Source dependency of exospheric sodium on Mercury

Icarus 216 (2011) 387–402

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Source dependency of exospheric sodium on Mercury Y.-C. Wang a,⇑, W.-H. Ip a,b a b

Institute of Astronomy, National Central University, No. 300, Jhongda Rd., Jhongli City, Taoyuan County 32001, Taiwan, ROC Institute of Space Science, National Central University, No. 300, Jhongda Rd., Jhongli City, Taoyuan County 32001, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 16 February 2011 Revised 21 September 2011 Accepted 22 September 2011 Available online 4 October 2011 Keywords: Atmospheres, Evolution Atmospheres, Structure Mercury Mercury, Atmosphere Terrestrial planets

a b s t r a c t Due to a large solar radiation effect, the sodium exosphere exhibits many interesting effects, including the formation of an extended corona and a tail-like structure. The current suite of observations allows us to study some physical properties of the sodium exosphere, such as the source rates and the interaction with the surface, both experimentally and theoretically. In order to quantify the complex variations in the sodium exosphere in more detail, we use an exospheric model with the Monte-Carlo method to examine the surface interactions of a sodium atom, including the surface thermal accommodation rate and the sticking coefficient. The source rates from different components, such as the photon stimulated desorption (PSD), the meteoroid impact vaporization (MIV), and the solar wind ion sputtering (IS), can be constrained by comparing our exospheric model calculations with the published observational data. The detected terminator to limb (TL) ratio on the disk and the tail production rate can be explained with no sticking effect and small thermal accommodation rates. We also examine the best fit of the MIV source evolution, through comparison with the disk-averaged emission. The resultant discrepancy between the observations and the model fit may reflect the surface variation in the sodium abundance. A comprehensive mapping of the surface geochemical composition of the surface by the MESSENGER and BepiColombo missions should give us more information about the nature of this surface-bound exosphere. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction During Mariner 10’s close encounter with Mercury on March 29, 1974, the ultraviolet spectrometer experiment onboard the spacecraft detected the emission of H, He and O in the thin atmosphere of this inner-most planet (Broadfoot et al., 1974, 1976). The total atmospheric density is so low that the atmosphere is basically collisionless (see review by Killen and Ip (1999)). Almost ten years later, the strong optical emission of the sodium D lines at 589.0 nm and 589.6 nm was discovered through ground-based observations of Potter and Morgan (1985). Potassium emission was subsequently detected (Potter and Morgan, 1986). These important discoveries have stimulated a number of observational works and theoretical investigations, providing a wealth of information on the physical nature of Mercury’s surface and its interaction with the solar radiation and solar wind plasma, even in the absence of new spacecraft observations. For example, the possible formation of an extended coma and tail-like extension of sodium atoms from surface ejection was explored by Ip (1986) and Smyth (1986). Ion sputtering (IS) and meteoroid impact vaporization (MIV) have been discussed by various authors (Johnson and Baragiola, 1991; Cheng et al., 1987; Ip, 1993; Hunten et al., 1988; Cintala, 1992). In

⇑ Corresponding author. E-mail address: [email protected] (Y.-C. Wang). 0019-1035/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2011.09.023

addition, it has been reported that photon simulated desorption (PSD) plays an important role in generating the observed sodium emission (McGrath et al., 1986; Yakshinskiy and Madey, 1999). It has been suggested that the MIV may produce sodium atoms up to 15–30% of the sodium ions seen, with the rest mostly coming from PSD (Killen et al., 2001; Wurz and Lammer, 2003; Leblanc and Johnson, 2003; Milillo et al., 2005; Domingue et al., 2007). Mouawad et al. (2011) also suggested that the MIV source alone cannot match the observed emission distribution on the dayside. However, according to Kameda et al. (2009) MIV could produce more sodium atoms than expected, because the averaged column densities seemed to be more closely correlated with the interplanetary dust distribution. Through the estimation of the meteoroid flux, Borin et al. (2010) also determined the production rates from asteroidal and cometary dust impacts to be about one order of magnitude higher than that previously reported. The ballistic transport of the sodium atoms driven by solar radiation pressure was first modeled by Ip (1990), and by Smyth and Marconi (1995). In a theoretical calculation, Ip (1990) showed that the production rate should be on the order of 5.4–8.5  1024 atoms s1 through a comparison with a diskaveraged column density of sodium atoms of about 1.07  1011 atoms cm2 (Potter and Morgan, 1987; Potter et al., 1999; Killen and Ip, 1999; Wurz and Lammer, 2003; Schleicher et al., 2004; Milillo et al., 2005; Domingue et al., 2007). Ip (1990) also investigated the variation of the disk-averaged column density

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(Nc) in different orbital phases, indicating a trend for Nc to increase with a decrease in the ratio of the radiation pressure to the surface gravity (Potter and Morgan, 1987; Potter et al., 2007). Later on, more attention was given to investigation of the details of the source mechanisms as well as setting the constraints of the emission velocity distributions of the sodium atoms (Yakshinskiy and Madey, 1999; Wurz and Lammer, 2003; Mura et al., 2007). Leblanc and Johnson (2003) used experimental and theoretical energy distributions relevant to sodium atom ejection to produce threedimensional models of Mercury’s sodium exosphere and its extended coma structure in the anti-sunward direction. On the other hand, Potter (1995) pointed out the chemical sputtering effect where sodium atoms and H2O molecules can be created by the surface chemical reaction of solar protons with sodium-silicate compounds (i.e., Na2SiO3). An indication of the presence of such a chemical sputtering effect was observed in a proton beam bombardment experiment by Nash et al. (1975). Mura et al. (2009) introduced this particular effect into their time-varying exospheric model, comparing their results to the limb profile of the sodium emission obtained in Schleicher et al. (2004) using solar disk occultation measurements. The main point is to match the observed dawn-to-dusk asymmetry and polar brightening. Sprague et al. (1997) pointed out that the dawn-todusk asymmetry obtained by previous ground-based observations could be the result of the temporary storage of sodium atoms on the cold night side, with a sudden release at the dawn side as a consequence of ballistic transport (Ip, 1990). Recent observations made by the MESSENGER spacecraft during its encounters with Mercury have shown enhancement on the dusk-side rather than on the dawn-side (McClintock et al., 2009; Vervack et al., 2010), suggesting potential conflicts with the theoretical predictions. The polar brightening of the sodium emission has also been the focus of numerous discussions. Enhancement of disk sodium emission in localized regions of Mercury has been reported before (Potter et al., 1999) with solar wind proton precipitation at the clefts of the magnetosphere considered to be a likely cause (Sarantos et al., 2001). Burger et al. (2010) constructed a detailed model invoking an increase of the PSD source rate at higher latitudes to explain the close-up observations of the sodium ‘‘fantail’’ mapped by the MESSENGER spacecraft during its first two encounters with Mercury. We use a Monte-Carlo method to simulate the sodium exosphere, as described in Section 2. In order to set constraints on the surface interactions and the three major sources, in Section 3, we compare our steady exospheric model using different parameters with some published observational analysis. The source variations, including the most probable orbital evolution of the meteoroid impact vaporization (MIV) source and the possible existence of inhomogeneous surface abundance distribution, are discussed in Section 4. A short summary is given in Section 5.

