# Rosetta’s Final Act: Free-Fall to the Surface of Comet 67P

The European Space Agency’s historic Rosetta mission is over.  On September 30, 2016, the spacecraft was allowed to fall ballistically (unpowered) to the surface of comet 67P/Churyumov-Gerasimenko, capturing numerous high-resolution photos of the comet’s surface during its 14 hour descent.  The final image, taken from an altitude of 20 meters, is shown in Figure 2.

Comets are presumed to be the primordial building blocks of the solar system, so by observing 67P at close range, Rosetta was investigating the origins of our solar system.

During its 12 year lifespan, Rosetta became the first spacecraft to orbit a comet and observe the chemical processes taking place as it passed through perihelion in August 2015, when the rate of gas and dust evolution peaked.  In November 2014, Rosetta was the first spacecraft to deploy a lander, Philae, to the surface of a comet.  Although Philae functioned only briefly, the two allied space probes made numerous discoveries about the chemical makeup and geology of comets.

Comet 67P has a period of 6.45 years and an aphelion distance of 5.7 AU.  At that distance, Rosetta‘s onboard heating system would have been incapable of keeping the spacecraft warm enough for its instruments to hibernate safely.  Consequently, flight directors opted to end the mission by crash-landing the spacecraft onto the comet, gathering as much data as possible while its instruments were still functioning.  Today, 67P is 4.3 AU from the Sun.  As it heads toward aphelion in November, 2018, it carries the remains of Rosetta and Philae, the two pioneering spacecraft that doggedly chased it for over a decade.

In this blog post, we present a set of student-ready exercises based on Rosetta‘s final maneuver and descent to the surface of 67P.  These exercises were inspired by Dr. Philip Blanco’s recent letter (Ref. 1) to the editor of The Physics Teacher.  Both Philip and I are deeply indebted to Pablo Muñoz, flight dynamics engineer at the ESA’s European Space Operations Centre in Darmstadt, Germany.  Pablo supplied indispensible data on the position and velocity of the spacecraft before and after its final thruster burn, plus the mass of Rosetta and the amount of fuel consumed during that final burn.  Without his input, none of these exercises could have been written.

To make the exercises suitable for introductory students, I have modified the details slightly, as indicated below.   To fully appreciate the complexity of Rosetta‘s dance with its cometary partner, follow the links in Ref. 4 and 5 to the beautiful ESA videos of the mission.  Have fun!

## A.  Problems

In these exercises, $\boldsymbol{ GM=666.15\mathrm{ m^{3}/s^{2}}}$, where
$\boldsymbol{ M}$ is the mass of the comet ($\boldsymbol{ M\simeq 1.0\times 10^{13}} \textup{ kg}$), and model the comet as a homogeneous sphere of radius 2.00 km.

Prior to its plunge to the comet surface, Rosetta was placed in an unpowered elliptical orbit with apocenter (farthest distance to the comet’s CM) $\boldsymbol{r_{a}=23.44}\textup{ km}$ and pericenter $\boldsymbol{r_{p}=13.58}\textup{ km}$.  This is shown nicely in Ref. 4.

1.  Calculate the specific energy $\boldsymbol{ E/m}$ (where $\boldsymbol{m}$ is Rosetta‘s mass) and the orbital period of the spacecraft.  (Ans: $\boldsymbol{E/m=-1.800\times 10^{-2}}\textup{ J/kg}$$\boldsymbol{ T=6.13\times 10^{5}}\textup{ s}\simeq 1\textup{ week}$)

2.  What was the speed $\boldsymbol{ v_{0}}$ of the spacecraft at apocenter? Compare this to the speed of a spacecraft in low Earth orbit.  Why the big difference?  (Ans: $\boldsymbol{ v_{0}=0.1444}\textup{ m/s}$$\boldsymbol{M_{67P}/M_{Earth}\sim 10^{-12}}$)

The final maneuver that sent Rosetta diving to the comet surface occurred at a true anomaly of 208º, i.e., 28º beyond apocenter (0º corresponds to pericenter).  (See Ref. 4.)  To streamline the following questions, assume instead that the maneuver took place exactly at apocenter.  This alters the results modestly.

