Atmospheric Entry Profiles from the Mars Exploration Rovers Spirit and Opportunity Paul Withers (a,b,*) and Michael D. Smith (c) (a) Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA. (b) Visiting Research Fellow, Planetary and Space Sciences Research Institute, The Open University, Walton Hall, Milton Keynes, MK7 6AA, Great Britain. (c) NASA Goddard Space Flight Center, Code 693, Greenbelt, MD 20771, USA. (*) Corresponding author email address: withers@bu.edu Formatted for submission to Icarus using LaTeX packages natbib, aasms4, and graphicx Editorial Correspondence to: Paul Withers Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA. Email: withers@bu.edu Phone: (617) 353 1531 Fax: (617) 353 6463 Abstract We report on accelerometer measurements made by Spirit and Opportunity during their entries through the martian atmosphere. Vertical profiles of atmospheric density, pressure, and temperature with sub-km vertical resolution were obtained from above 80 km altitude to below 10 km altitude. Spirit's temperature profile is about 10 K warmer than Opportunity's between 20 km and 80 km. Unlike all other martian entry profiles, Spirit's temperature profile does not contain any large amplitude, long wavelength oscillations. A moderate dust storm occurred on Mars shortly before these two atmospheric entries and differences in atmospheric dust loading may account for some of the differences between the two profiles. Both profiles are very consistent with Mars Global Surveyor Thermal Emission Spectrometer (TES) pressure/temperature profiles at pressures less than 30 Pa. The temperature profiles from Spirit and Opportunity are 4 K and 8 K warmer, respectively, than the corresponding TES profiles between 30 and 200 Pa. Some previous workers believe that previous Mars entry profiles were also too warm in this altitude region. The Spirit and Opportunity entry profiles below about 30 km contain a near-inversion and a strong inversion, respectively. A similar inversion was observed by Mars Pathfinder, but such inversions are inconsistent with TES profiles and other observations. These two problems with Mars entry profiles, high-altitude warming and low-altitude inversions, suggest that the characteristics of this measurement technique are not well-understood at present. This has implications for the operational use of real-time entry data to control a spacecraft's atmospheric entry. It is possible that errors in the treatment of spacecraft attitude are responsible for these problems. Keywords: Mars, Atmosphere; Data Reduction Techniques; Atmospheres, Dynamics 1. Introduction We have used data from the entry, descent, and landing (EDL) of the two Mars Exploration Rovers (MERs), Spirit and Opportunity, to obtain two profiles of martian atmospheric density, pressure, and temperature from >80 km to <10 km altitude. These are the first vertical profiles of martian atmospheric structure measured during dusty conditions that have good vertical resolution and good vertical coverage. The atmospheric processes that can be observed in such profiles were discussed by [MAGALHAESETAL1999], who also compared the advantages and disadvantages of this measurement technique to those of other techniques. The thermal structure of the martian atmosphere is sensitive to radiative forcing from suspended dust and to diabatic heating associated with atmospheric dynamics [ZUREKETAL1992, LEOVY2001]. It is also perturbed by a wide variety of waves and tides [LEOVY&ZUREK1979, BANFIELDETAL2000, WITHERSETAL2003A]. In situ atmospheric entry profiles currently provide our best knowledge of the thermal structure of the atmospheres of Venus, Jupiter, and Titan [SEIFFETAL1980, SEIFFETAL1998] [FULCHIGNONIETAL2000, FULCHIGNONIETAL2005], yet several groups have expressed scepticism about the accuracy of such profiles at Mars [CLANCYETAL2000, WILSON&RICHARDSON2000]. We compared the MER profiles to near-collocated and near-simultaneous Mars Global Surveyor (MGS) Thermal Emission Spectrometer (TES) profiles of atmospheric pressure and temperature to test whether a systematic bias exists in atmospheric profiles derived from in situ entry data. The MER project sent two nearly-identical rovers to Mars [GARVINETAL2003, SQUYRESETAL2004A, SQUYRESETAL2004B]. The "MER-2" rover, which was launched on the "MER-A" mission to Gusev Crater on 10 June 2003, was later renamed "Spirit". The "MER-1" rover, which was launched on the "MER-B" mission to Meridiani Planum on 7 July 2003, was later renamed "Opportunity". The positions and times of the two landings are shown in Table 1. [Table 1] In Section 2 we discuss the MER entry systems, entry measurements, data availability, and the entry states. In Section 3 we discuss the trajectory reconstruction process. In Section 4 we discuss the atmospheric structure reconstruction process. In Section 5 we discuss the spacecraft attitude during entry. In Section 6 we extend the atmospheric structure reconstruction to higher altitudes. In Section 7 we discuss our results. In Section 8 we discuss possible applications of these results. In Section 9 we discuss potential improvements to these results. Since the MER entry measurements were not made by the scientific payload, the archived datasets are not accompanied by extensive documentation, nor have papers been published describing the data processing. Therefore Sections 3-6 are intended to be comprehensive. 2. Atmospheric Entry 2.1. Entry Systems The design of the MER spacecraft for cruise and EDL was based on the successful Mars Pathfinder design [SPENCERETAL1999, CRISPETAL2003]. Each MER spacecraft consisted of a cruise stage, which was jettisoned at Mars arrival, and an axisymmetric entry capsule. The entry capsule consisted of a 70 degree sphere-cone ablative front heatshield with Viking-era design heritage and a conic backshell. Inside the entry capsule was a lander. Inside the lander was a rover. Each MER entry capsule entered the martian atmosphere after its interplanetary cruise at a speed of over 5 km/s without entering Mars orbit. Aerodynamic drag decelerated the entry capsule as it descended. A parachute was deployed from the rear of the backshell at 5-10 km altitude, when the speed of the entry capsule was a few hundred metres per second. The front heatshield was jettisoned shortly afterwards and the lander was then lowered beneath the backshell on a 20 m bridle. About one minute after parachute deployment, the lander was at about 100 m altitude and travelling at about 100 m/s. A cocoon of airbags around the lander then inflated as retrorockets on the backshell fired to decrease both the vertical and horizontal components of the lander velocity. The bridle joining the lander to the backshell and retrorockets was cut at about 10 m altitude when the lander had a speed of about 10 m/s. The lander fell to the ground and bounced many times, protected by its airbags, as it rolled to a stop. The airbags were later deflated in preparation for the departure of the rover from the now-useless lander. All these events during EDL were controlled autonomously by the MER computer. The EDL control algorithm required a steady stream of information to make its decisions, specifically measurements of the deceleration of the lander and changes in the attitude of the lander. We have used these measurements to reconstruct the entry trajectory of each MER and the atmospheric structure along each trajectory. 2.2. Entry Measurements Each MER carried two Litton LN-200S inertial measurement units (IMUs), one mounted on the backshell and one mounted on the rover [CRISPETAL2003, KASSETAL2004]. The datastream from the backshell IMU (B-IMU) to the MER computer ceased when the bridle was cut prior to impact, whereas the datastream from the rover IMU (R-IMU) continued after impact. Neither IMU was close to the centre of mass or the symmetry axis of the entry capsule. By contrast, atmospheric structure investigations on previous spacecraft were positioned as close to the centre of mass and symmetry axis as possible. Each IMU contained three identical silicon accelerometers and three identical fibre optic gyroscopes. The onboard processing of raw IMU data, which we now describe, has significant implications for the trajectory reconstruction process. Each accelerometer measured one component of the acceleration of a test mass with respect to the body of the IMU. Each test mass was held in a fixed position. Each accelerometer had a fixed dynamic range of -40 g to 40 g, where 1 g = 9.80665 m s^(-2). Each had a resolution of 2.4 mg and noise level of 1.6 mg when sampled at 400 Hz. Each gyroscope measured the angular velocity at a certain point within the IMU and in a certain direction. The dynamic range, resolution, and noise of the angular velocity measurements are not publicly available. We call these six quantities, which were measured at different points within the IMU, "Type 1 data". Type 1 data were sampled at 400 Hz and then transformed into the three dimensional acceleration vector at a reference position within the IMU and in a reference frame defined with respect to the body of the IMU, and into the three dimensional angular velocity vector at that position and in that reference frame. We call this transformed data "Type 