Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers

General-relativistic deflection of light by mass, dipole, and quadrupole moments of the gravitational field of a moving massive planet in the solar system is derived in the approximation of the linearized Einstein equations. All terms of order $1\mu \mathrm{as}$ are taken into account, parametrized, and classified in accordance with their physical origin. The monopolar light-ray deflection, modulated by the radial Doppler effect, is associated with the total mass and radial velocity of the gravitating body. It displaces the apparent positions of stars in the sky plane radially away from the origin of the celestial coordinates associated with the planet. The dipolar deflection of light is due to a translational mismatch of the center of mass of the planet and the origin of the planetary coordinates caused by the inaccuracy of planetary ephemeris. It can also originate from the difference between the null cone for light and that for gravity that is not allowed in general relativity but can exist in some of the alternative theories of gravity. The dipolar gravity field pulls the apparent position of a star in the plane of the sky in both radial and orthoradial directions with respect to the origin of the coordinates. The quadrupolar deflection of light is caused by the physical oblateness, $J_2$, of the planet, but in any practical experiment it will have an admixture of the translation-dependent quadrupole due to inaccuracy of planetary ephemeris. This leads to a bias in the estimated value of $J_2$ that should be minimized by applying an iterative data reduction method designed to disentangle the different multipole moments and to fit out the translation-dependent dipolar and quadrupolar components of light deflection. The method of microarcsecond interferometric astrometry has the potential of greatly improving the planetary ephemerides, getting unbiased measurements of planetary quadrupoles, and of thoroughly testing the null-cone structure of the gravitational field and the speed of its propagation in the near-zone of the solar system. We calculate the instantaneous patterns of the light-ray deflections caused by the monopole, the dipole, and the quadrupole moments, and derive equations describing apparent motion of the deflected position of the star in the sky plane as the impact parameter of the light ray with respect to the planet changes due to its orbital motion. We discuss the observational capabilities of the near-future optical (SIM) and radio (SKA) interferometers for detecting the Doppler modulation of the radial deflection, and the dipolar and quadrupolar light-ray bendings by Jupiter and Saturn.

Figure 1

Light propagates from the star to the observer in the direction shown by the unit vector $\overrightarrow{k}$. The planet is displaced from the origin of the coordinate system by vector $\overrightarrow{L}$ that is a parameter of the data analysis algorithm. Its rotation axis is specified by the unit vector $\overrightarrow{s}$. The impact parameter of the light ray is $d$. Two unit vectors, $\overrightarrow{n}$ and $\overrightarrow{m}$, are orthogonal to the vector $\overrightarrow{k}$ and form a “plane of the sky” that is perpendicular to the line of sight. $\chi$ denotes the angle between the star and the assumed position of the planet in the coordinate origin.

Figure 2

The gravitational field of a planet is a retarded solution of the gravity-field wave equation (11). In general relativity, gravitational field progresses on the hypersurface of a null cone from the past to the future. Directions of the propagation of the gravitational field of the planet are shown by arrows. The picture assumes that the coordinate-dependent effects associated with the dipole moment $I^i$ are excluded ($I^i=0$). The observer measures the gravitational field at time $t$ when the planet is located at the retarded position on its orbit at the retarded time $s=t-r$, which is a null characteristic of the gravitational field.

Figure 3

The gravitational field of a planet (the case of dipole $I^i=0$ is shown) affects only the particles lying on the hypersurface of the future gravity null cone. A photon emitted by a star at time $t_0$ arrives to the observer at time $t_1$ along a null direction of the past light cone with a vertex at the observer. Therefore, the future gravity null cone of the planet and the past light cone of the observer must coincide along the null direction that is a null characteristic of the retarded solution of the gravity-field wave equation (11). The photon detected at time $t_1$, is deflected by the planet’s gravity force from the planet’s retarded position taken at time $s_1=t_1-r_1$. This effect of the retardation of gravity can be observed by measuring the amount of gravitational deflection of light by a moving planet, and used to determine the speed of the gravity field propagating from the planet to the point of observation (see Sec. 5 and papers 21, 22). Note that the retardation of gravity would not be measurable if the planet were at rest with respect to the observer.

Figure 4

Light-ray deflection by a static (a) and moving (b) planet. In case (a) the distance between the planet and observer does not change as light propagates. Thus, measuring the deflection of light does not allow us to determine experimentally whether the gravity force of the planet acts with retardation from position (1), or instantaneously from position (2). In case (b) the distance between the planet and the observer varies as the photon travels toward the observer. The retarded interaction of gravity with light becomes apparent since measuring the angle of the gravitational deflection of light allows us to distinguish between positions (1) and (2) of the planet on its worldline making the causal structure of the gravity cone clearly visible and measurable.