3.   Immediately after the burn, Rosetta‘s velocity was $\boldsymbol{ \vec{v}{_{0}}'=-0.0211\hat{i}-0.3255\hat{j}}\textup{ m/s}$, where $\boldsymbol{ \hat{i}}$ and $\boldsymbol{ \hat{j}}$ refer to the radial and azimuthal directions (Ref. 2) at apocenter.  See Figure 3.  Show that the burn inserted the spacecraft into an unbounded hyperbolic trajectory.  (Ans: $\boldsymbol{ {E}'/m=2.478\times 10^{-2}}\textup{ J/kg}>0$)

4.  Note that $\boldsymbol{ {\vec{v}}'_{0}}$ is not perfectly radial, i.e., directed toward the comet’s CM.  Convince someone that the spacecraft will still strike the comet.  This question and the solution below were suggested by Philip Blanco.          (Ans:  Compare the angle subtended by the comet at apocenter to the angle between the velocity vector $\boldsymbol{{\vec{v}}'_{0}}$ and the radial direction.)

5.  With what speed $\boldsymbol{ v}$ will the spacecraft strike the surface?  Compare this to the surface escape speed.  (Ans: $\boldsymbol{ v=0.846}\textup{ m/s}$$\boldsymbol{ v_{esc}=0.816}\textup{ m/s}$)

6.  Find the spacecraft’s velocity $\boldsymbol{ \vec{v}=v_{x}\hat{i}+\vec{v}_{y}\hat{j}}$ just before it strikes the surface.  Hint: first conserve angular momentum.  (Ans: $\boldsymbol{ \vec{v}=-0.809\hat{i}-0.247\hat{j}}\textup{ m/s}$)

7.  Calculate the magnitude of the change in velocity caused by the thruster burn.  The mass of the spacecraft at this time was 1422.5 kg, and the burn consumed 0.2 kg of rocket fuel.  What was the effective exhaust velocity of the fuel (a mixture of monomethyl hydrazine (MMH) and nitrogen tetroxide ($\large \mathrm{N_{2}O_{4}}$ or NOX))?  (Ans: $\boldsymbol{ \Delta v_{0}=.365}\textup{ m/s}$$\boldsymbol{ v_{ex}=2.6}\textup{ km/s}$.  This is substantially less than the “published” value of 3.35 km/s (Ref. 3) for MMH/NOX because a fuel leak forced flight engineers to operate the thrusters at lower than optimal pressure.)

8.  In September 2016, comet 67P has a rotation period of 12.055 hours.  This is less than its period in 2014 (12.4 hours), when Philae was released, due to outgassing from the comet as it passed through perihelion.   Assume that the comet’s rotation axis was perpendicular to the plane of the spacecraft’s motion (Figure 3).  Rosetta‘s velocity $\boldsymbol{ \vec{v}_{0}{}'}$ was selected so that its impact velocity relative to the rotating surface was nearly vertical, i.e., perpendicular to the impact surface.  (Hence, an observer standing on the surface at the impact point would have seen the spacecraft descending vertically.)  Show that this is roughly consistent with your calculated value of $\boldsymbol{v_{x}}$ in Question 6.  (Note: because the comet has such an irregular shape, the local vertical direction was significantly different from the radial direction.) (Ans:$\boldsymbol{ v_{rot}=-\omega R=-29.0\textup{ cm/s} \approx v_{x}=-24.7\textup{ cm/s}}$)

The following question is suitable for users of Physics from Planet Earth who have studied Section 12.9: Application: the “Slingshot Effect” Revisited, or who have seen a comparable treatment elsewhere.

9.  Suppose Rosetta‘s final maneuver had been incorrectly executed, and the spacecraft just missed striking the comet, instead grazing its surface as it flew by.

a.  From Question 3, we know that Rosetta‘s speed after the burn was 0.3262 m/s at apocenter.  In the comet’s reference frame, what will be its speed after it has flown past the comet and is very far from it?  (Ans: $\boldsymbol{ v_{\infty }=0.2226\textup{ m/s}}$)

b.  By what angle Θ will the spacecraft be deflected after its close fly-by of the comet?  (Ans:  $\boldsymbol{ \tan (\Theta /2)=1.769}$$\boldsymbol{ \Theta =121^{\circ}}$.  See Eqn. 12.12 of PPE.)

## B.  Sphere of Influence of Comet 67P

Here is an additional “space-themed” topic that is quite suitable for discussion in an introductory mechanics course.  As far as I know, it is not discussed in any currently available textbook (including PPE).  I thank Philip Blanco for suggesting it.