2 data." This transformation was performed by the IMU before any data were output to the MER computer. The reference position and reference frame for each IMU are not publicly available. Type 2 data at 400 Hz were sent to the MER computer, which averaged consecutive blocks of 50 measurements to give an effective sampling rate of 8 Hz, thus generating "Type 3 data." Type 3 acceleration data has a resolution of 50 micro-g and a noise level of 300 micro-g. Type 3 acceleration and angular velocity data measured by the B-IMU before parachute deployment were transformed by the MER computer to acceleration and angular velocity at the entry capsule centre of mass and in the entry capsule reference frame. Type 3 R-IMU data were transformed to the same position and frame. These measurements were then subsampled from 8 Hz to a lower sampling rate, which varied during EDL. Accelerations derived from the B-IMU were not processed further, but accelerations derived from the R-IMU were processed into time-integrated accelerations or velocity changes. These should not be interpreted as actual velocity changes because they do not include all accelerations acting on the spacecraft. Angular velocities derived from the B-IMU and R-IMU were processed into quaternions describing the entry capsule attitude with respect to the J2000 reference frame. The initial entry capsule attitude was required for successful completion of this step. We call these accelerations/time-integrated accelerations and quaternions "Type 4 data." The backshell and lander separated along the bridle after parachute deployment. After this separation, Type 4 data from the B-IMU were referenced to the backshell centre of mass and backshell reference frame, but Type 4 data from the R-IMU were referenced to the rover centre of mass and rover reference frame. The positions of these three centres of mass (entry capsule, rover, and backshell) and orientations of these three reference frames are not publicly available. 2.3. Data Availability and Quality Each MER spacecraft transmitted simple tones directly to Earth during EDL, but these communications did not return any IMU data to Earth. Neither Type 1 nor Type 2 data were returned to Earth. An incomplete set of Type 3 data (8 Hz) was transmitted from each MER to MGS during EDL and later relayed to Earth. An essentially complete set of Type 4 data at 4 Hz was returned to Earth from each MER after its successful landing. The MER project released to the Planetary Data System (PDS) Type 3 and 4 data from the EDL phase of each MER mission in August 2004 [KASSETAL2004]. The incomplete 8 Hz Type 3 dataset (called HIGHRATE in the PDS archive) is not presently useful because the B-IMU and R-IMU reference positions and reference frames are not publicly available. The essentially complete 4 Hz Type 4 dataset (TRANSFORMED) is useful. The orientation of the entry capsule reference frame used by the Type 4 data is not publicly available, but we assume, based on an inspection of the data, that its z-axis is the symmetry axis of the entry capsule and that the centre of mass of the entry capsule is located on that axis. The mean and standard deviations of each of the pre-entry Type 4 B-IMU accelerations are shown in Table 2. The standard deviations, all about 0.003 m s^(-2), are consistent with the expected noise level (300 micro-g). The mean pre-entry accelerations are not zero. If these measurements were made accurately and transformed accurately to the centre of mass of the entry capsule, then the mean of each pre-entry acceleration should be zero. Since each Type 4 acceleration measurement is a complicated function of the measurements of three accelerometers and three gyroscopes, the positions and orientations of these six sensors, and the origin and orientation of the entry capsule frame, rather than the output of a single sensor, we cannot attribute the non-zero pre-entry mean to a simple offset in a sensor's zero level. We did not process the Type 4 B-IMU acceleration data to remove the non-zero pre-entry means. The uncertainty in each measurement must be greater than or equal to the measured noise level and the non-zero pre-entry means show that noise is not the only source of error. Based on these factors, we assigned normally-distributed 1-sigma uncertainties of 0.01 m s^(-2) to each Type 4 B-IMU acceleration measurement. [Table 2] 2.4. Entry States Initial conditions (3-component vector for position, 3-component vector for velocity, and a scalar for time) are required to reconstruct an entry trajectory from measured accelerations. The full entry states for Spirit and Opportunity are not in the PDS archive, nor have they been published elsewhere as a whole. The entry states can be determined from publicly available information, but some effort is required to do so. Entry is defined to occur when the spacecraft's radial distance from the centre of mass of Mars reaches a specified value, 3522.2 km [KASSETAL2004]. The radial distances to the landing sites are known (Table 1). The latitudes and longitudes at entry are not publicly available, but the landed latitudes and longitudes are (Table 1). The inertial velocities at entry were 5.63 km/s (Spirit) and 5.70 km/s (Opportunity) [DESAI&KNOCKE2004]. The inertial flight path angle at entry was 11.5 degrees [DESAI&KNOCKE2004]. The inertial flight path azimuths at entry are not publicly available, but the orientations of the landing ellipse, which should be similar, were 75 degrees (Spirit) and 85 degrees (Opportunity) [GOLOMBEKETAL2003]. The time intervals between entry, which is not observable in the time series of measured accelerations, and parachute deployment, which is, are 251.0 s (Spirit) and 250.3 s (Opportunity) [DESAI&KNOCKE2004]. We identified the time of parachute deployment in the time series of Type 4 B-IMU accelerations by a small decrease in az shortly before a simultaneous large change in ax, ay, and az [SPENCERETAL1999]. First, we assumed an entry state. We reconstructed the trajectory using this entry state and the Type 4 B-IMU accelerations and attitude quaternions. The quaternions were used to relate accelerations expressed in the entry capsule reference frame to accelerations in a Mars-fixed frame. We tested how well the resultant trajectory satisfied the constraints above. This process was repeated until we found an entry state whose subsequent trajectory was reasonably consistent with the constraints. This iterative process succeeded for Spirit, but not Opportunity. Suspecting a problem with the attitude quaternions, we neglected them, assumed that the aerodynamic deceleration parallel to the atmosphere-relative velocity, vector vrel, was the root-sum-square of the three orthogonal Type 4 B-IMU acceleration measurements, and repeated the iterative process [WITHERSETAL2003B]. The iterative process was successful for both Spirit and Opportunity, and the entry state for Spirit was about the same as that obtained using the quaternions. We conclude that the Type 4 attitude quaternions for Opportunity are unreliable, possibly due to an inaccurate initial attitude. In order to treat the two MER spacecraft consistently, we do not use any quaternions in the remainder of this paper. We use only the Type 4 B-IMU acceleration data. Our estimated entry states are shown in Table 3. [Table 3] 3. Trajectory Reconstruction 3.1. Method We used the x, y, and z-axis Type 4 B-IMU acceleration data. The sign convention in the PDS archive is such that az is negative at peak deceleration. We reversed the sign of az for our convenience. Figures 1-2 show time series of ax, ay, an = sqrt(ax*ax + ay*ay), az, and an/az for Spirit and Opportunity. The data in the PDS archive are given as a function of spacecraft clock (SCLK) time. The UTC and SCLK start and stop times that accompany the data in the PDS archive are inconsistent, so we were not able to convert SCLK times to UTC times. All times in this paper are based on SCLK times. az is a smooth function of t, time, but the other panels in Figures 1-2 contain high frequency oscillations, which are probably caused by oscillations in the spacecraft attitude. Differences between the two plots of an/az, which is closely related to the angle of attack, alpha, the angle between the symmetry axis of the entry capsule and vector vrel, suggest that the angles of attack of Spirit and Opportunity behaved differently during their respective EDLs. [Figure 1] [Figure 2] We used the spacecraft attitude, the measured accelerations, and a model for the martian gravitational field to determine the total vector acceleration acting on the spacecraft. This time series of accelerations was then used in the equations of motion to reconstruct the trajectory. We determined the spacecraft attitude by assuming that alpha=0 and the measured axial acceleration, az, was directed parallel to vector vrel. We do not know the direction of ax or ay in a Mars-fixed frame, but the effects of these accelerations on the trajectory should average to zero because the entry capsule rotates about its symmetry axis. We neglected ax and ay during the trajectory reconstruction. The entry capsule rotated about its symmetry axis at 2 rpm upon separation from the cruise stage and probably continued to