Variation of the Interplanetary Shocks in the Inner Heliosphere

Figure 1 shows examples of the interplanetary shocks and associated interplanetary drivers as detected by the Voyager 1 spacecraft. From top to bottom, the panels are the solar wind plasma speed (V_sw), density (N_sw), temperature (T_sw), IMF magnitude (B₀), and IMF components B_R , B_T , and B_N , respectively. The IMF data are in the spacecraft-centered radial tangential normal (RTN) coordinate system, where $\hat{{\boldsymbol{R}}}$ is the unit vector directed from the Sun to the spacecraft, and $\hat{{\boldsymbol{T}}}={\boldsymbol{\Omega }}\times \hat{{\boldsymbol{R}}}/| {\boldsymbol{\Omega }}\times \hat{{\boldsymbol{R}}}|$, where Ω is the Sun's spin axis. The right-hand system is completed by $\hat{{\boldsymbol{N}}}$. The high-resolution data are obtained from NASA's COHOWeb (https://omniweb.gsfc.nasa.gov/coho/), which provides access to the heliospheric magnetic field, plasma, and spacecraft position data for several spacecraft.

Zoom In Zoom Out Reset image size

During 1979 January 12–15 (days 12 through 15 of the year), when Voyager 1 was at R_h ∼ 4.96 au, it encountered an ICME (Figure 1, left panels). The fast (V_sw ∼ 524 km s⁻¹) ICME, propagating faster than the upstream plasma by more than the local magnetosonic speed (V_ms ∼ 61 km s⁻¹), generated an interplanetary shock at ∼15:07 UT on day 13 (shown by the vertical dashed–dotted line), identified by sudden increases in the solar wind plasma parameters. V_ms is estimated as $\sqrt{{V}_{{\rm{A}}}^{2}+{V}_{S}^{2}}$, where V_A is the Alfvén speed (${B}_{0}/\sqrt{{\mu }_{0}\rho }$), and V_s is the sound speed ($\sqrt{5{k}_{{\rm{B}}}({T}_{p}+{T}_{e})/3{m}_{p}}$), μ₀ is the free space permeability, ρ denotes the solar wind mass density, k_B is the Boltzmann constant, T_p and T_e are the proton and electron temperatures, respectively, and m_p is the proton mass. Considering the upstream and downstream intervals of ∼10 minutes around the shock, the shock is characterized by sharp increases in V_sw from ∼399 to ∼524 km s⁻¹, in N_sw from ∼0.1 to ∼0.3 cm⁻³, in T_sw from ∼0.1 × 10⁵ to ∼1.1 × 10⁵ K, and in B₀ from ∼0.6 to ∼1.8 nT. The magnetic field ratio (r_B ∼ 3.0) and plasma density ratio (r_N ∼ 3.0) between the downstream and upstream solar wind are defined as the shock compression. Due to the simultaneous increases in V_sw, N_sw, T_sw, and B₀ across the shock, it is classified as a FF shock.

The upstream and downstream plasma (speed) and IMF vectors are utilized to determine the shock normal $\hat{{\boldsymbol{n}}}$ using the mixed-mode technique (Abraham-Shrauner & Yun 1976):

$$\begin{eqnarray}&&\hat{{\boldsymbol{n}}}=\displaystyle \frac{({\boldsymbol{\Delta }}{\boldsymbol{B}}\times {\boldsymbol{\Delta }}{{\boldsymbol{V}}}_{\mathrm{sw}})\times {\boldsymbol{\Delta }}{\boldsymbol{B}}}{| ({\boldsymbol{\Delta }}{\boldsymbol{B}}\times {\boldsymbol{\Delta }}{{\boldsymbol{V}}}_{\mathrm{sw}})\times {\boldsymbol{\Delta }}{\boldsymbol{B}}| },\end{eqnarray} \tag{ 1 }$$

where Δ B ( = B _d − B _u ) and Δ V _sw ( = V _d − V _u ) are the IMF and V_sw jumps, respectively, subscripts u and d denote the upstream and downstream values across the shocks. The shock normal is found to be [0.92, 0.29, −0.26].

