Revisiting the Solar Wind Deceleration Upstream of the Martian Bow Shock Based on MAVEN Observations

The solar wind deceleration upstream of the Martian bow shock is examined using particle and magnetic field measurements obtained by the Mars Atmosphere and Volatile EvolutioN (MAVEN). Mars lacks a strong intrinsic magnetic field so its upper atmosphere extends beyond the Martian bow shock and interacts directly with the solar wind. Neutral atoms in the Martian upper atmosphere can be ionized through several physical processes and then start to move with the solar wind flow to form pickup ions. In return, the solar wind is expected to slow down due to the momentum transfer to the pickup ions. The present study surveys the MAVEN solar wind measurements between 2015 and 2019 to evaluate the solar wind deceleration upstream of the Martian bow shock. Different from previous studies of solar wind deceleration, our analysis carefully excludes the solar wind deceleration in the shock foot region. The average solar wind deceleration obtained is about 0.7% of the initial solar wind speed, much smaller than the values given by previous studies. Further calculation using several reasonable Martian upper atmosphere density profiles demonstrates that the deceleration observed is consistent with the pickup ion mass-loading scenario.


Introduction
The solar wind deceleration is a physical phenomenon commonly observed upstream of planetary and cometary bow shocks.It was first observed upstream of the Earth's bow shock in the 1960s (Formisano & Amata 1976).A series of studies have shown that the solar wind deceleration upstream of the Earth's bow shock is due to the interactions between the solar wind ions and the ultra-low-frequency waves excited by the shock-reflected ions (Bame et al. 1980;Bonifazi et al. 1980Bonifazi et al. , 1983;;Fu et al. 2009).The wave-particle interactions lead to an effective momentum exchange between the solar wind and shock-reflected ions.On the other hand, the solar wind deceleration upstream of the Martian bow shock was first observed by Phobos 2 in 1991 and exhibits different characteristics from that at Earth (Verigin et al. 1991).Verigin et al. (1991) found the solar wind deceleration to be about 100 km s −1 by analyzing the first three orbits of Phobos 2 measurements around Mars.A statistical study of 70 bow shock crossings showed that the average values of the solar wind deceleration in different regions upstream of the Martian bow shock are about 4%-7% of the undisturbed solar wind speed (Kotova et al. 1997).The values are higher than those observed upstream of the Earth's bow shock.In addition, solar wind deceleration is a common feature upstream of the Martian bow shock and occurs for both quasi-parallel and quasiperpendicular shocks (Barabash & Lundin 1993;Dubinin et al. 1994;Kotova et al. 1997).In contrast, upstream of the Earth's bow shock, the solar wind speed decreases mostly in front of quasi-parallel shocks (Zhang et al. 1995).Therefore, it is difficult to explain the solar wind deceleration upstream of the Martian bow shock using the aforementioned momentum transfer between the shock-reflected ions and the solar wind ions.Instead, the mass-loading mechanism was proposed to explain the difference in the solar wind deceleration between Mars and Earth (Verigin et al. 1991;Kotova et al. 1997;Zhang et al. 1997).