2. Exospheric model In the beginning of the simulation, a packet including information about position and velocity will be generated. The initial weighting factor (Wp(0)) defines the number of sodium atoms represented by the packet. The packet will be ejected with an isotropic angular distribution away from the surface for the photon stimulated desorption (PSD) and the meteoroid impact vaporization (MIV) sources. The cos ve distribution is used for the ion sputtering (IS) source in the simulation, where ve is the zenith angle of the emission position (Sieveka and Johnson, 1984; Wurz and Lammer, 2003; Mura et al., 2007; Burger et al., 2010). A 4th order Runge–Kutta integration is used to trace the trajectory of the packet at a chosen true anomaly angle of Mercury (hTAA). In

addition to the gravitational force of Mercury, at times, the sodium atoms will be driven by the solar radiation pressure, which can be comparable to the gravity of Mercury (gM). The resonant scattering rate of the solar D lines can be calculated at the same time. Therefore, the column densities can be simultaneously transferred into the observed emissions. (For a detailed discussion of the solar radiation pressure acceleration (brp), the resonant scattering rate (the g-value), and the detected emission of the sodium atoms, see Appendix A). While integrating the trajectories of the packets, the sodium atoms on the sunlit side of Mercury will be photoionized. This effect is represented by a gradual decrease of the weighting factor (Wp(t)),

  Dt W p ðt þ DtÞ ¼ W p ðtÞ exp  tph

ð1Þ

where tph is the photoionization lifetime of the sodium atoms, which is 9.2  103 s at 0.387 AU (Huebner et al., 1992), and Dt is the integration time step. We set the numerical minimum counting rate to be 0.1% of the initial value (Wp(0)). As the packet reaches the minimum value, it will be excluded from the calculation. This means that the duration of a simulation packet on the sunlit side is 18 h on average. If the packet returns to the surface, its energy will be partially thermalized to the surface temperature. However, the thermal accommodation effect of the sodium atoms on Mercury is still unknown. We use the surface thermal model of Mercury from Wang and Ip (2008), which takes into account the diurnal temperature variation and the 3:2 spin–orbit resonant relation. Fig. 1 shows the surface temperature distribution obtained from the model at perihelion and aphelion. The highest temperature shift from noon time is due to the subsurface thermal diffusion effect (Wang and Ip, 2008). We define the re-emitted speed of the packet (Vout) as

V 2out ¼ aV 2th þ ð1  aÞV 2in

ð2Þ

where a has a value between 0 and 1 and indicates the thermal accommodation coefficient, Vth is the thermal speed determined randomly with Maxwellian distribution of the surface temperature of the impact location, and Vin is the speed before impact. Different surface accommodation rates (a = 0, 0.1, 0.25, 0.5, and 1) are tested in our simulations. The sticking effect at the surface is considered with a sticking coefficient S. At each bounce on the surface, the weighting factor of the packet (Wp(t)) will decrease as follows:

W p ðt þ DtÞ ¼ W p ðtÞð1  SÞ

ð3Þ

The sticking coefficient is still unknown and several cases are used in the simulation. Two extreme cases and three intermediate ones are set as follows:

8 S¼1 > > > > > > > > > SðTÞ ¼ step function > > : SðTÞ ¼ linear function

ð4Þ

While the sticking coefficient has been suggested to be approximately unity at T = 250 K and decreasing with increasing temperature (Killen et al., 2004), we assume two inverse temperature relation cases as a step function,

8 as T < 300 K > < SðTÞ ¼ 1 SðTÞ ¼ 0:33 as 600 K < T < 300 K > : SðTÞ ¼ 0 as T > 600 K

ð5Þ

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(b) Aphihelion

700

Latitude [deg]

600 45

500 400

0

300 200

-45

100 -90 mid-night

dawn

noon

dusk

mid-night

dawn

noon

Local Time

Temperature [K]

(a) Perihelion 90

0 mid-night

dusk

Local Time

and a linear function,

8 as T < 300 K > < SðTÞ ¼ 1 SðTÞ ¼ 1  ðT  300 KÞ=350 K as 650 K < T < 300 K > : SðTÞ ¼ 0 as T > 650 K

ð6Þ

In order to scale the source rate to the observed brightness, we define the numerical production rate (QN) as follows (Ip, 1990):

QN ¼

Np 1 X W p ð0Þ ðatoms s1 Þ Dt p¼1

ð7Þ

where Np is the number of ejected packets. There are three major sources, including photon stimulated desorption (PSD), meteoroid impact vaporization (MIV), and ion sputtering (IS), which contribute to the sodium exosphere at different levels. In addition to the source rate of IS, which might vary depending on how the solar wind interacts with the magnetosphere of Mercury, we can also expect some orbital dependence of the source rates from PSD and MIV. For example, in the case of the production rate of PSD (QPSD), which depends mostly on the photon flux, it can be reasonably assumed to have an inversely square dependency on the heliocentric distance (R),

Q PSD ðhTAA Þ /

1 R2

ð8Þ

Similarly, we can also assume the dependence of cosn h on the dayside surface, where h is the zenith angle between the ejection position and the Sun–Mercury lines (n = 1 is used in the simulations). In contrast, we assume a uniform production rate across the surface for the MIV sources, since we ignore the asymmetry effect due to the orbital motion and any violent events caused by meteoroid enhanced fluxes (Borin et al., 2010). The orbital evolution of the source rate from the MIV might not be easy to estimate since we do not understand the meteoroid velocity distributions near Mercury very well. Besides the heliocentric distance of Mercury, there might be other factors which could influence the production rate. We assume that the interplanetary dust is concentrated near the ecliptic plane and that the distance from the interplanetary dust plane (Z) can be replaced by the distance from the ecliptic plane (Zec). The dependence of the source rate (QMIV) can be formulated as in Kameda et al. (2009),

"   #   2 1 Z Q MIV ðhTAA Þ / a exp b R R

ð9Þ

where a and b are the proportionality parameters. The proportionality parameters a and b are chosen to be 1.4 and 50, respectively (Kameda et al., 2009). In the present simulations, the orbital variation of the IS source as the R2 dependence is considered, since the solar wind flux will change with the heliocentric distance. Our assumptions for the orbital variations of the source rates are

Normalized Source Rate [atoms s-1 ]

Fig. 1. The surface temperature distribution of Mercury at perihelion and aphelion, respectively.