do so until parachute deployment. In order to relate velocities in a Mars-fixed frame to vector vrel, we assumed that the atmosphere rotated at the same fixed angular rate, angular vector Omega, as the solid body of Mars (Table 4). Winds are neglected. [Table 4] All latitudes and longitudes in this paper are areocentric. Altitude is not referenced to any equipotential surface. Instead, altitude, z, means r - r0, where r is radial distance from the centre of mass of Mars and r0 is the radial distance to the relevant landing site (Table 1). We used a second degree and order spherical harmonic model of the martian gravitational field: Equation 1 vector g = gradient of U Equation 2 U = GM/r * ( 1 + (Rref/r)^2 C20 P20(cos theta) ) Equation 3 P20(x) = sqrt(5) * 0.5 * ( 3*x*x - 1 ) where vector g is the acceleration due to the gravitational field of Mars in an inertial frame, U is the gravitational potential, GM is the product of the gravitational constant and the mass of Mars, Rref is a reference radius, P20 is the normalised associated Legendre function of degree 2 and order 0, theta is colatitude, and C20 is the tesseral normalised spherical harmonic coefficient of degree 2 and order 0 [TYLERETAL1992, SMITHETAL1993, TYLERETAL2000]. The sign and normalization conventions for U and P20 are defined by Equations 1-3. Values of C20, which is related to the oblateness of Mars, GM, and Rref are given in Table 4. The use of higher order models of the gravitational field does not significantly alter the reconstructed trajectory or atmospheric structure [MAGALHAESETAL1999]. Note that this expression for vector g does not contain any centrifugal terms. Our reconstructed trajectories cannot be continued beyond the cutting of the bridle, which occured about 10 m above the surface, because we have used accelerations derived from the B-IMU, not the R-IMU. Some portions of the Type 4 dataset have a sampling rate of faster than 4 Hz. We neglected some measurements to ensure a constant 0.25s sampling rate throughout EDL. We also replaced several missing datapoints by linearly interpolating between neighbouring points. Thus we obtained a complete time series of ax, ay, and az at 4 Hz from entry to bridle cut. The trajectory was reconstructed from entry until bridle cut, but we shall not discuss results after parachute deployment. The trajectory reconstruction was performed using the procedures described in [WITHERSETAL2003B]. We assumed that the centre of mass of Mars was at rest in some inertial frame. We used a simple, first-order numerical integration routine. The two reconstructed trajectories are shown in Figures 3-4. Conditions at parachute deployment are shown in Table 5. Both MER spacecraft travelled 800 km horizontally between entry and parachute deployment. [Figure 3] [Figure 4] [Table 5] 3.2. Error Analysis We used a Monte Carlo error analysis to quantify the uncertainties in our derived trajectories and assumed that all uncertainties were normally distributed. We assumed that the uncertainties in the entry states of Spirit and Opportunity were the same as for Pathfinder (Table 3) [MAGALHAESETAL1999]. We generated n entry altitudes, where n=1000, by selecting values from a normal distribution specified by the relevant mean and standard deviation in Table 3. We generated the remaining components of n complete entry states in a similar manner. We generated n values of az at time t by selecting values from a normal distribution specified by the mean and standard deviation shown graphically in Figures 1-2 for that time. Repeating for all times, we generated n complete time series of az. We then reconstructed an entry trajectory for each one of these n entry states and time series of az using the techniques described above. We calculated standard deviations of the results as functions of time based on this set of n trajectories. We did not consider uncertainties due to our zero angle of attack assumption, due to truncation of Equation 2 at second order, due to the neglect of winds, due to the assumption that the centre of mass of Mars is at rest in some inertial frame, or due to errors introduced by our chosen numerical techniques. 4. Atmospheric Structure Reconstruction 4.1. Method Atmospheric density, rho, is related to the deceleration by the following equation [MAGALHAESETAL1999]: Equation 4 m az = rho A vrel^2 CA / 2 where m is the spacecraft mass, A is the reference area of the spacecraft, and CA is the axial force coefficient, which is usually on the order of 2. Values of m and A are given in Table 4. We used the trajectory from Section 3.1, the assumption that CA=2, and Equation 4 to calculate an initial estimate for the atmospheric density at each point along the entry trajectory. The magnitude of az is small at high altitudes due to the low density. The sign of az is sometimes negative at high altitudes, which implies unphysical negative densities, due to measurement errors and noise. We began the atmospheric structure reconstructions at a time corresponding to about 120 km altitude, about 10 km below the entry altitude, to minimize this problem. Given this initial density profile, we calculated an initial estimate of the collocated pressure, p, profile assuming hydrostatic equilibrium: Equation 5 dp/dr = rho * ( gr + cr ) where gr, which is negative and a function of position, is the radial component of Equation 1, and cr is the radial component of -1 * angular vector Omega x (angular vector Omega x vector r ). This centrifugal term, which is not included in Equation 1, is small and |cr / gr| is about 4E-3. The boundary condition applied at the top of the atmosphere will be discussed later. The ideal gas law leads to an estimate of the collocated temperature, T, profile: Equation 6 mu p = rho R T / NA where mu is the mass of one mole of the martian atmosphere, R is the universal gas constant, and NA is Avogrado's number (Table 4). We assumed that the mean molecular mass of the martian atmosphere was uniform. We now have initial estimates of rho, p, and T at each point along the entry trajectory. The actual values will differ from these initial estimates because CA is not exactly 2. We improved the initial estimates using the following iterative process. MER Project engineers estimated the likely entry trajectory and atmospheric structure before EDL occurred. They used numerical simulations to determine CA(alpha) at a finite number of points along the likely trajectory [SCHOENENBERGERETAL2005B]. Interpolation of CA(alpha) between these points required vrel, the Mach number, Ma, or the Knudsen number, Kn: Equation 7 Ma = vrel / sqrt(gamma p / rho) Equation 8 Kn = 1 / ( sqrt(2) pi d^2 nnd D ) where gamma is the ratio of the heat capacity at constant pressure to the heat capacity at constant volume of the martian atmosphere, d is the diameter of an atmospheric molecule, nnd, which satisfies nnd = NA rho / mu, is the atmospheric number density, and D, which satisfies pi D^(2) /4 = A, is the diameter of the entry capsule (Table 4). We assumed that gamma was uniform throughout the atmosphere and used values of d and gamma for pure CO2. The normal force coefficient, CN, is also relevant. It satisfies: Equation 9 m an = rho A v^2 CN / 2 Dividing Equation 9 by Equation 4 gives an / az = CN / CA. The aerodynamic database also specifies CN(alpha) at the same points along the likely entry trajectory [SCHOENENBERGERETAL2005B]. We used the published aerodynamic database and the initial estimate of rho, p, and T to find CA(alpha), CN(alpha), and CN/CA(alpha) as functions of time along the reconstructed trajectory. Since CN is zero when alpha is zero, CN / CA is a linear function of alpha for sufficiently small values of alpha. CN / CA is a monotonic function of alpha for all the trajectory points and values of alpha given in the published aerodynamic database. We determined alpha at each point along the trajectory by comparing the measured an / az, which equals CN / CA, to the relevant expression for CN/CA(alpha) at each point along the trajectory. We used these values of alpha to find the corresponding values of both CN and CA at each point along the trajectory. This completed the first atmospheric structure iteration. We returned to Equation 4 and repeated the cycle using our improved estimate for CA at each point. The estimated profiles of rho, p, T, Ma, Kn, alpha, CA, and CN all changed slightly. We repeated this iterative process until the greatest change in the reconstructed density from one iteration to the next was below a set threshold, given in Table 4, when we declared that the process had converged. This process should always converge on the correct solution because CA and CN are very weakly dependent on rho, p, and T. This iterative process does not give meaningful results if an/az is inaccurate, which occurs at high altitude as shown in Figures 1 and 2. To minimize this problem, we assumed that alpha = 0 before a time corresponding to about 80 km altitude. We label quantities on the upper boundary with a subscript "0". The density scale height at the upper boundary, H0, can be estimated from an exponential fit to the upper 10 km of the density profile. If the atmosphere is isothermal, then p0 = rho0 g0 H0 and T0 = mu g0 H0 / R. Since