The mass-loading mechanism was first proposed to explain the solar wind deceleration upstream of comets (Biermann et al. 1967).Neutral atoms originating from comets are ionized through processes such as photoionization, charge exchange, and electron impact in the solar wind.After ionization, the newborn ions start to move together with the solar wind due to the motional electric field and the interplanetary magnetic field (IMF) in the solar wind and/or wave-particle interactions.They are, therefore, referred to as pickup ions (PUIs).The PUIs gain momentum from the solar wind during this mass-loading process and cause the solar wind to slow down.The massloading mechanism is believed to operate upstream of the Martian bow shock as well (Verigin et al. 1991;Kotova et al. 1997;Zhang et al. 1997).The Martian bow shock is relatively close to Mars because it does not have a strong intrinsic magnetic field.The Martian upper atmosphere (corona) extends beyond the Martian bow shock and interacts directly with the solar wind.PUIs are expected to arise when the neutral atoms in the Martian corona are ionized and, subsequently, slow down the solar wind flow.However, there are also studies showing that the mass-loading mechanism plays at most a minor role upstream of the Martian bow shock, because the densities of the hot oxygen corona and the hydrogen corona of Mars are too low to provide the observed values of solar wind deceleration.Some evidence indicates that rather than mass loading, large-amplitude Alfvén waves play the primary role in generating the observed signatures of solar wind deceleration from Phobos 2 (Dubinin et al. 2000a(Dubinin et al. , 2000b)).Zhang et al. (2006) used a gas dynamic model to estimate the solar wind deceleration caused by the PUI mass loading from the hot oxygen corona.Even considering extreme oxygen density profiles, the deceleration along the solar wind streamline was found to be only about 10-15 km s −1 (∼2%-3%).Halekas et al. (2017) analyzed data from the Solar Wind Ion Analyzer (SWIA) on board MAVEN, covering the period from 2014 to 2016, to study the solar wind deflection caused by the PUI mass loading.They found that the solar wind deflection ranges from −2.9 to −4.9 km s −1 (with a reference value of 0 if no deflection occurs), indicating a very weak mass-loading effect upstream of the Martian bow shock.
Clearly, whether the PUI mass loading can produce the solar wind deceleration observed upstream of the Martian bow shock is still an open question.The present study revisits this problem by examining the MAVEN measurements between 2015 and 2019.In the rest of the paper, Section 2 describes the relevant MAVEN instruments and the data selection criteria.Section 3 presents the observation results of solar wind deceleration in comparison with the previous studies.In Section 4, the solar wind deceleration due to the PUI mass loading is estimated along the MAVEN orbit using both the momentum conservation equation and the gas dynamic model of Zhang et al. (2006), and the results are then compared with the observations.Finally, Section 5 concludes the study and provides further discussions.

Instrumentation and Data Selection Criteria
MAVEN was launched by NASA in 2013 November and entered its Mars orbit in 2014 September.It is a polar orbit satellite with an orbital inclination of 75°and an orbital period of 4.5 hr.The apoapsis and periapsis of the MAVEN orbit are 6200 km and 150 km above the Martian surface, respectively.Thus, MAVEN can cross the Martian bow shock twice during each orbit and flies for a long time in the upstream solar wind (Jakosky et al. 2015), which makes it suitable for studying the solar wind deceleration upstream of the Martian bow shock.
The present study investigates the solar wind deceleration using the measurements of the Magnetometer (MAG) and SWIA instruments on board MAVEN.MAG provides threecomponent magnetic field data with a time resolution of 32 Hz or 1 Hz (Connerney et al. 2015).SWIA measures ions in the energy range of 25 eV-25 keV.Depending on the operating mode, SWIA returns three-dimensional ion velocity distribution data with different energy and angular resolutions.Additionally, SWIA measurements can be used to derive ion moment parameters (density, velocity, and temperature).Note that SWIA typically operates in the "fine" and "coarse" modes when MAVEN is in the solar wind and the Martian magnetosheath, respectively (Halekas et al. 2015).The mode transition can lead to variations in the obtained plasma moment data.The changes in the measured solar wind velocity caused by the mode transition will contaminate the calculation of the solar wind deceleration upstream of the Martian bow shock if SWIA happens to switch its operating mode in the solar wind.These events need to be carefully excluded from our analysis.
