change the spacecraft’s speed (relative to Saturn) by 850 m/s?  Assume that the maneuver totally depleted the fuel supply.  (Ans: 750 kg)

9.  In its new trajectory (after T126), Cassini made 22 full orbits with period $\boldsymbol{5.565\times 10^{5}\: \mathrm{s}}$ before a long distance encounter with Titan caused it to plunge into the Saturnian atmosphere.  Why 22 orbits?  (Hint: compare the periods of Cassini and Titan.)

The following two exercises can be completed after studying Chapter 12 of PPE or similar material.

10.  Prove that in the reference frame of Titan, the spacecraft was deflected by 9.1° due to T126.  (This is the angle between $\boldsymbol{{\vec{u}}'}$ and $\boldsymbol{\vec{u}}$.

11.  On September 12, Cassini came within $\boldsymbol{1.2\times 10^{5}\: \mathrm{km}}$ of Titan, and its relative speed at closest approach was 5.2 km/s.  Estimate the angle by which the spacecraft was deflected (in Titan’s reference frame), and the maximum change of speed relative to Saturn.  This change in speed was enough to ensure that the spacecraft entered Saturn’s atmosphere at periapse.

### References

1.  Brent Buffington et al, IAC-10-C1.9.2, 61st International Astronomical Congress, Prague, CZ (2010)

3.  Entering Final Orbits: Cassini‘s Grand Finale.  Video is available courtesy of NASA/JPL at https://saturn.jpl.nasa.gov/resources/7508/?category=videos .