1.4 1.2 1 0.8 0.6 0.4

QPSD( QMIV( QIS(

0.2 0 0

TAA ) TAA ) TAA )

90

180

270

360

TAA [deg]

Fig. 2. The source rate evolution at different orbital positions, i.e., true anomaly angle (hTAA), for the photon stimulated desorption (PSD), meteoroid impact vaporization (MIV), and ion sputtering (IS) sources. Note that the source rates are normalized by the values at perihelion.

summarized in Fig. 2. Note that the double-minima feature of the MIV production rate comes from the orbital inclination of Mercury itself. We use the initial speed distribution of the PSD with a Maxwellian distribution of temperature T = 1500 K as measured in the laboratory by Yakshinskiy and Madey (1999). The MIV sources are hotter, with temperatures of 3000–5000 K (Wiens et al., 1997; Milillo et al., 2005; Domingue et al., 2007). The energies for IS are higher with a high-speed tail distribution (Sieveka and Johnson, 1984; Wurz and Lammer, 2003; Mura et al., 2007). The most probable speed is 4 km s1. The initial speed distributions are shown in Fig. 3. The source rate and surface distribution from IS are more complicated, involving the interactions between the time-variable solar wind and the magnetosphere of Mercury. We use a 3D hybrid model as described in Wang et al. (2010) to simulate the situations of the first two MESSENGER flybys (Slavin et al., 2008, 2009). The same interplanetary magnetic field (IMF) strength and solar wind number densities were used for flyby cases 1 (M1) and 2 (M2) (see Table 1). The major difference between M1 and M2 is the magnetic field line orientation (i.e., northward IMF vs. southward IMF). Table 2 shows that the total ion precipitation rates are comparable for the two cases. Fig. 4 shows the surface precipitation distributions of the solar wind protons for these two cases. With a southward IMF for M2, the ions can reach the surface at lower latitudes and are also more heavily concentrated on the day-side. In addition to the IMF orientation, the IMF strength and the solar wind number density might also have an influence on the ion precipitation rate and distribution. For example, an interplanetary corona mass ejection (ICME)

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precipitation effect, and the PSD source. The chemical alternation of the sodium bearing regolith by the solar wind ion impact proposed by Potter (1995) was adapted as a potential source for PSD by Mura et al. (2009). Burger et al. (2010) also suggested that proton precipitation could induce rapid sodium diffusion in the regolith. Both schemes lead to more complicated surface source distributions for PSD, which would indirectly be stimulated by the solar wind ion precipitation. In the following, we simulate the ion-enhanced PSD (IE-PSD) as our fourth source, while mapping the surface distribution according to the hybrid simulation results (M1) on the dayside. The orbital evolution and the dependence on the solar photon flux have also been taken into account, being similar to the normal PSD sources.

1

10

Normalized Counts

PSD (1,500 K) MIV (4,000 K) IS 0

10

-1

10

-2

10

-1

0

1

10

10

10 -1

Speed [km s ]

3. Model calculations and observational comparisons

Fig. 3. The initial ejecting speed distributions for the photon stimulated desorption (PSD), meteoroid impact vaporization (MIV), and ion sputtering (IS) sources.

Table 1 Parameters for the hybrid simulation (Milillo et al., 2005; Glassmeier, 2000). Mercury Radius (RM) Magnetic moment

2440 km

Surface conductivity

300 nT R3M 104–106 S m1

Solar wind properties Interplanetary magnetic field (IMF) strength Solar wind number density (nsw0) Solar wind velocity Solar wind ram pressure Solar wind flux (Fsw0)

21 nT 32 cm3 430 km s1 9.88 nPa 1.38  109 cm2 s1

Table 2 Surface precipitation rates from hybrid simulations of the first two MESSENGER flybys.

IMF orientation hB with respect to z-axis (°) /B with respect to x-axis (°)

M1

M2

53 27

118 37

Solar wind H+ (95% nsw0) Total precipitation rate ðQ ½Hþ  Þ ð1025 s1 Þ h i Q ½Hþ  = 4pR2M  95%F sw0 Day-side precipitation rate

1.59 ± 0.9

1.525 ± 0.095

1.6 ± 0.9%

1.5 ± 0.07%

0.905 ± 0.015

1.365 ± 0.055

57 ± 2%

90 ± 2%

3.55 ± 0.09

3.285 ± 0.345

0.68%

0.64%

1.465 ± 0.105

2.795 ± 0.205

41 ± 2%

85.5 ± 2.5%

ðQ d½Hþ  Þ ð1025 s1 Þ Q d½Hþ  =Q ½Hþ  Solar wind He++ (5% nsw0) Total precipitation rate ðQ ½Heþþ  Þ ð1023 s1 Þ R2M

Q ½Heþþ  =½4p

 5%F sw0 

Day-side precipitation rate 23

ðQ d½Heþþ  Þ ð10 Q d½Heþþ  =Q ½Heþþ 

s

1

Þ

event can have a large impact on the configuration of Mercury’s magnetosphere as a whole. As a result, the corresponding sodium source from IS might be drastically different from that of the normal solar wind (Potter et al., 1999; Leblanc et al., 2003). We obviously need to simulate more cases with different solar wind parameters in order to better understand the relations. We will use the surface proton precipitation distribution of M1 case from Wang et al. (2010) as a representative one in our steady exospheric model. There might be some interesting interconnections between the IS source, which is directly influenced by the solar wind ion