the atmosphere is not perfectly isothermal, these estimates of p0 and T0 are slightly incorrect. However, due to the exponential dependence of pressure on altitude, the effects of these errors on p(r) and T(r) two or more scale heights below the upper boundary are very small. This estimate of p0 supplied the boundary condition in Equation 5. The results of the atmospheric structure reconstructions are shown in Figures 5-6 and tabulated in Tables 6 and 7 at 12.5s time intervals. [Figure 5] [Figure 6] [Table 6] [Table 7] The assumption of a uniform molecular mass, mu, might not be valid at the highest altitudes [NIER&MCELROY1977, BOUGHERETAL1990, MAGALHAESETAL1999]. Since T is proportional to mu, readers can easily scale our T results to those that would be obtained using their preferred mu(z). The rho and p results are independent of mu. The atmospheric structure results will only be reliable if the reconstructed trajectories and atmospheric structures are similar to the "likely" trajectory and atmospheric structure assumed by the engineers who generated the aerodynamic database [SCHOENENBERGERETAL2005B]. We assume that they are sufficiently similar. The surface pressure, ps, can be estimated from: Equation 10 ps = pp exp( (rp - rs) / Hp ) Equation 11 Hp = (R Tp) / (mu gp) where the subscript p indicates values at parachute deployment. The estimated surface pressures are 720 +/- 110 Pa for Spirit and 610 +/- 110 Pa for Opportunity, which is consistent with the 1.8 km altitude difference between the two landing sites. 4.2. Error Analysis We used a Monte Carlo error analysis, similar to that of Section 3.2, to quantify the uncertainties in our derived atmospheric structures and assumed that all uncertainties were normally distributed. We slightly modified the atmospheric structure reconstruction technique from Section 4.1, changing the altitude of the upper boundary and using a fixed T0 instead of a fitted density scale height. As a consequence, the uncertainty envelopes in Figures 5-6 do not extend as high as the entry interface. We used each of the n trajectories generated in Section 3.2, where n=1000, the nominal aerodynamic properties, and n values of T0 to generate an set of n atmospheric structure profiles. Since the reconstruction process fails if any of the axial acceleration measurements are negative (what are the aerodynamics of an entry capsule in a fluid of negative density?), we altered the upper boundary from a time corresponding to about 120 km to a time corresponding to about 100 km. The temperature on the upper boundary, T0, was selected from a distribution with a mean value of Tx, shown in Table 4, and a standard deviation of 50 K. Tx was chosen based on the results from the nominal atmospheric structure reconstruction. The standard deviation was estimated. Each of the n trajectories was assigned a constant value of T0. The n profiles of atmospheric properties thus obtained do not reflect the effects of uncertainties in the aerodynamic database. A conservative estimate of the 1-sigma uncertainty in the values of CA in the aerodynamic database is 5% [MAGALHAESETAL1999, DESAIETAL2003]. Values of CN will also be uncertain. It is challenging to incorporate these uncertainties into the error analysis. We accounted for these uncertainties by modifying the results of each of our n atmospheric structure reconstructions after they had converged. We multiplied each value of CA in each profile in our set by (1 + x), where x is a normally distributed random variable with mean 0 and standard deviation 0.05. We did not modify our results for CN. alpha is proportional to CN/CA for all possible flow conditions, so we multiplied each value of alpha in each profile in our set by 1/(1 + x). rho is inversely proportional to CA, so we multiplied each value of rho in each profile in our set by 1/(1 + x). Kn is inversely proportional to rho, so we multiplied each value of Kn in each profile in our set by (1+x). p is related to the height-integrated value of rho, so we multiplied each value of p in each profile in our set by (1+x). T is proportional to p/rho and is quite insensitive to the value of CA, so we did not modify our results for T [WITHERSETAL2003B]. Ma is proportional to sqrt(T), so we did not modify our results for Ma. Due to the statistical nature of this approach, some of the values of az in the n time series were negative. If a trajectory had a negative value of az below the upper boundary of the atmospheric structure reconstruction, which implies a negative density, then we did not use the results derived from that trajectory in the atmospheric structure error analysis. Only a small fraction of our n trajectories were thus neglected (<1% for Spirit, <10% for Opportunity), so this does not significantly affect the error analysis. If the altitude of the upper boundary were higher, then significantly more trajectories would be neglected. We did not consider uncertainties due to our assumption of zero angle of attack at high altitudes, to our assumption of hydrostatic equilibrium, to our assumption of a constant molecular mass, to errors in CN, m, or A, or to the issues discussed at the end of Section 3.2. 5. Does alpha = 0? We assumed that alpha = 0 in the trajectory reconstruction, then permitted alpha to not equal 0 in the atmospheric structure reconstruction. Was that assumption, which separates the trajectory and atmospheric structure reconstructions, justified? If alpha does not equal 0, then the acceleration parallel to vector vrel, aparallel, is az cos(alpha) + an sin(alpha) and the perpendicular acceleration, aperp, is an cos(alpha) - az sin(alpha). The trajectory reconstruction requires aparallel and neglects aperp, whereas the IMUs measured az and an. We define epsilon as the ratio of az to aparallel: Equation 12 epsilon = az / ( az cos(alpha) + an sin(alpha) ) For small alpha, this becomes: Equation 13 epsilon - 1 = 0.5 * alpha^2 - (an * alpha) / az Figures 7 and 8 show that the maximum value of epsilon-1 is 0.015 for Spirit and 0.010 for Opportunity. Figure 18b of [SPENCERETAL1999] shows that the maximum value of epsilon-1 is about 0.004 for Pathfinder. epsilon-1 is small at high altitudes for both MER spacecraft, but increases at low altitudes. Note that epsilon-1 is generally positive, implying that aparallel < az. The actual speed of Spirit or Opportunity at a given time during its EDL will be faster than the results of this paper and the actual altitude will be lower because we assumed that aparallel = az in Section 3. [Figure 7] [Figure 8] Does the assumption that alpha = 0 in the trajectory reconstruction have a significant effect on the derived trajectory and atmospheric structure? Suppose an entry vehicle has alpha=0 (cos(alpha) = ) above 60 km and alpha=5 degrees (cos(alpha) = 1 - 0.004) between 60 km and 10 km. Since (an*alpha)/az << 0.5 * alpha^2 for alpha on the order of 5 degrees and most atmospheric conditions, the effect of an sin(alpha) on aparallel can be neglected and aparallel assumed to equal az cos(alpha). The change in aparallel, 0.4%, seems small, but it can affect vrel significantly. Suppose vrel is 5000 m/s at 60 km and 500 m/s at 10 km in the aparallel = az reconstruction. The velocity change of 4500 m/s in this reconstruction becomes (1 - 0.004) * 4500 m/s in the aparallel = az cos(alpha) reconstruction, which makes vrel at 10 km 518 m/s instead of 500 m/s, an increase of 3.6%. According to Equation 4, density at about 10 km in the aparallel = az cos(alpha) reconstruction will be 7.2% smaller than in the aparallel = az reconstruction. Pressures and temperatures will also be affected. We also considered the specific cases of Spirit and Opportunity. We derived the trajectory and atmospheric structure for both spacecraft using az cos(alpha) + an sin(alpha) ("new"), instead of az ("old"), as aparallel. We neglected aperp. The "new" reconstruction is not completely self-consistent because it uses alpha(t) from the "old" reconstruction. Nevertheless, it should be sufficient to indicate general trends. At parachute deployment, Spirit's new altitude was 200 m lower than the old value, its new latitude was 0.006 degrees further north, its new longitude was 0.02 degrees further east, and its new vrel was 13.2 m/s (3%) faster. The change in vrel is significantly greater than the uncertainty in the original value, 0.9 m/s. Consequently, the new density at parachute deployment was 6% smaller, the new pressure was 3% smaller, and the new temperature was 8K (4%) hotter. Changes to the trajectory and atmospheric structure above 40 km were negligible. The original uncertainties at parachute deployment were 5% in rho, 5% in p, and 1K (0.5%) in T. At parachute deployment, Opportunity's new altitude was 100 m lower than the old value, its new latitude was 0.001 degrees further north, its new longitude was 0.02 degrees further east, and its new vrel was 12.7 m/s (3%) faster. The change in vrel is significantly greater than the uncertainty in the original value, 0.7 m/s. Consequently, the new density at parachute deployment was 6% smaller, new pressure was 2% smaller, and new temperature was 8K (4%) hotter. Changes to the trajectory and atmospheric structure above 30 km were negligible. The original uncertainties at parachute deployment were 5% in rho, 5% in p, and 1K (0.5%) in T. The rough estimate and the two specific estimates are consistent. They demonstrate that seemingly small systematic errors in the acceleration in a Mars-fixed frame can drastically affect the reconstructed atmospheric structure. The effects will be most significant at low altitudes because the fractional error in vrel will be greatest there. Corrections for these effects are clearly important, but they will not be attempted in this work because they require detailed coupling between the trajectory and atmospheric structure reconstruction processes that exceeds the present capabilities of our software. We have shown that the trajectory and atmospheric structure reconstruction processes are extremely sensitive to alpha - yet Figures 5 and 6 show that alpha is uncertain and rapidly varying. A fully self-consistent trajectory and atmospheric structure reconstruction process, with a rigorous error analysis, that includes alpha will alter the results of both the trajectory and atmospheric structure reconstructions at low altitudes. It will also increase the uncertainties at low altitudes. The uncertainties in the aerodynamic database will be critically important for such a correction. 