The moment parameters with a time resolution of 4 s from SWIA and magnetic field data with a resolution of 1 s from MAG between 2015 January 1 and 2019 December 31 were used in the present study.Each orbit of the spacecraft was divided into two segments: the outbound segment from the periapsis to the apoapsis and the inbound segment from the apoapsis to the periapsis.The data of different segments were then selected according to the following criteria: (1) the spacecraft has flown in the upstream solar wind for a duration greater than 70 minutes; (2) there is no SWIA mode switch within the interval for the subsequent solar wind deceleration analysis; (3) there are no abrupt solar wind velocity drops caused by interplanetary shocks or other transient phenomena; and (4) the apoapsis angle (the angle between the MAVEN apoapsis direction and the x-direction in the Mars Solar Orbital (MSO) coordinates) should be less than 45°.Note that the apoapsis angle varies with the MAVEN orbital precession.When the apoapsis angle is small, MAVEN moves a relatively long distance along the x-direction of the MSO coordinates during the orbital segment.Since the solar wind flows mostly along the negative x-direction, the solar wind deceleration due to the PUI mass loading would be more pronounced when the apoapsis angle is small.The above selection criteria yielded a total of 310 segments of data for the subsequent statistical analysis. ), where B is the total local magnetic field measured, and B u and B d represent the average upstream and downstream magnetic fields.In the present study, B u and B d are computed as the magnetic fields averaged over 10-25 minutes upstream and 5-15 minutes downstream of the first shock overshoot, respectively.Note that the numbers in the first line below the horizontal axis represent the relative times with respect to the moment of shock crossing, and the relative time has been flipped to positive values for the inbound case shown in Figure 1(f) for convenience.The numbers in the second line below the horizontal axis still give the original universal times.Additionally, to mitigate the influence of short-period fluctuations in the solar wind speed, the speed data were further smoothed with a 1 minute (15 data points) averaging window in Figures 1(c) and (f).Both Figures 1(c) and (f) demonstrate a gradual deceleration of the solar wind as it approaches the Martian bow shock.

Statistical Results
Since the solar wind speed far upstream is certainly not a constant and varies by itself in reality, the solar wind speeds obtained for different orbital segments (as shown in Figures 1(c) and (f)) have been averaged over all the 310 segments selected to make the solar wind deceleration signal of interest better stand out.The solid black line in Figure 2 shows the resultant average solar wind speed versus the relative time (flipped to positive values for the inbound segments).In the study of Kotova et al. (1997), the solar wind deceleration has been quantified as v v v ) , where v r is the undisturbed solar wind speed far upstream (defined as the average speed between 20 and 50 minutes of the relative time) and v s is the speed at the moment of shock crossing.The average solar wind deceleration calculated according to this definition is ∼7.8% in our data set.This value is slightly larger than but consistent with the numbers given by Kotova et al. (1997) based on Phobos 2 observations.It is important to note that the solar wind deceleration defined in Kotova et al. (1997) contains the solar wind slowing down inside the magnetic foot of the Martian bow shock, which is more due to the shock dynamics related to the shock-reflected ions rather than the PUI mass loading (Woods 1971).This probably explains why the PUI mass loading estimated in the previous studies (e.g., Zhang et al. 2006) was not sufficient to account for the solar wind deceleration calculated this way.
The width of the shock foot varies with several plasma parameters.Generally, a smaller shock normal angle (the angle between the shock surface normal vector and the upstream background magnetic field) allows the shock-reflected ions to return further upstream, resulting in a wider foot (Balikhin & Gedalin 2022).Additionally, shock-reflected ions with larger gyroradii can move further away from the shock front (Liu et al. 2022), so the width of the foot is also influenced by the upstream ion temperature.For the Martian bow shock, MAVEN observations indicate that the shock foot width is smaller than the upstream local proton convected gyroradius (r ci = v sw /ω ci , where v sw is the solar wind speed, and ω ci is the upstream proton cyclotron frequency) when the shock is quasiperpendicular (Burne et al. 2021).In order to focus on the solar  wind deceleration related to the PUI mass loading, the present study chooses to define the solar wind deceleration as the difference between the average speeds during 65-70 minutes and 20-25 minutes of the relative time with respect to the shock crossing moment.The relative time interval of 20-25 minutes has been chosen to ensure that MAVEN is at a distance exceeding r ci upstream of the bow shock for most of the data segments.With the new definition, the average solar wind deceleration is approximately 0.7% of the undisturbed solar wind speed.This is significantly smaller than the values given by Kotova et al. (1997), but consistent with the weak massloading results revealed by Halekas et al. (2017) in terms of the observed solar wind deflection.On the other hand, it needs to be clarified that the solar wind deceleration value would vary if one chooses different relative time intervals to evaluate the solar wind deceleration.This is indeed expected in the massloading mechanism because the mass-loading effect accumulates along the solar wind streamline.