3.1. Initial speed distribution The initial ejection speed distribution is a vital feature of different sources. Although different surface interactions can significantly influence the surface distribution of the exospheric atoms, the anti-solar tail production rate is determined mostly from the initial ejection speed (Vi). We use our sodium exospheric model to simulate the tail production rates at different orbital phases of Mercury (hTAA). When a packet reaches the anti-solar direction X > 5RM, where RM is the radius of Mercury, the weighting of the packet recorded at X  1RM will be added to the source rate of the tail. The acceleration in the solar radiation pressure allows the sodium atoms traveling with emission speeds below the speed needed to escape from the gravity of Mercury (4.2 km s1) to escape within one bounce. This could be at speeds as slow as 2 km s1 at the true anomaly angle (hTAA)  70°, if the particles are ejected at spots away from the sub-solar region on the day-side (Ip, 1986; Smyth, 1986). The ratios of the initial speed distributions for different sources and the resultant tail production rates (Qtail) from the model calculations are summarized in Table 3. We can see that about half of the sodium atoms produced by IS will escape to the tail. Since the observational results (Potter et al., 2002; Potter and Killen, 2008) showed that the tail production rate should be about 1–10% of the source rate, the IS source cannot be the dominant one supplying the sodium exosphere. By the same token, the consideration also indicates that its contribution to the disk emission will be relatively limited. Therefore, we first focus on the PSD, the IE-PSD, and the MIV as sources for the disk emission distribution and evolution. 3.2. Disk emission Potter et al. (2007) collected disk-averaged D2 line emission observations of Mercury from 1997 to 2003. We use our sodium exospheric model to simulate the disk-averaged D2 line emission inside 1.5 to 1.5 RM and compare the results with the observational data. In order to compare the observed brightness, we have to use different ‘‘scaling’’ source rates for the sodium exospheres for different sticking coefficient distribution cases. This is because, when the sticking coefficient increases, the weighting factor of the simulation packet will decrease faster; the same with the lifetime (see Eq. (3)). Therefore, even with the same source rate, the disk emission will be smaller if the sticking coefficient of the surface is larger. Table 4 summarizes the total source rates at perihelion (Qper) constrained by the observational data from Potter et al. (2007) for different patterns of sticking coefficients. Note that our test with different surface thermal accommodation rates shows that the disk-averaged brightness does not change significantly for the same surface sticking coefficient distribution.

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(a) MESSENGER flyby 1 (M1) 90

8

Latitude [deg]

6 5

0

4 3 2

-45

Flux [ x 108 cm-2 s-1]

7 45

1 0

-90

(b) MESSENGER flyby 2 (M2) 90

8

Latitude [deg]

45

6 5 4

0

3 2

-45

Flux [ x 108 cm-2 s-1]

7

1 -90 midnight

0 dawn

noon

dusk

midnight

Local Time Fig. 4. The solar wind proton precipitation distribution from hybrid simulations of MESSENGER’s first two flybys (M1 and M2). The black lines indicate the borders between the closed and opened field lines.

Table 3 Fractions of the sodium atoms escaping due to photon stimulated desorption (PSD), meteoroid impact vaporization (MIV), and ion sputtering (IS) sources, and the corresponding tail production rates (Qtail) calculated from our exospheric model.

PSD MIV IS

Vi P 2 km s1 (%)

Qtail (S = 0 and a P 0.1) (%)

Qtail (S > 0) (%)

6 42 94

0–20 5–20 35–60

0–5 5–10 35–60

Table 4 The scaling total source rates at perihelion (Qper) for different sticking coefficient distributions. Sticking coefficient

Total source rate at perihelion (Qper) (s1)

S=1 S=0 S = 0.33 S(T) = step function S(T) = linear function

9  1025 7  1024 5  1025 2.5  1025 2.5  1025

Fig. 5 shows the disk brightness distribution at perihelion (hTAA = 0°) derived from the photon stimulated desorption (PSD), the ion-enhanced PSD (IE-PSD), the meteoroid impact vaporization (MIV), and the ion sputtering (IS) sources with a thermal accommodation rate of a = 0.1 for different sticking coefficient distribution cases. Since the photoionzation lifetime is only about a few hours for the sodium atoms, the bouncing time across the surface is relatively short. Therefore, the sodium exosphere can preserve information of the surface source distribution more clearly. Fig. 6 shows the same brightness distribution as in Fig. 5, but with a

2 arcsec seeing effect as the diameter of Mercury is 5.5 arcsec. We have redistributed the brightness inside each grid node to have a Gaussian distribution with the full width at half maximum (FWHM) as the seeing smearing. The surface interactions, including the thermal accommodation rate and the sticking coefficient, can also control the surface distribution of the sodium atoms. If the thermal accommodation rate increases, the dependence of the re-emitted velocity on the surface temperature will also increase. Since the re-emitted speed is smaller in cooler regions, sodium atoms will accumulated more easily, especially on the nightside, where the lifetime of the sodium atom is longer. In contrast, if the sticking rate has an inverse dependence on the surface temperature, more sodium atoms will stick to the surface if the regolith temperature is lower. Consequently, the contribution to the exospheric column densities will decrease on the nightside. While the increase in the thermal accommodation rate will lead to a larger accumulation toward the nightside, the increase in the inverse dependence of the sticking rate will contribute lower column densities toward cooler regions. If the lifetime of the sodium atom is longer, as for the nonsticking case (S = 0), the thermal accommodation effect will be stronger. That is why we can have terminator enhancement for the S = 0 case and brighter limbs for the S(T) cases (see Figs. 5 and 6). As shown in Potter et al. (2006) (see their Figs. 1–4), the surface brightness distributions should not have very strong local enhancement on average, i.e., the missions at the north, south, and sunward limbs are approximately equivalent to each other. Therefore, we can expect the global sources (such as the PSD and the MIV) to be the dominant ones for the sodium exosphere. According to Killen et al. (2004), the source rate from the PSD is limited by the diffusion rate in the regolith, which is suggested

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Fig. 5. The disk brightness at perihelion (true anomaly angle hTAA = 0°) from the photon stimulated desorption (PSD), the ion-enhanced PSD (IE-PSD), the meteoroid impact vaporization (MIV), and the ion sputtering (IS) sources, with a thermal accommodation rate of a = 0.1 for different sticking coefficient (S) cases.

to be 107 cm2 s1 (Burger et al., 2010; Mouawad et al., 2011). If we take this diffusion limitation into consideration, the maximum total source rate from PSD is only about 2.5–3.75  1024 s1 for a cos h and a uniform distribution on the dayside. If the sticking rate at the surface is larger than 0, the sticking particles can also supply the PSD source. For the largest sticking case (S = 1), the PSD source rate shows an increase of about two times that of the diffusionlimited one. However, the dominant source of the sodium exosphere cannot be from the diffusion-limited PSD (DL-PSD) for S > 0 cases, since the scaling source rates are on the order of 1025 s1. In the following, we constrain the PSD source rate as a diffusion-limited one as QDL-PSD = 2.5  1024 s1 at perihelion, and its orbital variation is still taken into account. Meteoroid or ion impacts can over-turn the surface soil, or even feed the regolith with more sodium. Therefore, we do not set limitations on the MIV, IEPSD, and IS sources in our model. 3.3. Terminator to limb ratio After the analysis of separate source mechanisms, we now combine the surface brightness distributions at perihelion (hTAA = 0°) (as shown in Fig. 6) into one single exospheric structure, including those from the diffusion-limited PSD (DL-PSD), IE-PSD, and MIV