6. High Altitude Atmospheric Structure 6.1. Method The atmospheric structure reconstruction in Section 4 determined atmospheric properties every 0.25 s along the reconstructed trajectory between atmospheric entry and parachute deployment. Results at high altitude, though formally obtained, were so uncertain as to be practically useless. Since we have assumed that all uncertainties are normally distributed, we can use averages of data to determine atmospheric properties at high altitudes with reduced uncertainties, but at the cost of reduced vertical resolution. Neither spacecraft's velocity changed significantly until below 60 km altitude. In this case, according to Equation 4, ln(az) should change linearly with time if rho changes exponentially with z. We used the trajectory from Section 3.1, which has a sampling rate of 4 Hz. We calculated the mean of the logarithm of the first 10 consecutive acceleration measurements made after the time of entry, then used the anti-logarithm of this mean as our averaged acceleration. We also found the mean value of vrel for this block of 10 datapoints. At these high altitudes, the spacecraft's aerodynamics depend only on alpha and atmospheric density. We assumed that alpha=0, since an / az is not known reliably at high altitudes, then iterated as before to determine the atmospheric density that corresponds to each averaged acceleration. We also found the mean altitude and latitude for each block of 10 datapoints. We repeated this for sequential blocks of 10 acceleration measurements. We determined the density scale height for the ith block, Hi, by: Equation 14 - Hi = ( z_(i+1) - z_(i-1) ) / ( ln rho_(i+1) - ln rho_(i-1) ) We did not calculate Hi=0 for the first and last blocks. Atmospheric pressures and temperatures were obtained from rho and H under the assumption of an isothermal atmosphere: Equation 15 Ti = mu gi Hi / R Equation 16 pi = rhoi gi Hi The isothermal assumption is not strictly accurate, but it makes the relationship between density and other atmospheric properties very simple. The isothermal assumption is reasonable given the relative weakness of the critical assumption that all uncertainties are normally distributed. The derived upper atmospheric properties for each spacecraft are shown in Tables 8 and 9. Spirit's results compare reasonably well with those determined in Section 4.1, though Opportunity's do not. [Table 8] [Table 9] 6.2. Error Analysis We used a Monte Carlo error analysis, similar to that of Section 3.2, to quantify the uncertainties in our derived high altitude atmospheric structures and assumed that all uncertainties were normally distributed. We obtained n profiles, where n=1000, of high altitude atmospheric properties from each trajectory in the set of n from Section 3.2 as described above, then calculated standard deviations in the atmospheric properties based on the variability within the set. Since some values of az in the n time series were negative, corresponding to unphysical negative densities, we screened the data as follows. We selected a block of 10 consecutive acceleration measurements from a given trajectory, then discarded any negative values. We calculated the mean of the logarithm of the acceleration measurements within this block using only the remaining positive accelerations, then determined the corresponding density. If 3 or more accelerations in the block of 10 were negative, then we did not determine an atmospheric density for that block nor did we determine a density scale height for the blocks immediately before and after that block. We only determined the standard deviation of density for a given 2.5 s block of 10 data points if we had determined densities for that block in 80% or more of the n trajectories. Standard deviations in density scale heights, temperatures, and pressures were also subjected to this 80% threshold. These results should be used with caution. This approach reduces the uncertainty in the atmospheric structure due to normally-distributed errors, but its results can be biased by systematic errors. Opportunity's unusual results could be more revealing about such systematic errors than about the martian atmosphere. Nevertheless, these results are useful because they probe the martian upper atmosphere where observations are sparse. 7. Discussion of Results 7.1. General Characteristics Spirit's temperature profile does not contain any large amplitude, long wavelength oscillations. Small amplitude, short wavelength oscillations occur below about 25 km, but they are no larger than the error bars. These oscillations are not present in the pressure profile because p = the integral of rho g dr. These oscillations are present in the temperature profile because T is proportional to p/rho. Spirit's temperature profile is a relatively smooth quadratic function of altitude above 30 km, but the shape of the temperature profile changes abruptly around 30 km. Opportunity's temperature profile, which is about 20 K colder at 80 km than Spirit's is, has a large amplitude, long wavelength oscillation around 60 km. Small amplitude, short wavelength oscillations occur below about 30 km, similar to those in Spirit's profile. The temperature in Opportunity's profile decreases by 15 K from 12 km to 8 km. 7.2. Dust The dust loading in the martian atmosphere can increase significantly from its background level within a few days during the onset of a regional/global dust storm. Micron-sized dust particles, which can be lifted 10-20 km by a dust storm, take days to fall one kilometre and the decay time of a large dust storm is on the order of months [POLLACKETAL1979, MURPHYETAL1990, SMITH2004]. The atmosphere can become hotter by about 15 K over a broad vertical range during a large dust storm [SMITHETAL2001]. Atmospheric dynamics are modified and some atmospheric tidal modes, especially the semidiurnal migrating tide, become stronger [ZUREKETAL1992, BRIDGERETAL1998]. The effects of dust storms extend at least as high as 160 km [KEATINGETAL1998]. The effects on the atmosphere may have a global extent even if the region of high dust opacity is relatively small. A large regional dust storm began on Mars in December 2003, which raised significant amounts of dust near the Opportunity landing site. The spatial distribution of dust in the atmosphere on local, regional, and global scales at the time of each entry will affect the thermal structure of the atmosphere. The Spirit and Opportunity entry profiles are the first measured on Mars in the immediate aftermath of a moderate dust storm. They are also the first datasets with sub-scale height vertical resolution to probe the martian middle atmosphere under dusty conditions. Figure 9 shows infrared dust opacities, tau, measured at the landing sites of both Spirit and Opportunity by the nadir-looking MGS TES instrument in December 2003 and January 2004. The values have been corrected for topographic differences between the two sites. The LST of all measurements was about 13.5 hrs. The longitudes of the Spirit measurements are between 170 and 200 degrees E; the longitudes of the Opportunity measurements are between -10 and 20 degrees E. The latitudes of each series of measurements are close to the latitudes of the respective landing sites. This is due to MGS's near-polar orbit, which has a period of about 2 hours. The 12 ground tracks that cross the equatorial region each day therefore have a longitudinal spacing of about 30 degrees. [Figure 9] Values of tau at the landing sites of both Spirit and Opportunity were about 0.2 from 1 December to 10 December. Values of tau at both sites increased slightly over the next few days, then tau at Opportunity's landing site tripled in less than one day, reaching 0.8 on 15 December. It remained extremely high, but variable, until 25 December, when it started to decrease steadily. The rate constant for the exponential decay in tau between 25 December and 8 January was about 1/(23 days). The rate of decay in tau was three times slower than this between 8 January and 31 January. Meanwhile, tau at Spirit's landing site increased from 0.2 on 1 December to 0.3 around 21 December and remained between 0.30 and 0.35 until around 25 January. Dust opacities at both landing sites were very similar before 14 December. They were also very similar after 19 January, although the dust opacities were 50% greater in late January than in early December. On the day of Spirit's EDL, tau at Spirit's landing site was 0.34 and tau at Opportunity's landing site was 0.42. On the day of Opportunity's EDL, tau at Spirit's landing site was 0.30 and tau at Opportunity's landing site was 0.28. The local and global-scale dust content of the atmosphere was greater for Spirit's EDL than for Opportunity's, which may account for some of the differences between the two profiles, such as the differences in middle atmospheric temperatures and the differences in temperature oscillations. 