Light Deflection under the Gravitational Field of Jupiter — Testing General Relativity

To take full advantage of the weak influence of plasma in Jupiter’s atmosphere relative to the Sun, the observational strategy revolves around Jupiter as a gravitational lens. The selected CES is J1925-2219 from ICRF3 (Charlot et al., 2020). Figure 1 presents the predicted deflection angle (see Section 3.2.1) of J1925-2219 and its two nearby CESs (i.e., J1923-2104 and J1928-2035, also from ICFR3) by Jupiter on 2020 October 23, 24, and 25, and 2021 February 5, GST. An observing day is considered as an epoch, where each are numbered 1 to 4 sequentially. The relative positions between the line of sight of the three CESs and Jupiter are also plotted. The first three days are selected around peak deflection angles (see panel (a) in Figure 1). The component of deflection in R.A.(J2000) almost reaches a maximum and minimum during epochs 1 and 3, respectively, while that in Decl.(J2000) reaches a maximum during epoch 2. We have also added epoch 4 on 2021 February 5. In this epoch, the calculated deflection angle caused by Jupiter is $<$ 5 $\mu$as. The effect of plasma on Jupiter is less than 0.3 $\mu$as in all four epochs.

We chose CESs J1923-2104 and J1928-2035 as background sources from ICRF3 (Charlot et al., 2020). Table 1 lists the positions of these two CESs and J1925-2219. Using the National Radio Astronomy Observatory’s (NRAO’s)¹¹1NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. VLBA, we observed them at 15 GHz from 2020 October 23 to 2021 February 5 for a total of four epochs (see above), under program BL277. Each epoch contains a 3.5 hr track.

Three geodetic blocks were put at the start, middle and end of each track, and phase-referenced observations were inserted between them. The observational setup and calibration procedures were similar to those used by Reid et al. (2009), except that: only continuum sources are included in this work; the frequency is 15 GHz; and four dual-polarized intermediate-frequency (IF) bands of 64 MHz, correlated with 128 channels per IF, were employed for the observations. All of the data were processed with the DiFX²²2DiFX is the Swinburne University of Technology’s software correlator for VLBI, which was developed as part of the Australian Major National Research Facilities Programme, and is operated under licence. correlator (see a brief description of the principles in Section 3.2.1) at the VLBA correlation facility in Socorro, New Mexico, USA (Deller et al., 2007).

We used J1925-2219 as the phase-reference source. After mapping the three CESs, the task JMFIT in the NRAO’s Astronomical Image Processing System (AIPS; van Moorsel et al., 1996) was used to fit the two-dimensional Gaussian brightness distributions, yielding their relative positions.

The observed relative positions between the two CES pairs (i.e., J1925-2219 and J1923-2104, C1–C2, as well as J1925-2219 and J1928-2035, C1–C3) are shown in Table 2 and Figure 2. These relative positions are relative to the coordinates used to correlate the VLBA data (see Table 1), where the errors of these coordinates in R.A. are denoted as $C_{C1,C2}$ for C1–C2 and $C_{C1,C3}$ for C1–C3 (see the details in Section 3.3). Because these three sources have low Decl. (i.e., as low as $\sim-22^{\circ}$; see Table 1), the components in this direction have poor relative positions, largely due to systematic errors from unmodeled atmospheric delays. Therefore, only the components in R.A. were chosen to analyze the parameter $\gamma$. The formal accuracy in R.A. is about 20 $\mu$as. The standard deviations of the changes in the relative positions throughout all the epochs are 51.0 $\mu$as for CES pair C1–C2 and 29.7 $\mu$as for C1–C3 in R.A.

To further analyze the value and error of $\gamma$, we should set up a new physical quantity, $V_{\mathrm{new}}$, whose value is the sum of the observed relative positions in R.A. and the theoretical ones with $\gamma=1$. The reasons for setting up $V_{\mathrm{new}}$ are as follows: (1) in the correlation process, it assumed $\gamma=1$ (Petit & Luzum, 2010) in all of the general relativity computations (in the form of the PPN approximation; e.g., Will, 1993, 2015; Ni, 2017); and (2) the calculation of the ephemerides was not conducted in this work. We used the positions of the Sun, the planets, and the satellites as calculated by PyEphem (Rhodes, 2011), and the ephemerides as provided by JPL (Folkner et al., 2014). In their numerical calculations of the ephemerides, they assumed $\gamma=1$ when a gravity model was involved. However, the ephemerides should be calculated based on long-term monitoring, comprising both ground-based and space-based observations (Folkner et al., 2014; Pitjeva & Pitjev, 2018; Fienga et al., 2020). Therefore, the assumption of $\gamma=1$ should hardly influence the results of the positions of celestial bodies in the solar system in this work. For further discussion about the ephemerides, see Section 4.2. It is worth noting that the theoretical value with $\gamma=1$ is from those labeled as “All” in Table 3 (see Section 3.2.2), because the correlator model covered the Sun, the planets, the Moon, and the large satellites of Jupiter, Saturn, and Neptune (Petit & Luzum, 2010).