sources. To make a comparison with ground-based observations, we further add the 2 arcsec seeing effect into the model. The non-diffusion-limited part at perihelion (QnonDL = Qper  QDL-PSD) is taken to be the combination of the IE-PSD and the MIV sources. Fig. 7 shows the combined distributions with QMIV = 80%QnonDL and QIE-PSD = 20%QnonDL, cut through the center of the disk from limb to terminator (TL) and from south to north (NS) for different observational phase angles (hph). Since the IE-PSD source is concentrated in higher latitudes, the south/north enhancement will become stronger if this contribution increases. In order to confine the exosphere to have approximately equivalent limbs at the north, south, and, sunward sides, we can only have an IE-PSD source of 620%QnonDL for the S = 1 and 0.33 cases. As the scaling total source rate at perihelion (Qper) is smaller for the smaller sticking coefficient cases, the fraction of QDL-PSD can be larger, and the contributions from QnonDL will decrease. Since we have fixed the DLPSD source rates regardless of the surface interactions, larger fractions out of QnonDL for QIE-PSD cannot make larger contributions to the exosphere for smaller sticking cases, and therefore will not lead to too strong limb brightening in the south and north regions (650%QnonDL for S = 0, and 680%QnonDL for S(T) cases). As discussed in Section 3.2, we have different terminator to limb (TL) ratios for different combinations of surface thermal

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Fig. 6. The disk brightness with the 2 arcsec seeing effect, when the diameter of Mercury is 5.5 arcsec, at perihelion (true anomaly angle hTAA = 0°), from the photon stimulated desorption (PSD), the ion-enhanced PSD (IE-PSD), the meteoroid impact vaporization (MIV), and the ion sputtering (IS) sources, with a thermal accommodation rate of a = 0.1 for different sticking coefficient (S) cases.

accommodation coefficients (a) and sticking rates (S). We calculate the TL ratio after binning of the emission across the equator within Z = ±0.1RM (south–north direction) from the center of Mercury. The theoretical center of the bright disk for different observational phase angles (hph) can be determined, and the limb and the terminator emissions are computed from the center to ±1.2RM in the X direction (sunward–anti-sunward). Fig. 8 shows the resultant TL ratios for the exospheres from the combined sources (see Fig. 7). Since there is little contribution from the IE-PSD source near lower latitudes, the TL ratio is dominated by the MIV source for the S > 0 cases. The observational analysis by Potter et al. (2006) (see their Fig. 9) shows that the TL ratio is on average about 1 (in the range of 0.4–1.6). The TL ratios for the S(T) cases are only about 0.5 or below, therefore, the observed sodium exosphere distributions cannot be explained by this kind of sticking coefficient distribution. If the PSD source is not limited by the diffusion rates of the regolith and can make a larger contribution to the exosphere, there will be no significant change in the TL ratio. It could become even smaller if the surface distribution varies according to the cosn h relation. Consequently, we can conclude that the surface sticking coefficient of sodium atoms is not strongly dependent on the surface temperature. The most probable surface interaction indicated from the TL ratio will converge to S = 0 in the case of a > 0.

3.4. Tail production rate We use the same source rates and ratios as those adapted in Fig. 7, in our calculation of the disk-averaged D2 line brightness for compare with the observational data from Potter et al. (2007). Fig. 9a–e shows the comparisons with different thermal accommodation rates (a) and sticking coefficients (S). The orbital evolutions for the S(T) cases will have smaller brightness at the two sides of aphelion (70° < hTAA < 140° and 230° < hTAA < 300°) and larger ones near the perihelion (hTAA < 70° and hTAA > 300°), which dose not fit the observed variations very well. This is consistent with the TL ratio analysis. Fig. 9f–i shows the same modeling results for the tail production rate evolution. The observational results from Potter et al. (2002, 2008), McClintock et al. (2009) and Schmidt et al. (2010) are also displayed in the figures. We can find that the measured tail production rate evolution is not symmetric about hTAA = 180°, but decreases about two times from the outlag (0° < hTAA < 180°) to inlag (180° < hTAA < 360°) orbital phases. Fig. 10 shows the integration parameter variations along the trajectory of a sodium atom with an initial speed of 1.5 km s1 at hTAA = 70° and 290°. If a sodium atom can live long enough, the acceleration of the radiation pressure will keep influencing the particle. The sodium atom will ultimately be accelerated to greater

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

Fig. 7. The surface emission distributions at perihelion (hTAA = 0°) with the 2 arcsec seeing effect, cut through the center of Mercury from limb to terminator (TL) and from south to north (NS) for different observational phase angles (hph). The combination of the exosphere includes QDL-PSD, QMIV (80%QnonDL), and QIE-PSD (20%QnonDL). Note that the limb of the bright disk is at X = 1RM.

(a)

(b)

(c)

(d)

(f)

(g)

(h)

(j)

(k)

(l)

(e)

(i)

(m)

Fig. 8. The terminator to limb (TL) ratios at different true anomaly angles (hTAA) with different thermal accommodation rates (a) and sticking coefficients (S).

than the speed needed to escape Mercury, except near the sub-solar region, where the particles can only be driven toward the surface. However, the surface thermal accommodation will act as a decelerating mechanism for the particle. Therefore, if the thermal accommodation rate is small enough (a < 0.5) and the particle can live longer (S < 1), the sodium atoms can escape toward the tail after bouncing several times, even when the initial speed is less than 2 km s1. Since the radiation pressure acceleration is proportional to the resonant scattering solar photon flux, which will

change as the velocity with respect to the Sun varies, the anti-sunward orbital motion in the outlag phase and the sunward motion in the inlag phase of Mercury will have different effects on the cumulative radiation pressure acceleration of a sodium atom, as shown in Fig. 10. This is why the tail production rate evolution about hTAA = 180° will become asymmetrical. If the initial speed is large enough (P2 km s1), the sodium atom will escape within a single bounce and the cumulative difference from the radiation pressure acceleration will not alter the tail

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(a)

(f)

(b)

(c)

(d)

(e)

(g)

(h)

(i)

(j)

Fig. 9. The disk-averaged D2 line emission and the tail production rate evolution at different orbital positions (hTAA) for different values of the thermal accommodation rate (a) and sticking coefficient (S) values. The observational data are taken from Potter et al. (2002, 2007), Potter and Killen (2008), McClintock et al. (2009), and Schmidt et al. (2010).