So far, we have ignored the influence of the Sun on the motion of Rosetta about comet 67P.  We might justify this by invoking Einstein’s Principle of Equivalence (see Section 7.5 of PPE) which states that in a free-fall reference frame, the local effects of gravity disappear.  Comet 67P’s heliocentric orbit is free-fall motion about the Sun, so in the comet’s reference frame, we can ignore the Sun’s gravitational force on the comet (which is obvious).  However, there is a small but important complication when we consider the motion of the spacecraft in the same reference frame.  Since the Sun’s gravitational force varies inversely with the square of the distance, the spacecraft’s acceleration toward the Sun varies as the distance between it and the Sun changes along its orbit.  Let $\boldsymbol{ R}$ be the distance from the Sun to the comet, and $\boldsymbol{ r}$ be the distance from the comet’s CM to the spacecraft.  See Figure 4.  When the spacecraft is at position a(b) in the figure, its distance from the Sun is a maximum (minimum), and its acceleration due to the Sun is

$\LARGE \boldsymbol{a=\frac{GM_{Sun}}{(R\pm r)^{^{2}}}\simeq \frac{GM_{Sun}}{R^{2}}\left ( 1\mp \frac{2r}{R} \right )}$.

(1)

The acceleration of the comet toward the Sun is $\boldsymbol{ {a}'=GM_{Sun}/R^{2}}$, so the maximum difference between the spacecraft’s acceleration and that of the comet is $\boldsymbol{ \Delta a_{max}=2GM_{Sun}\, r/R^{3}}$.  How close is the spacecraft’s motion to an unperturbed Keplerian orbit about the comet?  The answer depends on the ratio between $\boldsymbol{ \Delta a_{max}}$ and the centripetal acceleration of the spacecraft toward the comet, $\boldsymbol{ a_{c}=GM/r^{2}}$, where $\boldsymbol{ M}$ is the comet mass.  The smaller this ratio is, the smaller the influence of the Sun, and the better the motion can be described as an unperturbed Keplerian orbit about the comet.

Now let’s ask the opposite question.  Imagine the spacecraft is far enough from the comet that its motion is better described as a heliocentric orbit perturbed slightly by the comet.  How close is its motion to a pure heliocentric orbit?  The answer depends on the ratio between the acceleration $\boldsymbol{ a_{c}}$ toward the comet and the acceleration toward the Sun $\boldsymbol{ {a}'}$:

$\LARGE\boldsymbol{ \frac{a_{c}}{{a}'}=\frac{M}{M_{Sun}}\frac{R^{2}}{r^{2}}}$.

(2)

The smaller this ratio is, the smaller the influence of the comet, and the better the motion can be described as a Keplerian orbit about the Sun.  The “sphere of influence” of the comet is the region within which the motion of the spacecraft is only weakly perturbed by the Sun.  By convention, its radius $\boldsymbol{ r_{soi}}$ is found by equating the ratios in Eqns. 1 and 2, and solving for $\boldsymbol{ r\textup{ }\: (=r_{soi})}$:

$\LARGE\boldsymbol{\frac{2M_{Sun}}{M}\frac{r^{3}}{R^{3}}=\frac{M}{M_{Sun}}\frac{R^{2}}{r^{2}}}$

or

$\LARGE\boldsymbol{ \frac{r^{5}}{R^{5}}=\frac{1}{2}\left ( \frac{M}{M_{Sun}} \right )^{2}}$.

Ignoring the factor $\boldsymbol{ (1/2)^{1/5}=0.87\approx 1}$, we obtain the radius of the sphere of influence:

$\LARGE\boldsymbol{ r_{soi}=R\left ( \frac{M}{M_{Sun}} \right )^{\frac{2}{5}}}$.

(3)

10.  a.  Philae was released from Rosetta when comet 67P was about 3 AU from the Sun, from an initial distance 22.5 km from the comet’s CM.  Show that the spacecraft was well within the comet’s sphere of influence at that time.  (Ans:  $\boldsymbol{ r_{soi}=54\textup{ km}}$)

b.  At perihelion (August 13, 2015), 67P was 1.24 AU from the Sun.  To escape possible damage from the comet’s efflux, Rosetta was removed to a distance of 330 km.  Was it in an unpowered orbit about 67P at that time?  (Ans: No, $\boldsymbol{ r_{soi}=22\textup{ km}}$.  See Ref. 5, fast-forward to August 2015.)

## C. References

1a.  “Modeling Rosetta’s final descent,” Philip Blanco, The Physics Teacher 54, 516 (2016)

2a.  If students are familiar with polar coordinates, substitute $\boldsymbol{ \hat{r}}$ and $\boldsymbol{ \hat{\theta}}$ for $\boldsymbol{ \hat{i}}$ and $\boldsymbol{ \hat{j}}$.