Momentum Conservation and Gas Dynamic Models
In the mass loading mechanism, neutral atoms initially at rest (in the MSO reference frame) are ionized and then picked up by the solar wind, thereby gaining momentum.Due to the conservation of momentum, the solar wind must lose momentum and thus slows down.
The law of conservation of momentum is first used to estimate the solar wind velocity change from the PUI mass loading.The calculation starts with the undisturbed solar wind of a certain density of n sw,0 at x = 5 R M in the MSO coordinates, where R M is the Mars radius.PUIs are then gradually generated in the solar wind as it flows along the negative x-direction with an initial speed of v sw,0 .The solar wind region upstream of the Martian bow shock is numerically divided into uniform cubes of 20 km × 20 km × 20 km.For each cube, the incoming flow exchanges momentum with the PUIs newly generated inside the cube, and they then leave the cube with the same bulk flow velocity.Thus, the law of conservation of momentum leads to Here, m sw is the mass of the solar wind proton, n sw,i and v sw,i are the solar wind density and velocity flowing out of the ith cube, n i s PUI, is the PUI density of species s in the ith cube, m s PUI is the PUI mass of species s, and the summation is over different PUI species involved.It should be noted that the calculation assumes an instantaneous momentum exchange between the newborn PUIs in a cube and the incoming plasma flow.As will be further discussed in Section 5, this assumption is overly simplistic but should provide a reasonable upper limit for the solar wind deceleration caused by the PUI mass loading.
PUIs in the solar wind upstream of the Martian bow shock come from the neutral atoms in the Martian corona after they are ionized through photoionization, charge exchange, and electron impact.The PUI number density of species s newly generated in the ith cube is given by where n i s neu, is the neutral corona density of species s in the cube, Δx = 20 km is the size of the cube (along the x-axis), and f s ph , f i s ex, , and f s el are the photoionization, charge exchange, and electron impact frequencies, respectively.Only hydrogen and oxygen PUIs are considered in the present calculation because previous studies have shown that the mass loading is mainly contributed by these two species upstream of the Martian bow shock (e.g., Kotova et al. 1997).The hydrogen and oxygen corona densities are taken from Modolo et al. (2016).As shown in Figure 3(a), they decrease rapidly with altitude and differ between solar maximum and minimum.In addition, the photoionization frequency also changes between solar maximum and minimum.In the present calculation, the values adopted are f f 4.28 10 s , 31.25 ´-for solar minimum (Modolo et al. 2005).Moreover, the charge exchange between the solar wind protons and the neutrals in the Martian upper atmosphere leads to the increase of n i s PUI, (and the decrease of n sw,i ).The charge exchange frequency is given by f , where σ s represents the charge exchange cross section of species s.In this study, the charge exchange cross sections for oxygen and hydrogen are taken as 8 × 10 −16 cm 2 and 2 × 10 −15 cm 2 , respectively (Stebbings et al. 1964;Mott & Massey 1965).Finally, the electron impact frequency is usually much lower than the other ionization frequencies.In the upstream region of the Martian bow shock, the electron impact frequencies of oxygen and hydrogen are typically in the range of 10 −9 -10 −7 s −1 (Cravens et al. 1987), and the present study assumes a value of 10 −8 s −1 for both species.
