AN HIGH-PRECISTION SEMI-ANALYTICAL ON-HUGONIOT EOS FOR GEOLOGIC MATERIALS.
Seiji Sugita1, Kosuke Kurosawa2, Toshihiko Kadono3 and Takayoshi Sano3, 1Univ. of Tokyo (Kashiwa, Chiba,sugita@k.u-tokyo.ac.jp), 2JAXA/ISAS (Sagamihara, Kanagawa), Osaka Univ. (Suita, Osaka).
Introduction: The equation of state (EOS) of geogologic materials, such as silicates and ices, underhighly shock-compressed states play an important rolein planetary evolution. Extensive research has been
conducted for the EOS of solid materials. A number ofvery sophisticated thermodynamics-based EOS’s, such
as SESAME [1] and M-ANEOS [2], have been developed. These numerical EOS codes, however, use manymodel parameters to reproduce experimental data.
Thus, it is difficult to find the optimum model parameters uniquely, requiring extensive experiments covering a wide range of thermodynamic conditions.
Such costs for EOS’s may be necessary evil because physics behind the EOS of highly shockcompressed geologic materials is very complicated. Infact, recent experimental result using high-power laser
have revealed further intricate properties of silicates
under high compression conditions, such as large departure of isochoric specific heat Cv from Dulong-Petit
limit due to molecular dissociation and ionization [e.g.,3,4]. In order to incorporate such complex properties
into a thermodynamics-based EOS properly, physics
behind these materials needs to be modeled well.
Such thorough understanding of material properties is essential for building a versatile EOS for hydrocode calculations. However, there are many occasions
where the thermodynamic properties only along Hugoniot compression curve are needed. For example,estimation of the fractions impact melt/vapor and thefinal molecular composition of impact vapor plumerequires only the entropy gain due to initial impact
shock.
Furthermore, when a new range of shock pressures
is explored, thermodynamic measurements are usually
very limited. Also, the vast majority of shock compressed EOS data are taken only along Hugoniot
curves. However, many thermodynamic data including
both on-Hugoniot and off-Hugoniot conditions areneeded to revise sophisticated EOS’s accurately. Suchrevisions often take time and have to wait for data accumulation. Nevertheless, when new experimental data
are obtained, one would wish to examine the significance of such new data promptly. Thus, a simple EOS
that works only on Hugoniot curves but can describe
the experimental data accurately would be very usefulfor shock compression experimental data analysis.
In this study, we propose a semi-analytical formulaof on-Hugoniot EOS derived from the differential formof Rankine-Hugoniot equation and compare it withconventional EOS’s and experimental data.

Analytical Formulation: One of the outstanding
properties of condensed matter under shock compression is the linear velocity relation between particle
velocity Up and shock velocity Us:
U s = C o + s U p,
(1)
where Co, and s are bulk sound velocity and a constant,respectively. This relation is known to hold for a variety of materials over a wide range of impact velocity [5].
Despite the wide applicability of this relation, mostEOS’s do not take advantage of this relation. One of
the few exceptions is the analytical EOS by Tillotson[6]. It is simple and accurate (for compression) and has
been used widely. However, it is not thermodynamically self-consistent and cannot calculate entropy gain due
to shock heating.
In addition to the Up-Us relation (1), we use only
general thermodynamic relations, the differential formof Rankine-Hugoniot relations, and Grüneisen EOS.
From these relations, we obtain ordinary differential equations for temperature T and entropy S:(Us "U p )dT= Co ! oTdU p
U sq+1
2
dS sU p=dU p TU sq"1+sU p2
CvU s(2)(3)

Density and pressure can be obtained with frequently
used analytical solutions for Hugoniot curves:Us.
p = !oU sU p ! = !o(7),U s !U pThen, all the basic thermodynamic variables can becalculated for Hugoniot without having knowledge of
an off-Hugoniot EOS. This EOS needs only six model
parameters ρo, Co, s, Cv, Γo, and q.
Experimetal Validation: In order to examine thevalidity of the new on-Hugoniot EOS, we compared
the results of our calculations and literature values.
Fig. 1 shows comparisons between our EOS andice shock experiment data with single-stage powder
gun [7]. Because all the parameter values are well established, there is no room for adjustment for fitting.
However, the comparison indicates that our semianalytical EOS can reproduce shock temperature andentropy very well.
Similarly, our semi-analytical EOS are comparedwith shock data of both diopside and quartz obtainedby 2-stage light-gas gun [9,10] (Fig. 2). The shock
temperatures measured around 150 GPa are well reproduced by our EOS with Dulong-Petit assumption.