Fig. 10. The integration path along the trajectory of a sodium atom with an initial speed of 1.5 km s1 at hTAA = 70° and 290°. The initial location of the particle at noon is at 30° latitude, and the ejection angle is randomly chosen by the model. The thermal accommodation rate is set to be 0.1 and no sticking effect (S = 0) is applied in the simulation. (a) and (c) show the variations of the resonant solar photon flux for a tail-ward sodium atom. (b) and (d) show the tail-ward distance (Xp), the speed (Vp) divided by the escape speed of Mercury (Ves), and the radiation pressure acceleration (brp) divided by the gravity of the Sun (gs) of the sodium atom along the trajectory. The dips of brp along the integration path indicate that the sodium atom enters the shadowed region when brp = 0. The solar D2 line is adapted from the ESO website.

production rates. Only when the source speed is small and the radiation pressure is large enough (near hTAA = 70° and 290°) will there be a difference between the integrated speeds as well as the tail production rates from the outlag to inlag phases. Our model calculations shows that only PSD or IE-PSD sources can have the kind of orbital asymmetry in the tail production rates indicated by the measurements (Potter et al., 2007, 2002; Potter and Killen,

2008; McClintock et al., 2009; Schmidt et al., 2010). This implies that if the PSD is limited by the diffusion rate of the sodium in the regolith, the sticking rate should approach to 0. This explains both the observed disk brightness and the asymmetry in the evolution of the tail production rate. Otherwise, the dominant MIV source will lead to symmetrical evolution of the escape rates for S > 0 cases.

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(a) North Pole

(b) South Pole 7

7

10

D2 Brightness [R]

10

6

10

M3 Obs. fittings 24 -1 QDL-PSD(cos ) = 2.4 10 s QIE-PSD(M1) = 2.1 1024 s-1 24 -1 QMIV = 2.6 10 s QIS(M1) = 9.6 1022 s-1 QNa(total)

6

10

5

5

10

10

4

4

10

M3 Obs. fittings 24 -1 QDL-PSD(cos ) = 2.4 10 s 24 -1 QIE-PSD(M1) = 2.1 10 s 24 -1 QMIV = 2.6 10 s -1 QIS(M1) = 9.6 1022 s QNa(total)

500

1000

1500

Altitude [km]

10

500

1000

1500

Altitude [km]

Fig. 11. The sodium emission variation as a function of the altitude at hTAA = 331° at the (a) north and (b) south poles, respectively. The observational fittings present the efolding fit of MESSENGER’s third flyby measurements as obtained from Vervack et al. (2010).

Fig. 12. The D2 line brightness distribution of the sodium exosphere at hTAA = 331° for perfect (upper panels) and 2 arcsec seeing effects (lower panels) with different observational phase angles (hph). The combination of the source rates from different mechanisms are the same as in Fig. 11.

Unlike the disk-averaged emissions, the tail production rate can be greatly influenced by the surface thermal accommodation rates (a). As can be seen in Fig. 9g, the simulations with a = 0.1 fit the observations quite well. As suggested by the measurements of the sodium D2 line profiles, the thermal accommodation rate might

Table 5 The parameters for the orbital evolution of the MIV source.

be small on Mercury (Killen et al., 1999). Mouawad et al. (2011) also suggested that if the PSD speed distribution is a Maxwellian one, the thermal accommodation rate might be 60.3. As a result, we can conclude from both the TL ratio analysis and the comparisons with the tail production rate evolutions, that the surface sticking effects is very small (0), and the thermal accommodation rates is about 0.1 for the sodium atoms on Mercury. 3.5. Altitude distribution

Proportionality parameters

Least v2 fitting

Assumptions (Section 3)

a b MIV source rate at perihelion (s1)

3.2 57 2.28  1024

1.4 50 2.2  1024

Now we focus on the case where the sticking coefficient is S = 0 and the thermal accommodation rate a = 0.1. We compare our simulation results with the altitude distributions from the MESSENGER flyby 3 (M3) measurements at hTAA = 331°, as reported by

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(b) Model fitting (Disk Emission) 2.5

2.5

TAA ) TAA ) TAA ) TAA )

2 1.5 1 0.5 0

Potter et al. (2007) Model MIV

2

23

QDL-PSD( QIE-PSD( QPSD( QIS(

D2 Brightness [MR]

Source Rate [ 1024 atoms s-1]

3

1.5

1

0.5

0 0

90

180 TAA

270

[deg]

360

(c) Tail Production Rate -1

QMIV variation

atoms s ]

2

0

90

180 TAA

[deg]

270

360

Tail Production Rate [ 10

(a) Least

20 Model MIV Potter et al. (2002, 2008) McClintock et al. (2009) Schmidt et al. (2010)

15

10

5

0 0

90

180 TAA

2

270

360

[deg]

Fig. 13. The orbital evolution of the best v fitting curve for the MIV source and the corresponding comparisons between the model and the observations from Potter et al. (2002, 2007), Potter and Killen (2008), McClintock et al. (2009), and Schmidt et al. (2010).

Fig. 14. The orbital evolution of the sodium emission without the seeing effect at the dawn-side disk and the tail.

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Vervack et al. (2010). Fig. 11 shows the D2 line emission distribution along the altitude at the north and the south poles, respectively. Vervack et al. (2010) fitted the measurement data to two types of slopes, including one with a smaller scale height near the surface and a larger one for the outer part. This implies that the sodium exosphere is at least composed of two different sources, one with lower energies and the other with higher speeds. Since the IE-PSD is a lower energy source, and the MIV is a higher one, we can find their respective ratios through comparison with the M3 measurements. From the analysis, we can set an upper limit for the MIV sources, which is 2.6  1024 s1 at hTAA = 331°. At the same time, we can also constrain an upper limit for the energetic IS source, which is 1023 s1 at perihelion. The attribution from the IS source is minimized here since we would not like to construct a sodium exosphere with too strong local enhancements near the higher latitudes from both the IE-PSD and the IS productions. Since we have scaled the total source rate (Qper) to be 7  1024 s1 and constrained the QDL-PSD = 2.5  1024 s1 at perihelion, the correspond-

ing IE-PSD sources can now be deduced with assumed orbital evolutions (see Fig. 2). The resultant fitting curve is indicated by the thick black lines (QNa) in Fig. 11.

4. Source rate variation 4.1. Orbital variation of the MIV source We have confined the surface interactions and the source rate distributions at hTAA = 331° through comparison with the published observational detection and analysis. However, at the beginning of the deduction, it was assumed that the MIV source has an orbital evolution, which is not certain based on present knowledge. Therefore, with the reasonable limitations obtained based on analysis from Section 3, we attempt to find a best fit for the MIV source evolution to the disk-averaged D2 line brightness. There are two proportionality parameters a and b in the assumed MIV source evolution as given in Eq. (9). The proportionality

Fig. 15. The orbital evolution of the sodium emission with a 2 arcsec seeing effect at the dawn-side disk and the tail.