Equation ( 2) can be integrated/added over the cubes along a streamline of the solar wind, which is simply along the negative x-direction in the present study, to get the PUI density in a certain cube.The PUI density obtained can then be fed to Equation (1) to calculate the solar wind velocity v sw,i .The calculation is performed for all the cubes upstream of the Martin bow shock, and Figure 3(b) presents the solar wind velocity calculated with n sw,0 = 2.5 cm −3 , v sw,0 = 400 km s −1 , and other parameters corresponding to the solar minimum conditions.Since the solar wind has been simply assumed to flow along the negative x-direction and the neutral densities vary only with the altitude above the surface of Mars, the resultant solar wind deceleration due to mass loading is strictly axisymmetric about the x-axis through the subsolar point.Therefore, Figure 3(b) only displays the result in the x-z plane.The white area is the region downstream of the Martian bow shock, whose location is provided by Gruesbeck et al. (2018).As expected, the maximum solar wind deceleration occurs immediately upstream of the bow shock.Interestingly, the maximum deceleration values are similar (∼2%) at the subsolar location and in the flank regions.Although the deceleration rate (the deceleration over a certain distance along the negative xdirection) is most significant at the subsolar location, the deceleration occurs over a longer distance in the flank regions.On the other hand, the maximum deceleration of 2% in Figure 3(b) seems larger than the value of 0.7% when the deceleration is defined as the difference between the average speeds during 65-70 minutes and 20-25 minutes of the relative time with respect to the shock crossing moment (as described in Section 3).This is because 20-25 minutes of the relative time corresponds to regions at some distance from the immediate bow shock upstream.So the deceleration ratio obtained is naturally smaller.
In order to be better compared with the MAVEN measurements, the solar wind velocity along the MAVEN trajectory needs to be derived.However, the undisturbed solar density (n sw,0 ) and velocity (v sw,0 ) need to be first figured out for each of the 310 segments of data.In this regard, the most upstream point (with the largest x-value in the MSO coordinates) of the MAVEN trajectory during a certain data segment is identified.With the observed solar wind density and velocity at this location, Equation (1) is then used to trace backward along the solar wind streamline to the location of x = 5 R M to get n sw,0 and v sw,0 .Using the n sw,0 and v sw,0 derived, the solar wind velocities at the other locations along the MAVEN trajectory during this data segment can be calculated, similar to how the result shown in Figure 3(b) has been obtained.Figure 3(c) shows the calculation result for the segment of 15:24:57-16:34:57 on 2019 March 24 as an example.Here the black curve represents the spacecraft trajectory, and the color indicates the solar wind speed variation upstream.
The same calculation illustrated by Figure 3(c) has been done for all the 310 data segments.The solar wind velocities obtained, after being averaged over all the segments according to the relative time with respect to the shock crossing moment, are shown as the two blue lines in Figure 2. The solid and dashed blue lines correspond to the solar maximum and minimum conditions, respectively.The 5 yr data period from 2015 to 2019 falls between solar maximum and minimum (Petrovay 2020;Courtillot et al. 2021).Therefore, as expected, the black line obtained from observation lies between the solid and dashed blue lines except between 0 and 10 minutes of the relative time.Thus, the mass loading mechanism can largely explain the solar wind deceleration observed.The deviation from the theoretical results given by the simple conservation of momentum between 0 and 10 minutes of the relative time is not surprising, because the solar wind deceleration there is more due to the shock dynamics related to the shock-reflected ions rather than the PUI mass loading.On the other hand, the comparison of the two blue curves with the black curve between 20 and 70 minutes of the relative time suggests that the observation result (the black curve) agrees better with the theoretical result under the solar minimum conditions.This could be due to the assumption of instantaneous momentum exchange in the momentum conservation model, which likely overestimates the solar wind deceleration.
In addition, we also utilized the gas dynamic model given by Zhang et al. (2006) to estimate the solar wind deceleration caused by mass loading and compared the results with the simple momentum conservation model.The gas dynamic method still assumes that momentum exchange occurs instantaneously but takes into account the changes in energy and pressure of the plasma fluid during the mass-loading process.The calculation details of this model are consistent with the momentum conservation model described earlier, but the ratio of the solar wind speeds flowing into and out of a certain cube is determined by Equation (4) in Zhang et al. (2006; with the adiabatic index γ = 5/3 in our calculation).The results are shown in Figure 2 as the solid and dashed red lines, corresponding to the solar maximum and minimum conditions, respectively.Similar to the results from the momentum conservation model, the general trend matches well with the observation result.However, compared to the momentum conservation model, the gas dynamic model predicts higher deceleration values.