Assuming the conservation of mass flux in the shock rest frame, the shock speed V_sh is estimated as

$$\begin{eqnarray}&&{V}_{\mathrm{sh}}=\displaystyle \frac{\hat{{\boldsymbol{n}}}.({\rho }_{{\boldsymbol{d}}}{{\boldsymbol{V}}}_{{\boldsymbol{d}}}-{\rho }_{{\boldsymbol{u}}}{{\boldsymbol{V}}}_{{\boldsymbol{u}}})}{{\rho }_{{\boldsymbol{d}}}-{\rho }_{{\boldsymbol{u}}}}.\end{eqnarray} \tag{ 2 }$$

V_sh along the shock normal is determined to be ∼531 km s⁻¹. The shock angle (θ_Bn) between the shock normal $\hat{{\boldsymbol{n}}}$ and the upstream IMF vector is determined to be ∼82.8°, implying this to be a quasi-perpendicular shock. The magnetosonic Mach number M_ms, defined as the ratio of V_sh (relative to the upstream V_sw) to the upstream V_ms, is computed to be ∼2.2 (confirming it to be a shock, not a wave). Thus, the FF shock at ∼15:07 UT on day 13 detected at ∼4.96 au is found to be a quasi-perpendicular shock propagating at a relative speed ∼2.2 times of the ambient magnetosonic speed at an angle of ∼82.8° relative to the ambient magnetic field. Detail description of the shock analysis method by solving the Rankine–Hugoniot conservation equations (Rankine 1870; Hugoniot 1887, 1889) can be found in literature (e.g., Smith 1985; Tsurutani & Lin 1985; Tsurutani et al. 2011; Hajra et al. 2016, 2020; Hajra & Tsurutani 2018a).

The FF shock on day 13 is followed by a downstream interplanetary sheath from ∼15:07 to ∼20:00 UT on day 13 (shown by a green bar on the top, Figure 1, left panel). Following the sheath, the shaded region (marked by a red bar on the top, Figure 1, left panel), from ∼20:00 UT on day 13 to ∼05:36 UT on day 14, is characterized by an enhanced IMF B₀, smooth and slow rotation in B_T , and reduced T_sw. These are the typical signatures of a flux rope magnetic cloud (MC; Burlaga et al. 1981; Klein & Burlaga 1982; Tsurutani & Gonzalez 1997) in the interplanetary space.

A CIR was encountered by Voyager 1 when it was at R_h ∼ 8.30 au during 1980 June 26–29 (between days 178 and 181 of the year). The CIR (marked by a blue bar on the top, Figure 1, right panel) can be identified between a slow-speed stream with V_sw ∼ 419 km s⁻¹ on day 178 and a coronal hole emanated HSS with V_sw ∼ 552 km s⁻¹ on day 181. The CIR is a compressed plasma characterized by high plasma density (N_sw ∼ 0.7 cm⁻³), temperature (T_sw ∼ 2.2 × 10⁵ K), and magnetic field (B₀ ∼ 3.0 nT) compared to the ambient plasma. Most interestingly, the CIR is bounded by two shocks at its leading and trailing ends.

At ∼06:33 UT on day 178 (at the CIR leading edge), a sharp increase in V_sw from ∼427 to ∼484 km s⁻¹ is accompanied by increases in N_sw from ∼0.2 to ∼0.3 cm⁻³, in T_sw from ∼0.1 × 10⁵ to ∼0.2 × 10⁵ K, and in B₀ from ∼1.7 to ∼2.5 nT. By definition, this is an FF shock with the compression ratios of r_B ∼ 1.5 and r_N ∼ 1.5. Following the above-mentioned techniques, the FF is found to be propagating with a speed V_sh ∼ 557 km s⁻¹ at an angle θ_Bn ∼ 87.2° relative to the ambient IMF. The estimated M_ms is ∼1.3. The result implies that the FF shock detected at ∼8.30 au was quasi-perpendicular, propagating at a relative speed of ∼1.3 times the magnetosonic speed at an angle ∼87.2° relative to the ambient magnetic field.