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parameters suggest the dependence on the heliocentric distance (a) and on the vertical concentration distribution of the dust disk (b). We assume that the dust is populated on the ecliptic plane. We have simulated the disk-averaged emission only for the case with an observational phase angle of 90°. However, the disk-averaged brightness will change as the observational phase angle varies. Fig. 12 shows three different phase angle observations with perfect and 2 arcsec seeing effects, as well as the resultant diskaveraged emissions at hTAA = 331° as obtained from our model. The combination of the source rates from different mechanisms are the same as discussed in Section 3.5. We include the phase an-

gle effect in the comparison with the 107 observational values of Potter et al. (2007). The v2 fitting is used as the standard deviation (r) of the observational data to indicated the weighting factor,

v2 ¼

X ½D2 ðObs:Þ  D2 ðModelÞ2

ð10Þ

r2

i

We fix the source rates at hTAA = 331° (as deduced in Section 3.5) and include the orbital evolutions of other sources as well. Table 5 shows the best v2 fitting parameters and the previous assumptions made in Section 3 for the orbital evolution of the MIV source. The

(a) 2.5

Model Obs. Orbit 1 (dawn) Obs. Orbit 2 (dusk) Longitude

90

2

May 7, 2003 (dusk)

1.5

0

1 -90

Oct 6, 2008 (dusk)

0.5

Oct 6, 2008 (dawn) -180

0

180

3

(c) 2.5

D2 Brightness [MR]

(d)

Model Obs. Orbit 2 (dawn) Obs. Orbit 2 (dusk) Longitude

Model Obs. Orbit 2 (dawn) Obs. Orbit 1 (dusk) Longitude

90

2

May 7, 2003 (dawn)

1.5

0

1 -90

Center of Longitude [deg]

D2 Brightness [MR]

(b)

Model Obs. Orbit 1 (dawn) Obs. Orbit 1 (dusk) Longitude

Center of Longitude [deg]

180

3

0.5

0

0

90

180 TAA

270

360

0

90

[deg]

180 TAA

-180 360

270

[deg]

D2(Model) [MR]

Fig. 16. The four different types of disk-averaged sodium emission evolution. The data set from Potter et al. (2007) is divided into two resonant orbits where the observational orbit 1 started on April 5, 1997 and the observational orbit 2 started on July 15, 1997. The transit observation on May 7, 2003 (Schleicher et al., 2004) is located at observational orbit 2 and the MESSENGER flyby 2 detections on October 6, 2008 (McClintock et al., 2009) is at orbit 1. The center of longitudes in view of the bright disk with an observational phase angle of 90° at the dawn- and dusk-side are displayed for different resonant orbits to allow a better correlation of the detected surface regions.

2 Oct 6, 08 (dawn)

1.5 1

Oct 6, 08 (dusk)

May 7, 03 (dusk)

May 7, 03 (dawn)

0.5

D2(Obs.)

0 -0.5 -1 -180

-90

0

90

180

Longitude [deg] Fig. 17. The deviations of the observations from our modeling results with the best v2 fit as the orbital evolutions are transferred into the center of longitude in view of the bright disk.

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best fitted heliocentric distribution of the dust approaches to 1/R3 when a = 3.2. A larger value of b indicates that the dust is more concentrated in a plane. The fitting results indicate that the dependence on the heliocentric distance, and the vertical distance from the dust plane are important factors when considering the varying influence of the MIV source. Fig. 13a and b shows the orbital evolution of the best v2 fitting curve and the resultant comparisons between the model and the observations. Because we have larger proportionality parameters a and b, the distance away from the Sun and the dust plane will lead to larger decreases for the MIV source. Fig. 13c also displays the tail production rate with the best v2 fitting evolution, which is deduced from the disk-averaged observations. However, this best v2 fitting curve seems to underestimate the tail source rates at the outlag (0° < hTAA < 180°) orbital phases by about one tenth of the observational values near hTAA = 70°. There are two possible explanations for the difficulties in finding the best fitting tail production rate evolution as well as the disk-averaged emissions. There may be some other orbital dependences for the surface solar wind ion impact rates. Although we have included the approximate solar wind ion flux evolution with R2 relations in the IS source, the ion impact rates will also affect the IE-PSD source. However, because the variation of the disk-averaged brightness in the orbital region around 45° < hTAA < 120° is too large (it can range from about 500 to 2000 kR at very small orbital separation), the fitting at these orbital phases is not satisfactory. This kind of variation cannot be explained solely by the orbital evolution. There might be some other mechanisms that can influence the source rate even when Mercury is in the same position in relation to the dust disk and the Sun. We will describe one possible mechanism that could account for the non-orbital variations in the next section. To obtain a better understanding of our deducted results for the sodium exosphere, examine Figs. 14 and 15 showing the sodium emissions at different true anomaly angles (hTAA) on the disk and the tail, with and without the seeing effect, respectively. 4.2. Surface abundance variation So far we have assumed that the abundance and the diffusion rate of the sodium is uniform across the surface of Mercury. However, the regolith, soil, or rocks might not be so evenly distributed globally. Because the orbital motion of Mercury is in 3:2 spin–orbit resonant, one Mercury day is equal to two orbital evolutions or three sidereal days. Therefore, the observational data can be separated into two ‘‘resonant’’ orbits, since the observations will detect the sodium atoms emitted at different locations of the surface, even though Mercury is at the same orbital position. The measurements at the dawn-side and the dusk-side are also separated. Fig. 16 shows the four different types of time evolutions of the disk-averaged emissions. The observational data are arranged in such a way that the detections of about one quarter of Mercury can be characterized. The center of longitudes in view of the bright disk with an observational phase angle of 90° at dawn- and duskside for different resonant orbits are displayed for better correlation of the detected surface regions. The sodium emission evolutions obtained from the model, with an observational phase angle of 90° is also plotted for comparison. The large differences in disk-averaged emissions around 45° < hTAA < 120° can be clearly seen for the four different types of evolutions. We can conclude that this kind of variation in this orbital range can be characterized as due to the differences in the surface abundance or diffusion rates. Fig. 17 shows the deviation of the observations from our modeling results with the best v2 fittings, after transferring the orbital evolution into the center of longitude in view of the bright disk. If we exclude any source rate changes due to the influence of violent

alternations of the solar wind ion or meteoroid fluxes, the variations may indicate a longitudinal variation of the surface abundance. Although the bright disk in view may cover a large range of longitudes depending on the observational phase angle, a clear trend in the distribution can be identified. We can see that there may be a possible sodium-enriched region near longitudes of 90–180°. An area of depletion can also be found around longitude 45°. We may need more observational data to exclude the influence of non-orbital variations of the source rates by external impacts in future.

