Mercury’s resonant rotation from secular orbital elements

Abstract

We used recently produced Solar System ephemerides, which incorporate 2 years of ranging observations to the MESSENGER spacecraft, to extract the secular orbital elements for Mercury and associated uncertainties. As Mercury is in a stable 3:2 spin-orbit resonance, these values constitute an important reference for the planet’s measured rotational parameters, which in turn strongly bear on physical interpretation of Mercury’s interior structure. In particular, we derive a mean orbital period of $(87.96934962 \pm 0.00000037)\,\hbox {days}$ and (assuming a perfect resonance) a spin rate of $(6.138506839\pm 0.000000028){}^{\circ }/\hbox {day}$. The difference between this rotation rate and the currently adopted rotation rate (Archinal et al. in Celest Mech Dyn Astron 109(2):101–135, 2011. doi:10.1007/s10569-010-9320-4), corresponds to a longitudinal displacement of approx. 67 m per year at the equator. Moreover, we present a basic approach for the calculation of the orientation of the instantaneous Laplace and Cassini planes of Mercury. The analysis allows us to assess the uncertainties in physical parameters of the planet, when derived from observations of Mercury’s rotation.

Appendices

Appendix 1

The Keplerian orbital elements derived from the INPOP13c ephemeris (Fienga et al. 2014) are given in Table 2. We find very little difference of the values when comparing to the DE432 ephemeris (see Table 1). The deviation for the trend $a_1$ of the semi-major axis is about 1.7 meter per century.

Table 2 The same as Table 1 but derived from the INPOP13c ephemeris and a time span of 2000 years (09.06.973 AD—23.06.2973)

In Table 3 we give values for reference frame dependent orbital elements with respect to the ecliptic (ECLIP, inclination 23.439291$^\circ$), Mercury orbital plane (OP), and Mercury Laplace plane (LP) at J2000.0. The rotation matrices for the transformation to the these reference frames from the ICRF are given by

$$\begin{aligned} \mathbf {R}_{\text {\tiny ECLIP}}=\left( \begin{array}{r r r} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0.91748206 &{}\quad 0.39777716 \\ 0 &{}\quad -0.39777716 &{}\quad 0.91748206 \end{array}\right) , \end{aligned}$$

(38)

$$\begin{aligned} \mathbf {R}_{\text {\tiny OP}}=\left( \begin{array}{r r r} 0.98166722 &{}\quad 0.19060290 &{}\quad 0 \\ -0.16742216 &{}\quad 0.86227887 &{}\quad 0.47795918\\ 0.09110040 &{}\quad -0.46919686 &{}\quad 0.87838205 \end{array}\right) , \end{aligned}$$

(39)

$$\begin{aligned} \mathbf {R}_{\text {\tiny LP}}=\left( \begin{array}{r r r} 0.88845611 &{}\quad 0.43672271 &{}\quad 0.14113473 \\ -0.45838720 &{}\quad 0.82896828 &{}\quad 0.32045711 \\ 0.02295468 &{}\quad -0.34940643 &{}\quad 0.93669004 \end{array}\right) . \end{aligned}$$

(40)

The precession of the pericenter of Mercury is given by $\varpi _1^{\mathrm{OP}}=\varOmega _1^{\mathrm{OP}}+\omega _1^{\mathrm{OP}}=575.3\,\text {arc sec/cy}$. The inclination of the orbital plane with respect to the Laplace plane $I_0^{\mathrm{LP}}=\iota =8.58^{\circ }$ remains constant $I_1^{\mathrm{LP}}=I_2^{\mathrm{LP}}\approx 0$. The precession of the orbit around the Laplace plane is $|\varOmega _1^{\mathrm{LP}}|=\mu =0.109981^{\circ }/\text {cy}$.

