Landing spacecraft on Mars and other planets: An opportunity to apply introductory physics Paul Withers Citation: Am. J. Phys. 81, 565 (2013); doi: 10.1119/1.4811215 View online: http://dx.doi.org/10.1119/1.4811215 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v81/i8 Published by the American Association of Physics Teachers Additional information on Am. J. Phys. Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html Downloaded 19 Jul 2013 to 128.197.73.19. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Landing spacecraft on Mars and other planets: An opportunity to apply introductory physics Paul Withersa) Astronomy Department, Boston University, 725 Commonwealth Avenue, Boston, Massachusetts 02215 (Received 3 January 2012; accepted 31 May 2013) The Curiosity rover safely landed on Mars after “seven minutes of terror” passing through the Martian atmosphere. In order to land safely, Curiosity had to decelerate from speeds of several kilometers per second and reach zero speed exactly upon touching down on the surface. This was accomplished by a combination of atmospheric drag on the enclosed spacecraft during the initial hypersonic entry, deployment of a large parachute, and retrorockets. Here, we use the familiar concepts of introductory physics to explain why all three of these factors were necessary to ensure a safe landing. In particular, we analyze the initial deceleration of a spacecraft at high altitudes, its impact speed if a parachute is not used, its impact speed if a parachute is used, and the duration of its descent on a parachute, using examples from Curiosity and other missions. VC 2013 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4811215] I. INTRODUCTION the spacecraft per unit time, is qAv2. This is the aerodynamic force acting on the spacecraft, a result that is surprisingly The fiery entry of a spacecraft into a planetary atmos- accurate. Even with sophisticated numerical simulations, the phere—whether the return of astronauts to Earth, the landing aerodynamic force on a spacecraft is typically within a few of Curiosity on Mars (Fig. 1), or the descent of Huygens to tens of percent of qAv2. Neglecting the effects of gravity on Titan’s surface—attracts immense public interest. It also the spacecraft (mass m), which is reasonable in the initial offers an opportunity to enliven an introductory physics stages of atmospheric entry, gives the drag equation class, since many fundamental aspects of the behavior of such spacecraft can readily be explained by familiar physical dv m ¼qAv2: (1) principles. dt Here, we analyze the initial deceleration of a spacecraft at high altitudes (Sec. II), its impact speed if a parachute is not Balancing the forces acting vertically on a small parcel of used (Sec. III), its impact speed if a parachute is used (Sec. gas in the background atmosphere leads to IV), and the duration of its descent on a parachute (Sec. V). pðzÞ¼pðz þ DzÞþqgDz, where p is the atmospheric pres- Although the concepts used in this article are commonly dis- sure, z is the altitude, and g is the acceleration due to gravity. cussed in introductory physics classes, we note that the math- This consideration of forces, which is underpinned by the ematics is at a slightly higher level. principle of conservation of momentum, leads to the equa- tion of hydrostatic equilibrium, dp=dz ¼qg. For an iso- II. INITIAL DECELERATION thermal ideal gas in which the pressure is proportional to the density, both pressure and density decrease exponentially Spacecraft enter atmospheres at speeds comparable to with increasing altitude: q ¼ qs expðz=HÞ, where qs is the orbital speeds. In some instances, such as the return of astro- density at the surface and H is the atmospheric scale height; nauts from the space station, this means the speed of a low- this result can be used to eliminate q from Eq. (1). altitude orbit about the planet in question. In other cases, If the spacecraft travels at an angle / from the vertical, such as the Curiosity rover of the Mars Science Laboratory 1 then a pair of first-order differential equations governs mission, this means the speed at aphelion of an elliptical changes in z and v with time: heliocentric orbit that has its perihelion at Earth and its aphe- lion at the planet in question (known as a Hohmann transfer orbit). The two types of entry speeds can be calculated as a demonstration of Kepler’s laws of orbital motion and Newton’s law of gravity.2 Typical values are several kilo- meters per second, much greater than