7.3. Comparison to TES T(p) Profiles The MGS TES instrument observed temperature as a function of pressure between the surface and 10 Pa near the two landing sites before and after the landings [SMITHETAL2001]. We can validate atmospheric profiles derived from accelerometer data against independent observations for the first time since the PAET experiment in the terrestrial atmosphere in 1971 [SEIFFETAL1973]. Figures 10-11 compare MER entry profiles and TES profiles. One TES profile was selected from each day as being the closest in latitude, longitude, and LST to the EDL conditions. The 21 TES profiles in each of Figures 10-11 span a period from ten days before to ten days after the day of EDL. Both entry profiles are very consistent with the TES profiles at between 10 and 30 Pa. The trend of the TES profiles suggests that the entry profiles would be consistent with the TES profiles at even lower pressures if the TES profiles extended further. The gross shape of the Spirit entry profile between 30 and 200 Pa is not similar to the shapes of the corresponding TES profiles, whereas the gross shape of the Opportunity entry profile in this pressure range is quite similar to the shapes of the corresponding TES profiles. The log-pressure-weighted mean difference in temperature between the Spirit entry profile and the TES profile from the day of EDL over the 30-200 Pa pressure range is 4 K. The corresponding value for Opportunity is 8 K. At pressures greater than 200 Pa, the Spirit entry profile contains a near-inversion and the Opportunity entry profile contains a strong inversion. There are no hints of inversions in the TES profiles closest to EDL, although the two earliest TES profiles in Figure 10 have different lapse rates from the subsequent TES profiles. This change in the shape of the TES profiles is probably a temporal effect related to the dust storm, although spatial effects are also possible. There are significant regional variations in topography near the Spirit landing site and the TES profiles in Figure 10 are distributed over about 30 degrees in longitude [GOLOMBEKETAL2003]. The TES profiles compared to the Opportunity profile in Figure 11 exhibit much less variability, possibly due either to the flat regional topography at Meridiani compared to Gusev or the less rapid changes in atmospheric dust loading at the time of Opportunity's EDL compared to Spirit's [GOLOMBEKETAL2003]. The vertical resolution of the TES instrument is about one scale height [CONRATHETAL2000]. Uncertainties in its derived atmospheric temperatures at these altitudes are about 4 K [SMITH2004]. Uncertainties in the Spirit and Opportunity atmospheric temperatures at these altitudes are about 1 K (Tables 6-7). However, the angle of attack effects discussed in Section 5 will increase the uncertainties in the entry profiles. Considering the uncertainties in each measurement technique and the poor vertical resolution of TES, the entry T(p) profiles are reasonably consistent with the TES T(p) profiles. Without knowing how the Opportunity profile continues at lower altitudes, it is difficult to say whether the TES instrument would notice such an inversion. [Figure 10] [Figure 11] Figure 12 compares temperature-pressure profiles from Viking Lander 1, Viking Lander 2, Mars Pathfinder, Spirit, and Opportunity. Viking Lander 1 landed at 22 degrees N, 312 degrees N on 20 July 1976, when Ls was 96 degrees and LST was 16:13. Viking Lander 2 landed at 48 degrees N, 134 degrees E on 3 September 1976, when Ls was 117 degrees and LST was 09:49. Mars Pathfinder landed at 19 degrees N, 326 degrees E on 4 July 1997, when Ls was 143 degrees and LST was 02:58 [SEIFF&KIRK1977, MAGALHAESETAL1999]. Corresponding values for Spirit and Opportunity are shown in Table 1. All profiles, except Spirit's, have large amplitude, long wavelength oscillations around 1 Pa. The Opportunity and Pathfinder profiles are remarkably similar between 2 and 20 Pa. Spirit's profile stands out as being the warmest between 10 and 100 Pa. [Figure 12] 7.4. Unusual Aspects of Results There are two aspects of martian entry profiles that are potentially unusual. First, unusual behaviour at low altitudes (p > 100 Pa) is seen in the Pathfinder, Spirit, and Opportunity profiles that were derived from accelerometer data. None of these three spacecraft made successful measurements of atmospheric properties after parachute deployment, so these results cannot be verified against direct pressure/temperature measurements. Viking did not derive atmospheric properties from accelerometer data at pressures greater than 100 Pa. Several authors have discussed whether or not the inversion in the Pathfinder profile is an accurate measurement of the atmosphere [MAGALHAESETAL1999, HABERLEETAL1999, COLAPRETEETAL1999] [COLAPRETE&TOON2000, HINSON&WILSON2004]. The Opportunity inversion appears superficially similar to the Pathfinder inversion, but Spirit's is distinctly different. If these features are real, then the high dust content in the atmosphere during Spirit's entry could explain why Spirit's low altitude temperature structure is dissimilar to Pathfinder's and Opportunity's. If these features are not real, then the relatively high angle of attack of Spirit, by comparison to Pathfinder's or Opportunity's, could be responsible for the difference, as discussed in Section 5. Our current understanding of either the thermal structure of the martian atmosphere around about 10 km or the characteristics of this measurement technique is poor. In either case, temperatures derived from accelerometer data in the 10 or so kilometres before parachute deployment by Pathfinder, Spirit, and Opportunity deserve further study. Second, several workers believe, based on comparison with a wide range of models and other observations, that the Viking and Pathfinder profiles appear too warm by about 15 K over a vertical range of about 2 scale heights centred on 100 Pa [CLANCYETAL2000, WILSON&RICHARDSON2000]. Note that the Viking temperature profiles are not derived from accelerometer data at pressures greater than 50 Pa [SEIFF1976, SEIFF&KIRK1977]. The unusual sequence of measurement techniques used by the Viking Landers has not been repeated by any subsequent mission. A similar, but smaller, warming was observed in the comparison between the Spirit and Opportunity entry profiles and TES profiles (Figures 10 and 11) between 30 and 200 Pa. The first issue, the problem of unusual low altitude temperature inversions, is restricted to Mars Pathfinder, Spirit, and Opportunity. Atmospheric structure profiles determined from accelerometer data before parachute deployment were consistent with those determined from direct measurements immediately after parachute deployment on Pioneer Venus, Galileo, and Huygens [SEIFFETAL1980, SEIFFETAL1998, FULCHIGNONIETAL2002, FULCHIGNONIETAL2005]. The second issue, the problem of too-warm temperatures significantly above the altitude of parachute deployment, might be restricted to Viking, Mars Pathfinder, Spirit, and Opportunity - or it might be common to all atmospheric entry profiles. Our understanding of the thermal structures of the atmospheres of Venus, Jupiter, and Titan is sufficiently poor, and corroborating observations sufficiently rare, that spurious 15 K warmings in portions of the Pioneer Venus, Galileo, and Huygens entry profiles might not be detected. These two problems, low altitude inversions and warmings at higher altitudes, may or may not be connected. If atmospheric properties are being determined incorrectly by entry measurements in some cases, then operationally important properties, such as dynamic pressure, that are determined onboard in real-time during EDL and used to trigger critical events are also being determined incorrectly. Managers and engineers working on future planetary entry vehicles should be concerned about these potential problems with entry data. [WITHERS2005] considered various ways to remove the low-lying temperature inversion from the Pathfinder profile, and falsified all his hypotheses, except contamination of the measured accelerations by centrifugal terms. We have not yet been able to determine whether this effect is truly significant. [WITHERS2005] did not consider the issues discussed in Section 5. If the problems discussed in this Section are indicative of errors in the derived entry profiles, then the most likely causes are either (a) differences between acceleration measured at the accelerometer and acceleration at the centre-of-mass or (b) differences between axial acceleration and acceleration along the atmosphere-relative velocity vector. 