Conclusion and Discussion
Using the SWIA data from MAVEN between 2015 and 2019, the present study analyzes the solar wind deceleration upstream of the Martian bow shock.The observation results are consistent with the previous study that the deceleration measured at the bow shock crossing is ∼7.8% of the undisturbed solar wind velocity.However, this deceleration value contains the solar wind slowing down inside the magnetic foot of the Martian bow shock, which is more due to the shock dynamics related to the shock-reflected ions rather than the PUI mass loading.Excluding the influence of the magnetic foot, the deceleration is approximately 0.7%.Furthermore, both the simple momentum conservation model and the gas dynamic model are used to estimate the solar wind speed changes caused by the PUI mass loading.The results demonstrate good agreement with the observation in the range where the PUI mass loading is expected to dominate the solar wind deceleration.
In Section 4, the momentum exchange between the newborn PUIs and the solar wind has been assumed to complete instantaneously (or in a very short time).When the IMF in the solar wind is perpendicular to the solar wind velocity, newly ionized particles can be quickly accelerated by the motional electric field and the IMF in the solar wind, so momentum exchange can happen quickly.In the case that the IMF is parallel to the solar wind velocity, the momentum exchange between the solar wind and the newborn ions can only be achieved gradually through wave-particle interactions.This process takes minutes or tens of minutes (Cowee & Gary 2012;Cowee et al. 2012;Cheng et al. 2023), longer than the transit time it takes for the solar wind to pass through the region upstream of the Martian bow shock.In the more general situation that the IMF is at an angle to the solar wind velocity, the exchange of momentum perpendicular to the IMF might occur quickly, while the momentum exchange in the parallel direction takes time to complete.This implies that the real solar wind deceleration caused by the PUI mass loading would be smaller than the results presented in Section 4. Indeed, Dubinin et al. (2000b) have provided a correction factor for the solar wind deceleration caused by the PUI mass loading considering the IMF direction.For the typical conditions upstream of Mars, the correction factor is estimated to be ∼0.5-0.8.Including the correction factor will shift the theoretical curves in Figure 2 upwards and make the model and observational results better agree with each other.
For a newborn PUI initially at rest, it is first accelerated along the motional electric field in the solar wind and then gyrates around the local IMF to form a cycloidal trajectory.Over a time interval much longer than the PUI cyclotron period, the average PUI velocity is approximately the solar wind velocity (along the negative x-direction).However, since the scale of the PUI cycloidal trajectory can be large in comparison with the effective solar wind deceleration region upstream of the Martian bow shock and more PUIs are produced closer to the bow shock, the PUI velocities on average should have both a component in the negative xdirection and a component along the motional electric field.While the former can cause the solar wind to slow down in the negative x-direction, the latter is expected to lead to a lateral solar wind deflection (Halekas et al. 2017).Indeed, similar (but larger) plasma flow deflections have been shown to occur in the Martian magnetosheath and were explained using a two-fluid model with the assumption that the PUIs have velocities along the motional electric field (Dubinin et al. 2018;Romanelli et al. 2020).Such a two-fluid model should be generalized to assess the solar wind deflection and deceleration upstream of the Martian bow shock.It probably can give better results than the simple momentum conservation model used in the present study, but the assumption of the PUI velocities being along the motional electric field needs to be improved.Moreover, the role that wave-particle interactions play in causing the momentum exchange between the PUIs and the solar wind also warrants further investigation.