the speed of sound in planetary atmospheres.3 A spacecraft entering a planetary atmosphere at super- sonic speed decelerates due to atmospheric drag.4 Momentum is transferred from the spacecraft to the atmos- phere at a rate that can be estimated from the principle of conservation of momentum. The spacecraft of area A travel- ing at speed v sweeps through a volume AvDt of atmospheric gas in time Dt. The mass of this amount of air is qAvDt, where q is the local atmospheric mass density. Assuming that this amount of air is accelerated to the same speed as the 2 spacecraft, its momentum is qAv Dt. The momentum gained Fig. 1. Artist’s conception of the entry of Curiosity into the atmosphere of by the atmosphere per unit time, equivalent to that lost by Mars. NASA/JPL-Caltech image PIA 14835. 565 Am. J. Phys. 81 (8), August 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 565 Downloaded 19 Jul 2013 to 128.197.73.19. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission dv Àq Av2 spacecraft is sufficient to ensure a safe landing speed upon ¼ s expðz=HÞ; (2) dt m reaching the surface. We again neglect gravity. Combining Eqs. (2) and (3), we have dz ¼v cos /: (3) dv q Av dt ¼ s expðz=HÞ; (6) dz m cos / At the very first stages of atmospheric entry, when densities 5 and drag forces are very small, v is effectively constant. In and after integration this becomes such circumstances, the altitude satisfies ½qðzÞquHA v lnðv=vuÞ¼ : (7) z ¼ zu À ðt À tuÞ cos /; (4) m cos / where zu is the altitude at the top of the atmosphere when Here qu is the density at the top of the atmosphere, where the t ¼ tu. Using this result in Eq. (2) then leads to corresponding speed is vu. Because the location of “the top” of the atmosphere is arbitrary we can make q arbitrarily dv Àq Av2 u ¼ s expðz =HÞ exp½vðt À t Þ cos /=H: small, which leads to dt m u u (5) ÀqðzÞHA ÀpðzÞA lnðv=v Þ¼ ¼ : (8) u m cos / mg cos / The term in large square brackets is constant for the present assumptions, so the aerodynamic acceleration measured by Because atmospheric pressures and planetary gravities are an onboard accelerometer increases exponentially with time. more commonly used and tabulated than are densities and Figure 2 illustrates that this result held for the arrival of the scale heights, here we have replaced the atmospheric density 6 Pathfinder spacecraft at Mars during the interval 40–80 s af- by the pressure p using p ¼ qgH. Note that MA qsHA ¼ ter atmospheric entry, while the acceleration changed by psA=g is the mass of a vertical atmospheric column of area 7 more than four orders of magnitude. Determination of the A, and MA=cos / is the total mass of the atmospheric gas characteristic timescale for changes in acceleration, swept up by the spacecraft. 3 2 H=ðv cos /Þ, provides a way to estimate the atmospheric For the Curiosity landing, ps ¼ 10 Pa, g ¼ 3.7 m/s , 2 scale height H. For Pathfinder, the acceleration increases by A ¼ 16 m , m ¼ 2400 kg, and vu ¼ 6km=s. A vertical entry four orders of magnitude over 40 s, equivalent to a character- (/ ¼ 0) implies v ¼ 0:2vu ¼ 1:2km=s, while an angled entry istic timescale of 4 s. For the entry speed of 7.4 km/s and with / ¼ 60 implies a much slower speed, despite cos / / ¼ 75:6, the implied scale height is 8 km, which is consist- only changing from 1 to 0.5. This angled entry implies 8 ent with other observations. The brevity of this characteris- v ¼ 0:03vu ¼ 180 m=s, which is still 400 mph, far in excess tic timescale indicates that conditions during atmospheric of any plausible safe landing speed. Since the planet is entry can change very rapidly, which makes the safe passage round, the entry angle / cannot be increased all the way to of spacecraft through planetary atmospheres a formidable 90; at some point, the descending trajectory will become engineering challenge. merely a grazing trajectory and the spacecraft will skip out of the atmosphere and return to space.9 Thus, it is clear that III. IMPACT SPEED WITHOUT A PARACHUTE additional means of deceleration are required to ensure a safe landing speed for Curiosity, as described in Sec. IV. The intense heating that occurs during supersonic flight Figure 3 uses Eq. (8) to show how speed depends on alti- requires that a spacecraft be protected by a heat shield. Let tude for a range of possible values of MA=ðm cos /Þ.Asa us consider whether deceleration due to drag on the enclosed spacecraft descends its speed barely changes until MA= cos / > 0:1, and once MA= cos / > 1 the speed decreases precipitously.