8. Applications of Results Scientists use first principles-based numerical models of the martian atmosphere extensively, e.g. [BOUGHERETAL1990, HABERLEETAL1999, FORGETETAL1999]. Such models extend our understanding of martian atmospheric processes beyond what can be learned from studies of sparse, isolated observations of a handful of atmospheric properties. Global-scale models have often been validated by comparison to atmospheric measurements that either average over a large vertical (e.g., TES) or horizontal (e.g., radio science) distance, but they have rarely been validated against measurements with excellent spatial resolution, such as entry profiles, due to the scarcity of such observations. The vertical resolution of entry profiles is smaller than the spacing of grid points in present global-scale models, so such comparisons are especially useful for evaluating how accurately a model is parameterizing sub-gridscale processes. Mesoscale models for Mars desperately need to be validated against observations with a suitable spatial scale, such as entry profiles [RAFKIN&MICHAELS2003, TOIGO&RICHARDSON2003]. Numerical models of the martian atmosphere, whether empirical or based upon first principles, had a significant effect on the MER missions. They influenced the initial design of the entry vehicle and EDL systems, required the late addition of the DIMES/TIRS imaging/thruster system to the spacecraft, and influenced the site selection process. As a dust storm raged in the weeks before the two landings, predictions of the atmospheric state for each EDL were updated regularly in response to MGS TES and other observations. The changes in the predictions were so significant that changes were made to the onboard software that would control the EDL process. After two successful landings, it would be productive to compare all the atmospheric predictions that were used during the many stages of the MER missions against observations. This comparison would determine which models worked well and which did not, highlighting the strengths and weaknesses of each, so that obsolete models can be discarded and resources can be directed to where improvements are most urgently needed. The MER entry profiles can be analysed in conjunction with other datasets. Comparison to measurements of atmospheric properties that are repeated temporally or spatially, such as TES profiles, Mars Express SPICAM profiles, radio science profiles, or ground-based microwave measurements of atmospheric temperature, enables analyses that would not be possible using just one of these datasets. The possible benefits are greater when models are included as well. For example, TES measurements of atmospheric properties on a global scale over an extended period of time could be assimilated into a numerical model, and the model used to predict the atmospheric conditions experienced during the atmospheric entry of Spirit or Opportunity [MONTABONEETAL2005, MONTABONEETAL2006]. The benefits provided by the data assimilation can be quantified by comparison of these predictions to observations. The major advantages of entry profiles are their excellent vertical resolution, excellent vertical coverage, and absolute altitude scale. Many remote sensing measurements of atmospheric properties are made at known pressure levels that cannot be converted into accurate heights. A major disadvantage of entry profiles is that they only provide measurements at one place and one time. 9. Potential Improvements to Results In the past, entry profiles have been processed and analysed by a well-funded team of scientists working closely with the spacecraft engineers and managers [SEIFF&KIRK1977, SEIFFETAL1980, SEIFFETAL1998, MAGALHAESETAL1999] [FULCHIGNONIETAL2002, FULCHIGNONIETAL2005]. The entry profiles presented in this paper have not. If additional information is released by the MER Project, or the PDS datasets recalibrated, then the results in this paper might change. The most likely things that could happen to change these results are: (a) Release of actual entry states with uncertainties; (b) Release of IMU positions and orientations, making 8 Hz accelerations and angular rates useful; (c) Correction of attitude quaternions; and (d) Recalibration of datasets to remove the non-zero pre-entry means. The results in this paper could also be improved to account for the non-zero angle of attack during the trajectory reconstruction (Section 5). If the attitude quaternions are sufficiently accurate, then it may be possible to estimate horizontal wind speed and direction along the trajectory. If the coupled parachute-backshell-lander aerodynamics are known sufficiently accurately, then it may be possible to determine density, pressure, and temperature along the trajectory after parachute deployment from IMU data. 10. Conclusions We have presented measurements made by IMUs on Spirit and Opportunity during their descents into the martian atmosphere in January 2004. Relatively unprocessed, raw data (Type 3 data) are not available continuously during entry and, in the absence of additional information, are not currently useful. The archived attitude quaternions (Type 4 data) appear to be unreliable, so we have not used them in this work. We have used acceleration measurements (Type 4 data) to reconstruct the entry trajectories of both spacecraft and to derive profiles of atmospheric density, pressure, and temperature along these trajectories. These are the first high-resolution measurements of the extended vertical structure of the martian atmosphere made soon after a moderate dust storm. The impact of dust storms on the middle/upper regions of the martian atmosphere are not well-understood. The two MER temperature profiles show interesting differences in their middle atmospheric temperatures, the presence or absence of large-amplitude, long-wavelength oscillations, and their temperatures below about 20 km. Explanations of these features might require consideration of the local, regional, and global-scale dust loading in the atmosphere, the large-scale dynamics of the atmosphere, and local topography. The MER entry profiles are consistent with independent TES observations at pressures less than 30 Pa, are 4-8 K warmer than TES observations between 30 and 200 Pa, and have unusual temperature gradients at lower altitudes. Previous workers have suggested that the Viking and Pathfinder entry profiles are also about 15 K too warm around 100 Pa. An unusual temperature gradient was also present at low altitudes in the Pathfinder entry profile, although no such feature was observed in the Pioneer Venus, Galileo, or Huygens entry profile. These two problems merit further study and may be related to inadequate knowledge of the angle of attack of the entry vehicles. Together with other observations, the Spirit and Opportunity entry profiles can be used to test the many scientific and engineering models of the martian atmosphere that were used for spacecraft hardware design, landing site selection, and EDL software design. This work is completely reproducible. The results presented here and all the software used to generate them are publicly available from http://www.buimaging.com/withers/. Since this personal website is not as stable an archive as the Planetary Data System, interested readers are encouraged to make copies of these resources. Acknowledgments We acknowledge the efforts of the MER EDL and Atmospheric Advisory Teams that contributed to two successful landings and the delivery of the IMU datasets to the PDS. References Allison, M., McEwen, M., 2000. 