The shock at the trailing edge of the CIR, at ∼00:59 UT on day 181, is characterized by a sharp increase in V_sw from ∼507 to ∼532 km s⁻¹, accompanied by simultaneous sharp decreases in N_sw from ∼0.3 to ∼0.2 cm⁻³, in T_sw from ∼0.3 × 10⁵ to ∼0.1 × 10⁵ K, and in B₀ from ∼1.1 to ∼0.6 nT. This is defined as an FR shock. The FR shock propagates toward the Sun, but because the solar wind carrying the shock has a higher speed, the FR shock is also convected in the anti-sunward direction. The FR has the compression ratios of r_B ∼ 1.8 and r_N ∼ 1.5. The FR shock is estimated to be propagating with a speed V_sh ∼ 476 km s⁻¹ at an angle θ_Bn ∼ 58.3° relative to the ambient IMF. The estimated M_ms is ∼1.3. Thus, the FR shock at the CIR trailing edge, detected at ∼8.30 au, was a quasi-perpendicular shock propagating at a speed ∼1.3 times of the local magnetosonic speed at a comparatively smaller angle of ∼58.3° relative to the ambient magnetic field.

Using the method discussed above (Section 2.1), all the interplanetary shocks encountered by the Voyager 1 and 2 spacecraft in the ecliptic plane (in the heliolatitude range of ±10°) were identified. The method can be summarized as follows. First, from the temporal variations of the solar wind plasma and IMF, the abrupt increases in V_sw with simultaneous increases/decreases in N_sw, T_sw, and B₀ were identified as potential shocks. Second, values of the solar wind plasma and IMF across the potential shocks were used to estimate V_A, V_S , V_ms, and M_ms as defined in Section 2.1. The discontinuities with only M_ms > 1 were confirmed as shocks. Voyager 1 encountered 44 shocks from 1977 September to 1980 June in the R_h range of ∼1–8.3 au. Voyager 2 identified 76 shocks from 1977 September to 1984 October between R_h of ∼1 and ∼15.2 au. Beyond theses periods, Voyager 1 did not encounter any shock, Voyager 2 encountered only two shocks, at ∼29 au on 1989 February 25 and at ∼32 au on 1990 March 16, with M_ms of ∼4.5 and ∼4.8, respectively. Thus, these were not included in the statistical analysis. While the spacecraft data coverage did not change much beyond ∼15 au, several causes, like movement of the spacecraft to the off-ecliptic plane, data quality, change in the solar wind character etc., might be responsible for the drastic decrease in the shock detection in the ecliptic plane beyond ∼15 au. It may be mentioned that with much relaxed criteria, Richardson & Wang (2005) identified quite a significant number of discontinuities (encountered by Voyager 2) beyond ∼15 au; however, they also confirm a large decrease in the number.

The heliocentric and solar cycle dependencies of the 120 shocks (with M_ms > 1) identified between ∼1 and ∼15 au are explored in Figure 2. According to the monthly mean smoothed sunspot numbers, the solar cycle 21 started during 1976 March, attained the maximum during 1979 December, and ended during 1986 September. Thus, the period under study (1977 September through 1984 October) covers the ascending, maximum, and descending phases of the solar cycle. This is also reflected in the F_10.7 solar flux variation (Figure 2(a)).

Zoom In Zoom Out Reset image size

The shock occurrence (N_sh) shows a clear peak around 5 au, after which the occurrence decreases with the increasing R_h (Figure 2(b), (c)). However, owing to varying speed of the spacecraft at different R_h , the observation period (or the spacecraft time spent TS in each R_h bin) varied largely at different locations (Figure 2(b), (c), red horizontal steps, legend on the right). To account for this varying observation period, N_sh at each R_h bin was normalized by the observation period at that bin (N_sh TS⁻¹). The result is shown in Figure 2(d). This shows the peak occurrence at ∼1 au, a second and comparable peak around 5 au, followed by a sharp fall in the shock occurrence rate. However, on the average, normalized N_sh exhibits a linear decrease with the increasing R_h (Table 1). Normalized N_sh and R_h are strongly anticorrelated (correlation coefficient r = −0.86) at a confidence level of >99% (Student's t-test; Student 1908).

As the solar activity (F_10.7) largely varied during the study interval, the shock occurrence was also normalized by the F_10.7 solar flux (Figure 2(e)) to separate the solar activity contribution from the radial variation of the shocks. The solar flux normalized N_sh (N_sh F_10.7 ⁻¹) also exhibits a similar shock occurrence pattern as above: the peak occurrences at ∼1 and ∼5 au, followed by a sharp fall in the shock occurrence at the larger R_h .