Specifically, $V_{\mathrm{new}}$ reads:

The indexes of $i=1,2,3,4$ correspond to the four epochs of CES pair C1–C3, and $i=5,6,7,8$ correspond to those of C1–C2. $V_{\mathrm{obs}}$ denotes the observed relative positions (see Table 2). $C_{C1,C2}$ and $C_{C1,C3}$, which remain constant across the four epochs, should also be considered here. $C_{C1,C3}=0$ for $i=5,6,7,8$ and $C_{C1,C2}=0$ for $i=1,2,3,4$. $V_{\gamma=1}$ represents the theoretical relative positions assuming $\gamma=1$ (see the “All” column in Table 3).

The theoretical relative position with a value of $\gamma$ is $V_{\gamma}$ (similar to $V_{\gamma=1}$, but with different values of $\gamma$). Markov Chain Monte Carlo (MCMC) trials (Metropolis et al., 1953; Goodman & Weare, 2010) were used to estimate the maximum likely values of $\gamma$, $C_{C1,C2}$, and $C_{C1,C3}$ with the evaluation function, $\ln(\mathcal{L})$, as:

Similar to $V_{\mathrm{obs}}$, the formal errors of the observed relative positions, $\sigma_{\mathrm{obs}}(i)$, are listed in Table 2. Consequently, we calculated 10 MCMC sampling chains, where the first 40% were discarded. Finally, the total number of samples was 300 065. Figure 5 shows the MCMC result for $\gamma=0.984\pm 0.037$, which is consistent with the value of $1.01\pm 0.03$ in Fomalont & Kopeikin (2003, 2008). For the results from Fomalont & Kopeikin (2003, 2008), the formal accuracy of the measurement is also $\sim$ 20 $\mu$as and the smallest $b$ (with the lens being Jupiter) is $\sim$ 13 $R_{J}$ (slightly smaller than that in this work, i.e., $\sim$ 19 $R_{J}$).

The deflection angle caused by the plasma in the Sun’s corona could be larger than that caused by the solar gravitational field at 1 GHz when light grazes the solar surface (Shapiro, 1964). The electron density, $N_{e}$, in the solar corona or Jupiter’s atmosphere is assumed to have the form: $N_{e}(r)=N_{0}(R/r)^{A+2}$, where $r$ is the distance from the Sun or Jupiter and $R$ is their radius in units of m. $A$ is a number and $N_{0}$ is a constant with units of m^-3. The maximum deflection angle caused by plasma, $\alpha_{p}$, in units of $\mu$as reads as (see Kopeikin & Fomalont, 2002; Kopeikin & Makarov, 2007):

where $\nu$ is in units of GHz and $b$ is in units of m, and the electron integral column density, $N_{I}$, along the line of sight is:

where $b$ is the impact parameter, and $r_{0}$ and $r_{1}$—both are much greater than $b$—are the distances from the Sun or Jupiter to the CES and the Earth, respectively.

The plasma of the solar corona has been well studied, but knowledge of plasma in Jupiter’s atmosphere is scant. For the Sun, $N_{0}=0.57\times 10^{12}$ m^-3 with $A=0$, which was measured during 2011 and 2012, and this value can be considered a worst case scenario (Soja et al., 2014; Titov et al., 2018). To describe the electron content within 4 $R_{\odot}$ (corresponding to $\beta\sim 1.1^{\circ}$) from the solar center, additional terms are required (Verma et al., 2013; Soja et al., 2014). In this work, however, the targeted CESs are far from the solar center (the nearest $b$ is $\sim$ 94 $R_{\odot}$), thus the additional terms can be ignored. The deflection caused by the solar corona is plotted in Figure 6. We note that the plotted values are overestimated by at least a factor of two when $\beta$ approaches $90^{\circ}$.

For Jupiter’s atmosphere, it is hard to obtain comparatively accurate information about its plasma content from the few available observations. In this work, we have adopted the results reported by Ansher (2001), who used data from the plasma wave instruments on board the Galileo spacecraft. In that work, Jupiter’s atmosphere was divided into two parts and was bounded by $\sim$ 20–30 $R_{J}$, where $R_{J}$ is the mean radius of Jupiter. In the inner region, $5<r<30$ $R_{J}$, $N_{0}=2.74\times 10^{8}$ cm^-3 and $A=4.55\pm 0.05$, while in the outer region, $20<r<140$ $R_{J}$, $N_{0}=250$ cm^-3 and $A=0.14\pm 0.01$. The change in density values may deviate from power-law profiles by at least a factor of two. To estimate the maximum bending of light caused by plasma, we extended the two profiles to $r>1$ $R_{J}$ and summed $\alpha_{p}$ from them. The result is mapped in Figure 6.