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Table 1 Locations and Times of MER Landings Spirit Opportunity Date (a) (UTC) 4 January 2004 25 January 2004 Time of first impact (a) (UTC hrs) 04:26 04:55 Ls (b) (degrees) 327.7 339.1 Latitude (a),(c) (degrees N) -4.571892 -1.948282 Longitude (a),(c) (degrees E) 175.47848 354.47417 Radial distance (d) (km) 3392.3 3394.1 Local Solar Time (b) (hrs) 14:16 13:13 (a) [KASSETAL2004] (b) From Mars24 Sunclock, http://www.giss.nasa.gov/tools/mars24/, based on [ALLISON&MCEWEN2000] (c) Final landed position, not position of first impact (d) [SMITHETAL2003] Table 2 Characteristics of Pre-Entry Data Spirit Opportunity Mean ax (1E-3 m s^(-2)) -1.4 9.6 Mean ay (1E-3 m s^(-2)) 0.7 -2.0 Mean az (1E-3 m s^(-2)) 3.7 -3.9 Sqrt( Mean(ax*ax) - (Mean ax)^2 ) (1E-3 m s^(-2)) 3.0 3.0 Sqrt( Mean(ay*ay) - (Mean ay)^2 ) (1E-3 m s^(-2)) 2.9 2.8 Sqrt( Mean(az*az) - (Mean az)^2 ) (1E-3 m s^(-2)) 3.2 3.2 Table 3 Estimated Entry States Spirit Opportunity 1-sigma Uncertainty Time - tref (SCLK seconds) 2085.625 8194.625 0 tref (SCLK seconds) 126460000.000 128270000.000 0 Radial distance (km) 3522.2 3522.2 1.7 Areocentric latitude (degrees N) -17.7 -2.9 0.04 Areocentric longitude (degrees E) 161.8 340.9 0.01 Speed (a) (km/s) 5.63 5.70 7E-4 Flight path angle (b) (degrees) 11.5 11.5 0.02 Azimuth (c) (degrees) 79.0 86.5 0.02 (a) Relative to an inertial frame [SPENCERETAL1999, WITHERSETAL2003B] (b) Angle below horizontal of velocity vector in inertial frame (c) Angle east of north of velocity vector in inertial frame Table 4 Relevant Physical Properties and Constants GM (a) (m^3 s^(-2)) 4.2828E14 Rref (a) (km) 3394.2 C20 (a) -8.75981E-4 Omega (b) (rad/s) 2 pi / (24.6229 * 60 * 60) mu (c) (kg/mol) 43.49E-3 dCO2 (d) (m) 4.64E-10 gammaCO2 (e) 7/5 R (b) (J K^(-1) mol^(-1)) 8.31451 NA (b) (1/mol) 6.022E23 A (d) (m^2) 0.25 pi * 2.648^2 = 5.507 m (f) (kg) 827.0 (Spi.), 832.2 (Opp.) Tx (K) 160 (Spi.), 160 (Opp.) Convergence Threshold 1E-3 (a) [TYLERETAL2000] (b) [LODDERS&FEGLEY1998] (c) [MAGALHAESETAL1999] (d) [SCHOENENBERGERETAL2005B] (e) Ideal linear polyatomic gas [ATKINS2002] (f) [DESAI&KNOCKE2004] Table 5 Conditions at Parachute Deployment With 1-sigma Uncertainties Spirit Opportunity Time - tref (SCLK seconds) 2336.375 8444.625 Altitude (km) 7.5 +/- 1.7 6.2 +/- 1.8 vrel (m/s) 410.98 +/- 0.77 429.68 +/- 0.81 Latitude (degrees N) -14.528 +/- 0.039 -1.957 +/- 0.041 Longitude (degrees E) 175.411 +/- 0.013 354.413 +/- 0.013 Table 6 Reconstructed Atmospheric Structure With 1-sigma Uncertainties: Spirit tSCLK - tref z (km) rho (kg m^(-3)) p (Pa) T (K) 2110.625 103.5+/-1.7 2.79E-07+/-5.85E-08 6.34E-03+/-1.93E-03 118.7+/-52.8 2123.125 91.5+/-1.7 1.18E-06+/-8.20E-08 3.33E-02+/-2.60E-03 147.4+/-11.7 2135.625 80.4+/-1.7 4.91E-06+/-2.70E-07 1.39E-01+/-7.51E-03 148.4+/- 2.8 2148.125 70.2+/-1.7 1.80E-05+/-9.02E-07 5.14E-01+/-2.60E-02 149.4+/- 1.0 2160.625 60.8+/-1.7 5.49E-05+/-2.76E-06 1.63E+00+/-8.19E-02 155.5+/- 0.8 2173.125 52.5+/-1.7 1.36E-04+/-6.93E-06 4.30E+00+/-2.14E-01 165.6+/- 0.6 2185.625 45.3+/-1.7 2.88E-04+/-1.41E-05 9.63E+00+/-4.88E-01 175.0+/- 0.6 2198.125 39.3+/-1.7 5.11E-04+/-2.62E-05 1.80E+01+/-9.09E-01 184.1+/- 0.6 2210.625 34.5+/-1.7 7.92E-04+/-3.94E-05 2.94E+01+/-1.43E+00 193.8+/- 0.7 2223.125 30.6+/-1.7 1.10E-03+/-5.46E-05 4.27E+01+/-2.17E+00 204.0+/- 0.8 2235.625 27.3+/-1.7 1.46E-03+/-7.32E-05 5.79E+01+/-2.84E+00 206.8+/- 0.8 2248.125 24.5+/-1.7 1.88E-03+/-8.80E-05 7.51E+01+/-3.65E+00 208.4+/- 0.8 2260.625 21.9+/-1.7 2.38E-03+/-1.22E-04 9.52E+01+/-4.75E+00 208.9+/- 0.8 2273.125 19.5+/-1.7 3.03E-03+/-1.52E-04 1.19E+02+/-5.77E+00 205.6+/- 0.9 2285.625 17.1+/-1.7 3.75E-03+/-1.85E-04 1.49E+02+/-7.53E+00 207.1+/- 0.9 2298.125 14.7+/-1.7 4.68E-03+/-2.34E-04 1.85E+02+/-9.52E+00 206.6+/- 0.9 2310.625 12.4+/-1.7 5.78E-03+/-2.94E-04 2.30E+02+/-1.11E+01 207.7+/- 0.9 2323.125 10.0+/-1.7 7.03E-03+/-3.53E-04 2.85E+02+/-1.44E+01 212.3+/- 1.0 2335.625 7.6+/-1.7 8.57E-03+/-4.31E-04 3.53E+02+/-1.74E+01 215.6+/- 1.0 Table 7 Reconstructed Atmospheric Structure With 1-sigma Uncertainties: Opportunity tSCLK - tref z (km) rho (kg m^(-3)) p (Pa) T (K) 8219.625 101.4+/-1.8 1.71E-07+/-5.70E-08 7.51E-03+/-3.77E-03 229.9+/-140.9 8232.125 89.4+/-1.8 7.36E-07+/-6.55E-08 2.30E-02+/-3.80E-03 163.3+/- 29.4 8244.625 78.2+/-1.8 4.63E-06+/-2.45E-07 1.14E-01+/-6.86E-03 128.8+/- 4.8 8257.125 67.9+/-1.8 1.66E-05+/-8.20E-07 4.63E-01+/-2.44E-02 146.0+/- 1.5 8269.625 58.6+/-1.8 5.02E-05+/-2.55E-06 1.45E+00+/-6.87E-02 150.6+/- 0.7 8282.125 50.3+/-1.8 1.29E-04+/-6.74E-06 3.98E+00+/-1.96E-01 161.3+/- 0.6 8294.625 43.2+/-1.8 2.81E-04+/-1.44E-05 9.03E+00+/-4.57E-01 168.3+/- 0.6 8307.125 37.3+/-1.8 5.28E-04+/-2.63E-05 1.75E+01+/-9.03E-01 173.0+/- 0.7 8319.625 32.6+/-1.8 8.53E-04+/-4.23E-05 2.91E+01+/-1.51E+00 178.5+/- 0.7 8332.125 28.8+/-1.8 1.16E-03+/-5.86E-05 4.28E+01+/-2.14E+00 192.3+/- 0.8 8344.625 25.8+/-1.8 1.51E-03+/-7.70E-05 5.78E+01+/-2.90E+00 199.8+/- 0.8 8357.125 23.0+/-1.8 2.00E-03+/-1.06E-04 7.50E+01+/-3.87E+00 196.4+/- 0.8 8369.625 20.6+/-1.8 2.44E-03+/-1.19E-04 9.51E+01+/-4.79E+00 203.6+/- 0.8 8382.125 18.2+/-1.8 2.98E-03+/-1.54E-04 1.19E+02+/-5.95E+00 208.1+/- 0.9 8394.625 15.9+/-1.8 3.45E-03+/-1.77E-04 1.46E+02+/-7.28E+00 221.6+/- 0.9 8407.125 13.5+/-1.8 4.23E-03+/-2.22E-04 1.80E+02+/-9.23E+00 221.9+/- 1.0 8419.625 11.1+/-1.8 5.19E-03+/-2.71E-04 2.21E+02+/-1.13E+01 222.8+/- 1.0 8432.125 8.7+/-1.8 6.86E-03+/-3.37E-04 2.75E+02+/-1.39E+01 209.4+/- 1.0 8444.625 6.2+/-1.8 8.39E-03+/-4.25E-04 3.43E+02+/-1.75E+01 213.8+/- 1.0 Table 8 High Altitude Atmospheric Structure With 1-sigma Uncertainties: Spirit tSCLK - tref z (km) rho (kg m^(-3)) p (Pa) T (K) 2106.750 107.4+/-1.7 1.57E-07+/-2.36E-08 3.35E-03+/-1.04E-03 111.6+/- 33.5 2109.250 104.9+/-1.7 2.36E-07+/-2.30E-08 5.72E-03+/-1.30E-03 126.6+/- 27.2 2111.750 102.4+/-1.7 3.23E-07+/-2.51E-08 9.10E-03+/-1.57E-03 147.1+/- 23.1 2114.250 99.9+/-1.7 4.37E-07+/-2.87E-08 1.21E-02+/-1.50E-03 145.1+/- 15.5 2116.750 97.5+/-1.7 5.99E-07+/-3.42E-08 1.72E-02+/-1.79E-03 149.9+/- 12.4 2119.250 95.1+/-1.7 7.88E-07+/-4.28E-08 2.42E-02+/-2.14E-03 160.7+/- 10.9 2121.750 92.8+/-1.7 1.03E-06+/-5.81E-08 3.14E-02+/-2.32E-03 159.3+/- 8.0 2124.250 90.5+/-1.7 1.35E-06+/-7.24E-08 3.76E-02+/-2.50E-03 145.6+/- 5.2 2126.750 88.2+/-1.7 1.85E-06+/-9.54E-08 4.66E-02+/-2.74E-03 132.2+/- 3.4 2129.250 86.0+/-1.7 2.55E-06+/-1.28E-07 6.69E-02+/-3.72E-03 137.4+/- 2.7 2131.750 83.8+/-1.7 3.36E-06+/-1.72E-07 1.01E-01+/-5.44E-03 156.6+/- 2.6 2134.250 81.6+/-1.7 4.28E-06+/-2.12E-07 1.29E-01+/-6.51E-03 157.9+/- 2.1 2136.750 79.5+/-1.7 5.59E-06+/-2.78E-07 1.54E-01+/-7.65E-03 144.0+/- 1.5 2139.250 77.3+/-1.7 7.41E-06+/-3.82E-07 2.03E-01+/-1.06E-02 143.1+/- 1.2 2141.750 75.3+/-1.7 9.63E-06+/-4.78E-07 2.72E-01+/-1.43E-02 147.9+/- 1.1 2144.250 73.2+/-1.7 1.24E-05+/-6.37E-07 3.55E-01+/-1.79E-02 149.6+/- 0.9 Table 9 High Altitude Atmospheric Structure With 1-sigma Uncertainties: Opportunity tSCLK - tref z (km) rho (kg m^(-3)) p (Pa) T (K) 8213.250 107.9+/-1.8 1.14E-07+/-2.09E-08 3.59E-03+/-1.17E-02 164.6+/- 546.1 8215.750 105.3+/-1.8 1.48E-07+/-2.19E-08 6.15E-03+/-2.36E-02 217.2+/- 955.5 8218.250 102.8+/-1.8 1.76E-07+/-2.18E-08 1.00E-02+/-5.96E-02 298.7+/-1991.6 8220.750 100.3+/-1.8 2.02E-07+/-2.24E-08 1.14E-02+/-4.75E-02 294.8+/-1322.3 8223.250 97.8+/-1.8 2.39E-07+/-2.29E-08 8.91E-03+/-2.70E-03 194.6+/- 57.2 8225.750 95.4+/-1.8 3.21E-07+/-2.46E-08 8.94E-03+/-1.56E-03 145.6+/- 23.0 8228.250 93.0+/-1.8 4.41E-07+/-2.98E-08 1.18E-02+/-1.47E-03 139.8+/- 14.7 8230.750 90.6+/-1.8 6.02E-07+/-3.47E-08 1.41E-02+/-1.20E-03 122.2+/- 8.2 8233.250 88.3+/-1.8 8.97E-07+/-5.06E-08 1.62E-02+/-1.09E-03 94.7+/- 3.4 8235.750 86.0+/-1.8 1.48E-06+/-7.83E-08 2.77E-02+/-1.63E-03 97.5+/- 2.7 8238.250 83.8+/-1.8 2.13E-06+/-1.12E-07 4.95E-02+/-2.82E-03 121.8+/- 2.6 8240.750 81.5+/-1.8 2.93E-06+/-1.48E-07 7.51E-02+/-4.20E-03 133.9+/- 2.3 8243.250 79.4+/-1.8 3.92E-06+/-2.02E-07 1.06E-01+/-5.58E-03 141.0+/- 1.9 8245.750 77.2+/-1.8 5.20E-06+/-2.56E-07 1.38E-01+/-7.03E-03 138.7+/- 1.5 8248.250 75.1+/-1.8 6.95E-06+/-3.56E-07 1.88E-01+/-9.52E-03 141.3+/- 1.2 8250.750 73.0+/-1.8 9.03E-06+/-4.66E-07 2.61E-01+/-1.37E-02 151.4+/- 1.2 8253.250 71.0+/-1.8 1.15E-05+/-6.00E-07 3.49E-01+/-1.80E-02 158.4+/- 1.1 Figure Captions Figure 1: Time series of ax, ay, an = sqrt(ax*ax + ay*ay), az, and an / az for Spirit. 1-sigma uncertainties are shown on each panel by the grey envelope. The 1-sigma uncertainty of 0.01 m s^(-2) in az is also shown as a horizontal line. The data start at the entry interface and end at parachute deployment. Figure 2: As Figure 1, but for Opportunity. Figure 3: Spirit's entry trajectory between the entry interface and parachute deployment. 1-sigma uncertainties are shown on each panel by the grey envelope. Figure 4: As Figure 3, but for Opportunity. Figure 5: Reconstructed atmospheric structure for Spirit between the entry interface and parachute deployment. 1-sigma uncertainties are shown on each panel by the grey envelope. Uncertainties were not calculated at the highest altitudes. Figure 6: As Figure 5, but for Opportunity. Figure 7: Time series of epsilon-1 for Spirit between about 80 km and parachute deployment. 1-sigma uncertainties are shown on each panel by the grey envelope. Figure 8: As Figure 7, but for Opportunity. Figure 9: TES infrared dust opacity during December 2003 and January 2004. Values for Spirit's landing site are shown by diamonds, values for Opportunity's landing site are shown by crosses. The times of the landings of Spirit and Opportunity are marked. Figure 10: Comparison of entry profile and TES profiles for Spirit. The thick solid line is a 5-point running mean of Spirit's results. The thin solid lines are 21 TES profiles from a +/-10 sol window centred on the sol of EDL. The TES profile from the sol of EDL lies close to the centre of the cluster of TES profiles. Uncertainties are not shown. Figure 11: As Figure 10, but for Opportunity. Figure 12: Entry profiles from Viking Landers 1 and 2, Mars Pathfinder, Spirit, and Opportunity. Viking data are taken from [SEIFF&KIRK1977], who tabulated their results at 4 km intervals. Viking pressure and temperature results below 28 km were obtained using a different measurement technique and are not shown here. Squares indicate Viking Lander 1, triangles indicate Viking Lander 2. Pathfinder data (unmarked solid line) are taken from PDS volume MPAM_0001, which has a 4 Hz sampling rate [MAGALHAESETAL1999]. Spirit (dashed line) and Opportunity (dotted line) data come from the present paper. 5-point running means of the Pathfinder, Spirit, and Opportunity profiles are shown to reduce distracting high frequency oscillations. Uncertainties are not shown.