De-orbiting a satellite

The same authors mentioned earlier, Philip Blanco and Carl Mungan, have written a stimulating paper entitled “Satellite splat: an inelastic collision with a surface-launched projectile,” (Eur. J. Phys. 36 (2015) 045004).  Blanco and Mungan consider the case of a totally inelastic collision between a satellite in circular orbit about a planet and a projectile launched vertically from the surface of that planet.  What are the conditions such that the combined object “de-orbits,” i.e., falls to the planet’s surface?  The paper contains a detailed analysis, but since our goal is to provide homework and exam exercises appropriate for introductory physics students, we will consider only the simplest cases.

Denote the satellite mass and orbital radius by m and r, while the planetary mass (pictured as Earth) and radius are M and R.  Let m’ be the projectile mass, and for simplicity, set the projectile launch speed to be the minimum speed sufficient to intercept the satellite; i.e., when m’ reaches the orbital radius r, its speed is zero.  See the figure. Blog 2 figure

  1. What is the satellite speed before the collision? What is its speed immediately after the collision?
  2. After the collision, the body follows an elliptical trajectory. Is the collision point its apogee, perigee, or neither?  Explain.
  3. Suppose m’ is just sufficient to bring down the satellite. Where would it land?  Describe the combined body’s trajectory from the collision in space to touchdown on the planet’s surface.

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Rosetta and Philae

Philae-landing-1_Ken-Kremer
The spaceprobe Philae descending to the surface of comet 67P after its deployment from the spacecraft Rosetta.  Image credit:  ESA/Rosetta/MPS

On March 2, 2004, the Rosetta spacecraft was lofted into space from the European Space Agency’s (ESA) launch facility in Kourou, French Guiana, located on the northeast coast of South America.  The spacecraft’s ultimate goal was to rendezvous with Comet 67P/Churyumov-Gerasimenko, land a scientific probe onto its surface, and monitor the comet’s activity as it approached the Sun.  After 3 fly-bys of Earth and 1 fly-by of Mars, all of which took 10 years to accomplish, Rosetta reached the comet in August, 2014, entering a circular orbit about it in October of that year.  Then, on November 12, it de-orbited and “dive-bombed” the comet, deploying the Philae lander at a distance of 22.4 km above the comet’s center of mass. Once released, the lander fell to the surface ballistically.  Philae was equipped with harpoons, ice screws, and a pressurized-gas thruster to help “stick” its landing.  But these systems failed, and the probe bounced twice before coming to rest at a shady spot far from its intended touchdown site, where the sunlight needed to charge its batteries is limited.  Since then, communication with Philae has been sporadic, and will soon cease altogether as comet 67P, Philae and Rosetta recede from the Sun.   In spite of the setback, Rosetta and Philae achieved the first-ever soft landing on a comet and discovered a rich brew of organic compounds and molecular oxygen, findings of great significance for understanding the origin of life on Earth.

For our purposes, the Rosetta mission and the flight of Philae offer a treasure trove of potential exercises for the introductory physics course.  The following exercises were inspired by the excellent article written by Philip Blanco: “Rosetta’s Mission’s ‘7 Hours of Terror’ and Philae’s Descent,” The Physics Teacher 53, 339 (September 2015).  See also the follow-up letters by Carl Mungan (TPT 53, 452 (November 2015)) and Blanco (TPT 53, 516 (December 2015) and the online supplementary materials referenced therein).  Blanco uses an Excel spreadsheet to calculate the velocity vs. time and the overall elapsed time of Philae’s descent, but numerical analysis is not necessary to determine the impact velocity of the lander.  We welcome your suggestions for other introductory-level problems based on Rosetta and Philae.