These good reproductions of well-establishedshock temperature data and entropy estimates support
the validity of our new EOS for Hugoniot conditions.
Planetary-Velocity Impacts: For extremely highpressure shocks, Cv is not approximated by a constantvalue well; it may become well above Dulong-Petit
limit [e.g., 3,4]. The effect of specific heats can becalculated easily with our new EOS.
Although there is good agreement among different
EOS’s at relatively low shock pressures (~150 GPa) inFig. 2B, different EOS’s yield significantly different
results at higher shock pressures (several hundred GPa)[4]. This scatter results from the fact that there are notmany experimental data available in the higher shockpressure range. Under such conditions, our EOS isuseful because it does not require many data points tomake accurate predictions along a Hugoniot curve. It
can also be used as an anchor for the more sophisticated EOS for Hugoniot conditions.
Furthermore, because Cv is a very important property to characterize condensed matter, the capability toderive Cv from temperature data is very useful. A couple of examples of comparisons between our recent
experimental data and our EOS predictions are shownin Fig. 2. The quartz data at 150 GPa requires Cv significantly larger than 3R, but the shock temperatures ofdiopside at ~300 GPa is consistent with the DulongPetit value. Such difference in Cv among different silicates is of great importance in planetary science.
Planetary Applications: Perhaps, the most important thermodynamic variable that an on-Hugoniot
EOS is expected to provide is entropy. Once shockinduced entropy gain is known, its final thermodynamic state after adiabatic decompression can be accuratelycalculated. There are a number of applications of suchentropy calculations in Earth and planetary science.
For example, the degrees of melting and vaporization,
post-decompression temperature of impact-induced
vapor can be calculated relatively easily once entropy
gain is given. Furthermore, this EOS is useful formatching thermodynamic conditions between laserinduced and impact-induced vapor plumes [12].
References: [1] Kerley, G.I. (2009), Kerly Tech. Serv.
Rep. KTS09-1., [2] Melosh, H.J. et al. (2007), MAPS, 42,2079–2098. [3] Hicks, D. et al. (2006), PRL, 97, 025502. [4]Kurosawa, K. et al. (2011) Proc. 17th APS SCCM Mtg. inpress. [5] Trunin, R. F. et al. (2001), Experimental Data on
Shock Compression and Adiabatic Expansion of Condensed
Matter, Academic Press. [6] Tillotson, J. H. (1962), Metallic
equations of state for hypervelocity impact. San Diego: General Dynamics. [7] Stewart, S. T. et al. (2008), GRL, 35,L23203. [8] Svendsen, B. and T. J. Ahrens (1990), JGR, 95,6943-6953. [9] Lyzenga, G. A. et al. (1983), JGR, 88, 2431–2444. [10] Celliers, P.M. et al. (2010) PRL, 104, 184503.
[11] Kurosawa, K. et al. (2010), GRL, 37, L23203. [12]Sugita, S., et al. (2003), LPS XXXIV, Abstract#1573.

Figure 1. Comparison between shock experimental data [7]and our proposed EOS calculations. The results of Both temperature and entropy are reproduced well by our semianalytical EOS.

Figure 2. Comparison between EOS calculations and shock
experimental data for diopside [8,11] and quartz [9,10,11].
Temperature and entropy for both diopside and quartz atshock pressures below 200 GPa are reproduced well by oursemi-analytical EOS and Dulong-Petit Cv, but temperature
measurements at much higher shock pressure for quartz withintense laser [11] requires higher Cv than Dulong-Petite (3R).
For comparison, M-ANEOS and SESAME calculations by[4] for the same conditions are also shown.