Figure 5

The local tangential coordinate system in the plane of the sky related to the global ecliptic coordinate system is shown. The unit vectors ${\overrightarrow{e}}_{\lambda }$ and ${\overrightarrow{e}}_{\beta }$ point in the direction of increasing ecliptic longitude and latitude, respectively. The angular distance of the undeflected position of the star from the origin of the coordinate system is $\chi$ (cf. Fig. 1), which is at position angle $\phi$ from the north direction. The total deflection $\overrightarrow{\alpha }$ has both radial, ${\alpha }_n$, and orthoradial, ${\alpha }_m$, components: $\overrightarrow{\alpha }={\alpha }_n\overrightarrow{n}+{\alpha }_m\overrightarrow{m}$.

Figure 6

The gravitational deflection of light by the multipolar fields of Jupiter. The magnitude of deflection is coded by colors in a logarithmic scale shown at the top. The vectors in each plot indicate only the direction of displacement, not the magnitude. The monopole deflection pattern is shown in plot (a). The artifactual dipolar deflection patterns caused by a positional displacement ${\overrightarrow{x}}_P=L\overrightarrow{z}$ of the planet’s center of mass from the origin of the coordinate system used for the multipolar expansion of the gravitational field, are depicted in the central row of plots, for $L_1=350$ (b), $L_2=3500$ (c), and $L_3=35000\mathrm{km}$ (d). The bottom row of plots (e–g) shows the combined quadrupolar deflection patterns generated by the intrinsic oblateness of the planet and by the coordinate-dependent translational moment, for the same values of $L$. The direction of rotation axis $\overrightarrow{s}$ is indicated with a thin black line. Note that the quarupolar deflection at $L_1=350\mathrm{km}$ is dominated by that from the intrinsic moment, whereas for $L_3=35000\mathrm{km}$, on the contrary, the translational quadrupole exceeds the light deflection caused by $J_2$.

Figure 7

The deflection of light by Saturn caused by its mass (a), dipole (middle row), and quadrupole (bottom row) moments. Magnitudes of the deflection are expressed by colors in the logarithmic color grade shown in the upper right corner, and the direction (not the magnitude) of the deflection is shown by small arrows. Vector displacement $\overrightarrow{L}=L\overrightarrow{z}$ of the center of mass of Saturn from the origin of the coordinate system takes values (from left to right) of $L_1=250$ (b), $L_2=2500$ (c), and $L_3=25000\mathrm{km}$ (d). The unit vectors $\overrightarrow{z}$ and $\overrightarrow{s}$ indicate the directions of the displacement $L$ and rotational axis of the planet, respectively.

Figure 8

Geometry of the central mapping used for computation of the dynamic patterns (curves) of the gravitational deflection of the apparent position of a star by a planet as a function of time. The origins of the two coordinate charts $(x,y)$ and $X$, $Y$ coincide and are placed at the undeflected position of the star. For the sake of graphical convenience, the $Y$ axis is displaced along the positive direction of the $X$ axis at the constant distance $X_0$. The available planetary ephemeris predicts that the planet moves continuously from left to right along the $Y$ axis with $X$ fixed as time passes, and two positions of the planet are shown to explain the mapping geometry. The dashed line is parallel to the $Y$ axis and shows the trajectory of the real planetary center of mass which may differ from the planetary ephemeris based on the other (independent) set of observations giving rise to the dipolar component of the light deflection. The minimal distance between the planet and the undeflected position of the star is $X=X_0$. The apparent position of the star is deflected from its unperturbed (catalogue) position in both radial and orthoradial directions that are ${\alpha }_n$ and ${\alpha }_m$, respectively. The total deflection $\overrightarrow{\alpha }={\alpha }_n\overrightarrow{n}+{\alpha }_m\overrightarrow{m}$. The mapping establishes a one-to-one correspondence $F$ between the planet’s coordinates $(X,Y)$ and the star’s deflected position $(x,y)$, that is $(F\mathrm{\text{:}}d\to \alpha ,\phi \to \theta )$.

Figure 9

Trajectories of apparent positions of a star deflected by Jupiter. Left, middle, and right graphs show, respectively, the monopolar (circle), dipolar (cardiod), and quadrupolar (the Caley’s sextic) light-ray deflection curves. These curves are obtained by central mapping transformation explained in Fig. 8 and Sec. 6c. The planet passes directly below the true position of the star at the coordinate origin, at a constant velocity. The impact parameter $X_0=40\mathrm{arcsecond}$, and the orbital error is $L=250\mathrm{km}$. The time step between individual points is 0.02 days for the monopole deflection curve, and 0.002 days for the other two. Dynamic light-ray deflection curves for Saturn have a smaller amplitude but similar shape.

Figure 10

Combined dipolar and quadrupolar deflection of light by Jupiter for the same parameters and configuration as in Fig. 9. Details of the quadrupolar deflection (the sextic curve) are hidden by the dipolar deflection (the cardioid). Separation of the two deflection terms is a nontrivial experimental problem, and generally will require multiple observations of a number of sources around the planet.