It should be mentioned that our analysis only discusses the different solar wind decelerations under the solar maximum and minimum conditions.A series of studies using Mars Express and MAVEN observations have shown that the neutral densities in the Martian atmosphere, particularly hydrogen, are strongly modulated by seasons on Mars (Zou et al. 2011;Dong et al. 2015;Yamauchi et al. 2015;Halekas 2017).These seasonal variations also have an influence on the solar wind deceleration.Furthermore, the apoapsis altitude of the MAVEN orbit is only about 6200 km.This has forced us to derive the unperturbed solar wind density and velocity from the observation at the most upstream point of the MAVEN trajectory.In contrast, the orbit of the Tianwen-1 mission has an apoapsis altitude of 12,500 km (Zou et al. 2021).The measurements made by Tianwen-1 should be more suitable to study the solar wind deceleration upstream of the Martian bow shock, as the solar wind deceleration caused by the PUI mass loading is a cumulative effect over long distances.

Figure 1
Figure 1 illustrates two example cases to demonstrate how the solar wind deceleration is quantified from the MAVEN measurements in the present study.The left and right columns present one outbound segment and one inbound segment, respectively.The first and second rows give the total magnetic field (Figures 1(a) and (d)) and the solar wind speed (Figures 1(b) and (e)) in terms of universal time.Figures 1(c) and (f) further display the solar wind speed during the 70 minute intervals after and before the shock crossing for the outbound and inbound cases, respectively.The moment of shock crossing is defined as when B B B 2 u d = + () , where B is the total local magnetic field measured, and B u and B d represent the average upstream and downstream magnetic fields.In the present study, B u and B d are computed as the magnetic fields averaged over 10-25 minutes upstream and 5-15 minutes downstream of the first shock overshoot, respectively.Note that the numbers in the first line below the horizontal axis represent the relative times with respect to the moment of shock crossing, and the relative time has been flipped to positive values for the inbound case shown in Figure1(f) for convenience.The numbers in the second line below the horizontal axis still give the original universal times.Additionally, to mitigate the influence of short-period fluctuations in the solar wind speed, the speed data were further smoothed with a 1 minute (15 data points) averaging window in Figures1(c) and (f).Both Figures1(c) and (f) demonstrate a gradual deceleration of the solar wind as it approaches the Martian bow shock.Since the solar wind speed far upstream is certainly not a constant and varies by itself in reality, the solar wind speeds obtained for different orbital segments (as shown in Figures1(c) and (f)) have been averaged over all the 310 segments selected to make the solar wind deceleration signal of interest better stand out.The solid black line in Figure2shows the resultant average solar wind speed versus the relative time (flipped to positive values for the inbound segments).In the study ofKotova et al. (1997), the solar wind deceleration has been quantified as v v v

Figure 1 .
Figure 1.Example outbound segment (left column) and inbound segment (right column): (a) and (d) the total magnetic field, (b) and (e) the solar wind speed, (e) the solar wind speed within the 70 minute interval after the shock crossing (corresponding to the time interval 03:24:04-04:35:04 in (b)), (f) the solar wind speed within the 70 minute interval before the shock crossing (corresponding to the time interval 09:37:46-08:27:46 in (e)).The horizontal axis represents the universal time in the first two rows, and the vertical red dashed lines mark the shock crossing moments.The horizontal axis in the last row is the relative time with respect to the shock crossing (the numbers in the second line still give the corresponding universal times).Note that the relative time has been flipped to positive values for the inbound segment in (e) for convenience.

Figure 2 .
Figure 2. The average solar wind speed vs. the relative time with respect to the shock crossing.The solid black line is from MAVEN observations, while the red and blue lines represent theoretical results as labeled.

Figure 3 .
Figure 3. (a) The hydrogen (black lines) and oxygen (red lines) corona density profiles adopted.The solid and dashed lines represent the results at solar maximum and minimum, respectively.(b) The solar wind velocity upstream of the Martian bow shock calculated using the law of conservation of momentum (Equation (1)) using typical parameters under the solar minimum conditions.The black curves are the contour lines and the color represents the solar wind speed according to the right color bar.(c) The solar wind velocity calculated for the MAVEN trajectory from 15:24:57 to 16:34:57 on 2019 March 24.The black curve represents the MAVEN trajectory and the color gives the solar wind velocity upstream.