Figure 6 indicates that the impact of plasma (i.e. $\alpha_{p}$) in Jupiter’s atmosphere is far less than that in the solar corona. Both of them decrease much more steeply than the deflection angle caused by the gravitational field as a function of $\beta$ (see Figure 4). The deflection by Jupiter’s atmosphere would be less than 1 $\mu$as as long as $\beta>5^{\prime}$. Therefore, choosing Jupiter as a gravitational lens can greatly reduce the limitation of the measured accuracy of the gravitational deflection angle posed by the solar corona.

In this work, the closest distance of light from the three CESs to Jupiter occurred in epochs 1–3, with an impact parameter $b\sim 19$ $R_{J}$ (see Figure 1). The corresponding value of $\alpha_{p}$ is $\sim$ 0.3 $\mu$as, less than 1 $\mu$as. For the Sun, in epochs 1–3, the impact parameter was $b>209$ $R_{\odot}$ for the three CESs, and the corresponding $\alpha_{p}$ was $\sim 1.5$ $\mu$as. In epoch 4, $94<b<95$ $R_{\odot}$ for the three CESs, with the corresponding $\alpha_{p}$ being about 7.3–8.0 $\mu$as. Moreover, these values are overestimated, and the relative bending angles of the CES pairs are likely much smaller. In addition, they are both less than the formal accuracy by a factor of at least $\sim$ 4. Therefore, they can be ignored.

Because plasma in Jupiter’s atmosphere near its surface is a controversial issue, a more detailed discussion of the electron content in the inner region is presented here. The data in the inner region from the work of Ansher (2001) are from a different study without citation. Additionally, $N_{0}=2.74\times 10^{14}$ m^-3, i.e., $N_{e}(r)=2.74\times 10^{8}(R/r)^{6.55}$ cm^-3, which is much larger than that at the solar surface. The results from the first Galileo mission and Voyager 2’s Radio Occultation Experiment showed that $N_{e}$ close to Jupiter’s surface (i.e., several hundred to a thousand kilometers above Jupiter’s surface) is $\sim 1\times 10^{11}$ m^-3 (Hinson et al., 1997, 1998). If we assume that $N_{e}\sim 1\times 10^{11}$ m^-3 at $r=1.0$ $R_{J}$, and $N_{e}=2.5\times 10^{8}(R/r)^{2.14}$ for $r>20$ $R_{J}$ (Ansher, 2001), and that profile of $N_{e}$ in the range $1<r<20$ $R_{J}$ obeys a power law distribution, $N_{e}(r)=1\times 10^{11}(R/r)^{4.1}$. The corresponding $\alpha_{p}$ value for the range $1<b<10$ $R_{J}$ would be 1–2 orders of magnitude lower than the value plotted in Figure 6. Therefore, the value of $\alpha_{p}$ plotted in Figure 6 for Jupiter may be significantly overestimated, especially when $b$ is small.

The orbital accuracies of tens of kilometers in JPL DE430 or DE431, released in 2014 for Jupiter and Saturn (Folkner et al., 2014), correspond to positional accuracies of from about several mas to a dozen mas. From DE430 to DE440, released in 2021, the positional accuracies have improved to below 1 mas (Park et al., 2021). The ephemerides adopted in this work are from DE438, released in 2018, which have been tuned for the flight project’s target body, e.g., the Juno mission (Bolton et al., 2017; Park et al., 2021). Therefore, the positional accuracies of the planets in DE438 are better than those in DE430 or DE431. Therefore, the accuracy of $\beta$ was set to 10 mas for the cases of both Jupiter and Saturn.

The orbit accuracies of the inner planets are $\sim$ 0.2 mas (Folkner et al., 2014; Folkner & Border, 2015), and Earth’s orbit is around the Sun to the first order (Park et al., 2021). Here, the accuracy of $\beta$ was set to 0.5 mas for the cases of the Sun and the terrestrial planets. The contribution from the uncertainty of $\beta$ with the value set above was roughly calculated and plotted in Figure 7.

It can be seen that the contribution from the uncertainty of $\beta$ to the deflection angle, $\alpha_{err,c}$, is small. For Jupiter, when it is farthest from the Earth, the contribution is less than 1 $\mu$as as long as the light is $\gtrsim 1.0^{\prime}$ (corresponding to $b\gtrsim 3$ $R_{J}$) from Jupiter. The corresponding $\beta$ is $\gtrsim 1.5^{\prime}$ (corresponding to $b\gtrsim 4$ $R_{J}$) when Jupiter is nearest to the Earth. For Saturn, the contribution is less than 1 $\mu$as as long as the light is $\gtrsim 1.0^{\prime}$ (corresponding to $b\gtrsim 7$ $R_{S}$, where $R_{S}$ is the mean radius of Saturn) away from Saturn, no matter where it is. The corresponding $\beta$ is $\gtrsim 22.0^{\prime}$ (corresponding to $b\gtrsim 1$ $R_{\odot}$) for the Sun. Therefore, in this work, the impact of the uncertainties of the positions of gravitational lenses can be ignored.