The occurrence peak of the shocks at ∼5 au from the Sun, and decrease at the larger distances are consistent with previous results (e.g., Hoang et al. 1995; Luhmann 1995; González-Esparza et al. 1996, 1998; Richardson & Wang 2005; Neugebauer 2013; Echer 2019a, 2019b, and references therein). However, most of the previous reports are obtained from compilation of different spacecraft observations at different heliocentric distances, and during varying solar activity condition. However, the present work separates the solar activity contribution from the radial variation of the shocks.

The shocks identified above are classified into two groups: (1) the FF shocks characterized by the sharp increases in V_sw, N_sw, T_sw, and IMF B₀, and (2) the FR shocks characterized by a sharp increase in V_sw with simultaneous decreases in N_sw, T_sw, and B₀. The interplanetary drivers are identified for each of the shocks as ICME or CIR following the method described in Section 2.1. From the temporal variations of the solar wind plasma and IMF, a CIR is identified as the region of the compressed (high) plasma density N_sw and IMF magnitude B₀ between a slow stream and an HSS (Smith & Wolfe 1976). The shock driver is classified as an ICME when the shock is followed by a sheath characterized by the downstream high T_sw, N_sw and B₀ (Kennel et al. 1985; Tsurutani et al. 1988). This is sometimes followed by an MC characterized by a high B₀, a smooth and slow rotation in the IMF component(s), and a reduced T_sw (Burlaga et al. 1981). The shock characteristic parameters V_sh, θ_Bn, M_ms, r_B , and r_N , as shown in Section 2.1, are determined for each of the shocks. Figure 3 shows the variations of the shock types (FF and FR, Figure 3(a)), their interplanetary drivers (ICME and CIR, Figure 3(b), (c)), shock parameters V_sh (Figure 3(d)), θ_Bn (Figure 3(e)), M_ms (Figure 3(f)), r_B (Figure 3(g)), and r_N (Figure 3(h)), and the F_10.7 solar flux (Figure 3(i)) with R_h .

Zoom In Zoom Out Reset image size

Among all 120 shocks, ∼76% are FF and only ∼24% are FR type. The relative occurrence of the FF and FR shocks at different R_h is shown in Figure 3(a). While a larger percent of the shocks are of the FF type at any R_h , the FR percent increases gradually from ∼1 au and attains a peak around 7–8 au, after which it decreases again. The relative occurrence of the FF and FR shocks, and their heliocentric variation shown in Figure 3(a) are also consistent with previous reports. For example, at ∼1 au, ∼66%–88% of the shocks are reported to be FF depending on the solar cycle phase (Echer et al. 2003; Kilpua et al. 2015). At ∼5 and ∼10 au, the FF shocks are ∼69%–72% (Echer et al. 2010; Echer 2019a) and ∼75% (Echer 2019b) of all shocks, respectively.

The relative role of the interplanetary drivers (ICMEs and CIRs) is shown in Figures 3(b) and (c). As the ICMEs and CIRs exhibit the distinguished solar cycle variations (Sheeley et al. 1976; Burlaga et al. 1981; Gosling et al. 1990; Tsurutani et al. 1995, 2020; Gopalswamy et al. 2004; Obridko et al. 2012; Hajra et al. 2013, 2014; Hajra & Tsurutani 2018b; Hajra 2021), results in Figures 3(b) and (c) should be discussed with reference to the solar activity (F_10.7, Figure 3(i)) variation. The solar maximum (and a few years around the maximum) is dominated by sunspots, active regions, solar flares, CMEs, or ICMEs. The occurrence rate of ICMEs decreases with the decreasing F_10.7. On the contrary, the interplanetary space (in the ecliptic plane) is dominated by HSSs and CIRs during the descending and solar minimum phases when coronal holes extend to the lower solar latitudes and expand in size. For the entire period of observation (irrespective of the solar activity variation), ICMEs are found to be the major driver of the FF shocks, driving ∼89% of the FF shocks, while only ∼11% of the FF shocks are driven by CIRs. However, ∼59% of the FR shocks are associated with ICMEs and ∼41% with CIRs. Thus, CIRs are a significant driver for the FR shocks. Interestingly, around 1–2 and 10 au, all FR shocks are driven by CIRs (Figure 3(d)). As can be found from the F_10.7 variation (Figure 3(i)), these locations were traveled by the spacecraft during the solar minimum-to-ascending and the descending phases, respectively. In addition, around 8 au (after the solar maximum), CIRs are found to be the dominating driver of the FF (∼67%) and FR (∼50%) shocks, compared to ICMEs.