5. Summary We use a Monte-Carlo method to simulate the orbital variable sodium exosphere of Mercury. Through the comparison with the observations, we have obtained the following results:  From an analysis of the terminator to limb (TL) ratios, we find that the surface sticking coefficient cannot be strongly dependent on the surface temperature.  Both the TL ratio analysis and the tail production rate fits indicate that the surface sticking coefficient should be close to 0 and the thermal accommodation rate is about 0.1. The diffusionlimited PSD and the IE-PSD sources are required to have a larger contribution than the MIV source.  Through a comparison with the altitudinal distribution from MESSENGER flyby 3, we can set an upper limit for the energetic source including the MIV and the IS. The MIV source is about 2.6  1024 s1 and the IS source is about 1023 s1 at the true anomaly angle hTAA = 331°.  The best v2 fitting for the orbital evolution of the MIV source leads to a larger dependence of the heliocentric (/ R3.2) and the vertical distance from the dust disk (/ exp[57(Z/R)2]) than expected.  If the temporal variations of the dust and solar wind ion flux can be ignored, the deviations of the observations from our model fittings may indicate a sodium-enriched region near longitude 90–180°, and a depletion region near longitude 45°. The temporal variation in the source rates is also an important issue for the study of the sodium exosphere. More observational data are needed combined with the modeling fits to gain a better understanding of the short term effects in future. Acknowledgments We thank the reviewers for their useful comments which have greatly aided in improvement of this manuscript. This work is supported in part by an NSC Grant: NSC 96-2752-M-008-011-PAE and by the Ministry of Education under the Aim for Top University Program, NCU. Appendix A. Radiation pressure acceleration We use the solar Fraunhofer D line flux to determine the momentum transfer rate between the sodium atoms and the photons (Keller and Thomas, 1975; Smyth, 1979),

brp ¼

  1 hmD1 hmD2 g D1 þ g D2 mNa c c

ðA:1Þ

where brp is the solar radiation pressure acceleration, mNa is the mass of the sodium atom, m is the frequency of the photons at line D, and g is the resonant scattering rate of a sodium atom. We can calculate the resonant scattering rate (the g-value) as follows:

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(a) Mercury’s Orbit

des

V Vr

Mercury R TAA

Sun Aphelion

Perihelion AP

0.3 0.06 0.03 0 -0.03 -0.06

0

90

180

270

360

θ TAA [deg]

(c) brp on Na at rest (VNa = 0) 200 0.5 150

0.4 0.3

100 0.2

b rp / g M

Vr [km s-1 ]

0.4

Zec [AU]

R [AU]

(b) Orbital Motion of Mercury 10 5 0 -5 -10 0.5

Radiation Pressure Acceleration [cm s-2]

as

50 0.1 0

0

90

180

270

0 360

θ TAA [deg]

Fig. A.18. (a) Mercury’s orbit, where V is the orbital velocity, Vr is the relative velocity to the Sun, R is the heliocentric distance, and hTAA is the true anomaly angle. The dashed line represents the crossing of the orbital plane through the ecliptic plane, where xAP is the argument angle of perihelion, and Xas and Xdes are the ascending and descending nodes, respectively. (b) The orbital motion of Mercury, where Zec is the distance from the ecliptic plane. (c) The radiation pressure acceleration (brp) on sodium atoms at rest (VNa = 0).

g ¼ rNa F photon

ðA:2Þ

where rNa is the photo absorption cross-section of the sodium atom for photons at frequency m, and Fphoton is the corresponding photon flux. Due to Mercury’s eccentric orbit (e  0.21), the photon flux will vary with different true anomaly angles (hTAA) by a factor of 1/R2, where R is the heliocentric distance of Mercury. Furthermore, the frequencies of the solar photons will be shifted as the sodium atoms move. This can also alter the resonant photon flux for the sodium atom. Given the dependence of Mercury’s orbital distance (R) and its relative velocity to the Sun (Vr) (Fig. A.18a and b), we can estimate the g-value for a sodium atom at rest on Mercury. Similarly, the resultant radiation pressure acceleration brp can be determined as shown in Fig. A.18c. When Mercury’s speed relative to the Sun (Vr) is greatest at hTAA = 90° and 270°, the g-value and the radiation pressure forces on the sodium atoms should be larger, since the solar D lines are absorption lines. However, the heliocentric distance of Mercury (R) increases as Mercury approaches aphelion. This will cause the largest g-value and the radiation pressure forces on a sodium atom at rest on Mercury to shift toward perihelion, located at around hTAA  67° and 290° (Fulle et al., 2007; Potter et al., 2007; Killen et al., 2009). Note that in our exospheric model, the g-value and the radiation pressure acceleration will vary with the velocity of each particle at each time step. Not only the velocity of Mercury is taken into ac-

count. The column density (Nc) and observed emissions (E) can be calculated as follows:

Nc ðx; y; zÞ ¼

Np X

W p ðtÞ

ðA:3Þ

W p ðtÞgðtÞ

ðA:4Þ

pðx;y;zÞ¼1 Np X

Eðx; y; zÞ ¼

pðx;y;zÞ¼1

where Wp(t) is the weighting factor of a simulation packet, and Np is the total number of ejected packets. We can convert the emissions into the omnidirectional intensity in Rayleighs (1 R = 106 photons cm2 s1 emitted into 4p steradians) (McClintock et al., 2009; Burger et al., 2010). References Borin, P., Bruno, M., Cremonese, G., Marzari, F., 2010. Estimate of the neutral atoms’ contribution to the Mercury exosphere caused by a new flux of micrometeoroids. Astron. Astrophys. 517, A89 (1–5). Broadfoot, A.L., Kumar, S., Belton, M.J.S., McElroy, M.B., 1974. Mercury’s atmosphere from Mariner 10: Preliminary results. Science 185, 166–169. Broadfoot, A.L., Shemansky, D.E., Kumar, S., 1976. Mariner 10 – Mercury atmosphere. Geophys. Res. Lett. 3, 577–580. Burger, M.H., Killen, R.M., Vervack, R.J., Bradley, E.T., McClintock, W.E., Sarantos, M., Benna, M., Mouawad, N., 2010. Monte Carlo modeling of sodium in Mercury’s exosphere during the first two MESSENGER flybys. Icarus 209, 63–74. Cheng, A.F., Johnson, R.E., Krimigis, S.M., Lanzerotti, L.J., 1987. Magnetosphere, exosphere, and surface of Mercury. Icarus 71, 430–440.

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