  1. Kourou is located at latitude 5°10′ N. Why is this location nearly ideal for the launch of spacecraft, especially geosynchronous satellites?  (Hint: rockets are usually launched to the east.)
  2. Comet 67P orbits the Sun with period 6.45 yr and eccentricity \large \varepsilon =0.640. Find its perihelion and aphelion distances.  What is its speed at perihelion?  Using only these results, find the speed at aphelion.  (Ans: \large r_{p}=1.24 \textup{ AU}\large r_{a}=5.68\textup{AU}\large v_{p}=34.4\textup{ km/s}\large v_{a}=7.55\textup{ km/s})
  3. On August 6, 2014, when Rosetta first caught up with 67P, it was 3.601 AU from the Sun. What was the speed of the comet at that time?  Find the angle θ between the comet’s velocity and its position vector \large \vec{r} relative to the Sun.   (Ans:  \large v=15.8\textup{ km/s}, \large \theta =131^{\circ})

Rosetta triangular trajectory

Before entering unpowered orbit about the comet, Rosetta traced out a triangular trajectory, firing its onboard thrusters at the end of each 100 km leg to change its direction by 120°.  (See the figure to the right.)  These maneuvers allowed scientists and flight controllers to transition to an unpowered (Keplerian) orbit about the comet, and also to determine its mass.  In the following questions, take the comet mass to be M = 1.0 x 1013 kg, and assume that the triangle is centered on 67P.

  1. Rosetta was moving at approximately 60 cm/s along each leg of the triangle. Show that this was above the escape velocity everywhere along the leg.  (Ans: \large r_{min}=29\textup{ km}\large v_{esc}(max)=\sqrt{2GM/r_{min}}=21\textup{ cm/s})Presentation1
  1. Difficult. The comet’s mass M can be computed by measuring the change in direction of the spacecraft as it traverses one leg of the triangle, due to the gravitational pull of the comet.  Using the figure shown, calculate the deflection angle φ in radians or degrees.  (Hint:  Assume the deflection is small, so the spacecraft travels along a nearly straight line at constant speed: \large v_{x}\simeq \textup{const}=0.6\textup{ m/s}, and \large y\simeq \textup{const}=50\cot 60^{\circ}=29\textup{ km}.  Use  \large y=r\sec \theta and \large x=y\tan \theta to describe Rosetta’s location.)  (Ans: \large \varphi =0.11\textup{ (rad)} = 6.3^{\circ})

In the following questions, assume for simplicity that 67P is spherically symmetric with radius 2.393 km.

  1. Prior to deploying Philae, Rosetta was in a circular orbit of radius 30 km. Calculate the spacecraft’s speed and its orbital period.  (Ans: \large v=15\textup{ cm/s}, \large T=14.6\textup{ d})
  2. Philae was released from a point 22.63 km above the comet’s CM with an initial radial velocity \large v_{r0}=-.766\textup{ m/s} and azimuthal velocity \large v_{\theta 0}=.055\textup{ m/s}.  Calculate \large v_{r} and \large v_{\theta } at touchdown.  (Ans: \large v_{r}=-.902\textup{ m/s}, \large v_{\theta }=.525\textup{ m/s})

The Philae lander can be modeled crudely as a cylinder of mass 100 kg, radius ~0.5 m and height ~0.8 m. Its orientation was stabilized about its central axis by a rapidly spinning “reaction wheel” with angular momentum 5.17 N-m/s.  Just before reaching the comet surface, the spacecraft was rotating about its central axis at a rate 3.3 mHz (\large \omega =6.6\pi \times 10^{-3}\textup{ s}^{-1}).  After its initial touchdown, the reaction wheel was switched off, increasing Philae’s rotation rate to 77 mHz during its first bounce.

  1. Calculate the moment of inertia of the lander. (Ans: \large I_{Ph}=11 \textup{ kg-m}^{2})
  2. Check your answer by comparing it to the moment of inertia of a solid cylinder having the mass and dimensions given above. (Ans: \large I_{cyl}=12.5 \textup{ kg-m}^{2})
  3. During its descent, Philae deployed its landing gear: three legs each with approximate length 2 m. This slowed its rotation rate to 2.0 mHz.  Estimate the mass of each leg.  (Ans: \large m_{leg}\approx 2\textup{ kg})