The above results clearly contradict the results of Echer (2019a), reporting ∼90% of the FR shocks being driven by CIRs, and more or less equal distribution of the FF shocks between ICMEs (∼55%) and CIRs (∼45%) at ∼5 au. While the FR shock formation in association with ICMEs is quite rare, they may result from the speed differences between the fast solar wind following slower stream (Gosling et al. 1988), or owing to an overexpansion of an ICME (Gosling et al. 1994). In addition, the present results are interesting, also because it was previously reported that only few CIRs are bounded by shocks at ∼1 au (e.g., Tsurutani et al. 1995; Jian et al. 2006). As shock formation through the nonlinear steepening of the large-amplitude magnetosonic waves theoretically requires several nonlinear steepening times, shocks typically form at the leading and trailing edges of CIRs only at the large R_h (>1.5–2.5 au).

In the entire R_h range of the study, the shock speed V_sh exhibits a large variation (Figure 3(d), right panel): from ∼226 to ∼733 km s⁻¹ with an average V_sh of ∼458 km s⁻¹ for the FF shocks, and from ∼68 to ∼593 km s⁻¹ with an average V_sh of ∼323 km s⁻¹ for the FR shocks. The statistical significance of this result is assessed by the Student's t-statistics and estimation of the corresponding probability factor p (Reiff 1990; Press et al. 1992). A p value of <0.0001 was estimated, implying that the higher speed of the FF shocks compared to the FR shocks is statistically significant. At any R_h , the FF shocks seem to have higher speed than the FR shocks. On the average, V_sh exhibits a slow increase at a rate of ∼10.57 km s⁻¹ au⁻¹ with the increasing R_h (Table 1). V_sh is correlated (correlation coefficient r = 0.67) to R_h at a high significance level (>99%).

The shock propagation angle θ_Bn does not exhibit any clear dependence on R_h (Figure 3(e), left panel). The overwhelming majority, ∼88% of the FF shocks and ∼93% of the FR shocks, are found to be quasi-perpendicular (with θ_Bn > 45°), and only ∼12% of the FF and ∼7% of the FR shocks are quasi-parallel (θ_Bn ≤ 45°) (Figure 3(e), right panel). It is interesting to note that after ∼6 au, only two shocks are observed to be quasi-parallel. These results are consistent with previous results reporting dominance of the quasi-perpendicular shocks at R_h of ∼1 au (Bavassano-Cattaneo et al. 1986; Neugebauer 2013; Kilpua et al. 2015), ∼5 au (Echer 2019a), and ∼10 au (Echer 2019b). It can be noted that IMF, from an average orientation angle of ∼45° to the ecliptic plane at ∼1 au, becomes more and more tangential at the larger R_h (e.g., Parker 1965; Behannon 1978). This makes the shocks more likely to be quasi-perpendicular.

The heliocentric variations of the shock strength parameters (M_ms, r_B , and r_N ) are shown in Figures 3(f)–(h). No clear distinction can be made between the FF and FR shocks. From the linear regression analysis, M_ms is found to increase slowly with the increasing R_h at a rate of ∼0.10 au⁻¹, while r_N decreases with the increasing R_h at a rate of ∼−0.03 au⁻¹ (Table 1). However, the corresponding correlation (r = 0.58) and anticorrelation (r = −0.47) coefficients, respectively, of the linear regression analysis are not high. It is interesting to note that Echer (2019a) compared the shock compression ratio and M_ms at ∼1, ∼5, and ∼10 au based on data from several independent works, and concluded that the shock strength increases from ∼1 to ∼5 au, and then decreases from ∼5 to ∼10 au. However, the present work, with a much continuous and larger database, reveals no prominent systematic shock strength variation with R_h . This was also concluded by Richardson & Wang (2005) using the upstream to downstream density ratios across the interplanetary discontinuities.