Table 3 Orbital elements of Mercury as derived from the DE432 ephemeris at epoch J2000.0 with respect to the following reference frames: Ecliptic and Earth equinox of J2000; Mercury orbital plane of J2000.0 and ascending node with respect to the ecliptic; Mercury Laplace plane and ascending node with respect to the Mercury orbital plane of J2000.0

Appendix 2

In Table 4 we list the first ten periodic terms, which were identified in the osculating orbital elements time series. Some of the periods can be assigned to planetary perturbations, e.g., Venus: $(\lambda _V)$ 0.62 years; $(2\lambda _V)$ 0.31 years; $(2\lambda _M-5\lambda _V)$ 5.66 years; $(\lambda _M - 3\lambda _V)$ 1.38 years; $(\lambda _M - 2 \lambda _V)$ 1.11 years; $(2\lambda _M-4\lambda _V)$ 0.55 years; Earth: $(\lambda _M-4\lambda _E)$ 6.58 years; Jupiter: $(\lambda _{J})$ 11.86 years; $(2\lambda _{J})$ 5.93 years; $(3\lambda _{J})$ 3.95 years; Saturn: $(2\lambda _{S})$ 14.73 years, where $\lambda =M+\varpi = M + \varOmega + \omega$ denotes the mean longitude of the planet, respectively.

Table 4 Ten leading terms of the decomposition of the time series of the osculating orbital elements in $\sum _i A_i\cos (\nu _i\,t +\phi _i)$ with $\nu _i = 2\pi /T_i$

Appendix 3

The obliquity of the spin axis $i_c$ introduces small changes in the precession and resonant rotation rates. To stay within the Cassini plane the spin axis has to precess slightly faster than the orbital plane normal. In order to compute the corrections we expand equation Eq. 33 to first order in the obliquity $i_c$. The declination $\delta$ and right ascension $\alpha$ of the spin axis are then given by

$$\begin{aligned} \delta (t) = \frac{\pi }{2}-I + \frac{\dot{\varOmega }\sin I}{\sqrt{\dot{I}^2+(\dot{\varOmega }\sin I)^2}}i_c=\delta _0 + \delta _1 t \end{aligned}$$

(41)

and

$$\begin{aligned} \alpha (t) = \varOmega -\frac{\pi }{2}+ \frac{\dot{I}/\sin I}{\sqrt{\dot{I}^2+(\dot{\varOmega }\sin I)^2}}i_c=\alpha _0 + \alpha _1 t. \end{aligned}$$

(42)

By deriving the series in t we obtain the precession rates at J2000.0

$$\begin{aligned} \delta _1 = I_1\left( -1 + \frac{\varOmega _1 I_1^2 \cos I_0 + 2(\varOmega _2 I_1-I_2\varOmega _1)}{\sqrt{(I_1^2+(\varOmega _1\sin I_0)^2)^3}}i_c\right) \end{aligned}$$

(43)

and

$$\begin{aligned} \alpha _1 = \varOmega _1 - \frac{(I_1^2\cot I_0 + \varOmega _1^2\sin 2I_0)I_1^2/\sin I_0}{\sqrt{(I_1^2+(\varOmega _1\sin I_0)^2)^3}}i_c. \end{aligned}$$

(44)

For $i_c=2.04\,\text {arc min}$ (Margot et al. 2012) this results in

$$\begin{aligned} \delta _1 = -0.00486^{\circ }/{\text {cy}} \text { and }\alpha _1 = -0.03291^{\circ }/{\text {cy}}. \end{aligned}$$

(45)

The rotation rate is also slightly modified due to the obliquity. For small $i_c$ we get

$$\begin{aligned} \varphi (t) = \frac{3}{2}M+ \omega - \frac{\dot{I}\cot I}{\sqrt{\dot{I}^2+(\dot{\varOmega }\sin I)^2}}i_c=\varphi _0^{(3/2)} + \varphi _1^{(3/2)} t. \end{aligned}$$

(46)

The resonant rotation rate is consequently

$$\begin{aligned} \varphi _1^{(3/2)} = \frac{3}{2}n_0 + \omega _1 + \frac{(I_1 \varOmega _1)^2(3+\cos 2 I_0)/2 + (\varOmega _2 I_1-I_2\varOmega _1)\varOmega _1\sin 2I_0+I_1^4/(\sin I_0)^2}{\sqrt{(I_1^2+(\varOmega _1\sin I_0)^2)^3}}i_c \end{aligned}$$

(47)

and with $i_c=2.04\,\text {arc min}$ this amounts to $6.138506841^{\circ }/\text {day}$. The introduced correction is not significant when compared to the error of the resonant rotation rate in Eq. 21.