Binary microlensing with plasma environment – Star and planet

In gravitational microlensing, the mass of the lens is small. This implies that the lensing effects, such as image separation of multiple images and the image distortions are difficult to detect. The microlensing in the Local Group (or Galactic microlensing) refers to the temporal brightening of a background star due to intervening compact objects (e.g., Mao 2012). Since the first discovery of microlensing event by Alcock et al. (1993); Udalski et al. (1993), this field has made enormous progress. In particular, the gravitational microlensing has been developed into a powerful technique in the detection of exoplanets (Mao & Paczynski 1991). More and more planets have been found through large scale optical surveys, e.g., OGLE (Udalski et al. 2015); MACHO (Alcock et al. 2000) and KMT (Kim et al. 2016). By means of various other techniques, several thousand exoplanets have been confirmed to date¹¹1https://exoplanetarchive.ipac.caltech.edu. Microlensing not only provides a complementary to these methods in searching for exoplanets, but also detects planets around all types of stars (e.g. Schneider et al. 2006; Mao 2012; Gaudi 2012), and even planets around binary system (which can be called ’triple system’) (e.g. Kuang et al. 2022).

The microlensing light curves are usually achromatic, because the gravitational deflection of light is independent of wavelength. However, there can be reasons leading to chromatic effects in microlensing. First, there exists the well-known phenomenon of chromatic microlensing in strong lensed quasars (e.g. Kayser et al. 1986; Wambsganss & Paczynski 1991). The source can have different size at different wavelength, thus will be lensed with different magnifications. Another possibility of chromatic effects is attributed to the propagation of light in the plasma (e.g., review by Bisnovatyi-Kogan & Tsupko 2017). Chromatic refractive deflection can be caused by the ionised gas in the interstellar medium (ISM), or by more concentrated plasma near astrophysical objects.

The light propagation in presence of medium in the gravitational field was considered by Synge (1960), and further investigated for deflection in presence of plasma by Perlick (2000); for related discussion, see Bičák & Hadrava (1975); Tsupko (2021). Gravitational lensing in Schwarzschild metric with presence of non-homogeneous plasma has been extensively studied (e.g. Bliokh & Minakov 1989; Bisnovatyi-Kogan & Tsupko 2009, 2010, 2015; Tsupko & Bisnovatyi-Kogan 2013; Crisnejo & Gallo 2018; Crisnejo et al. 2019). Er & Mao (2014, 2022) presented the effects due to plasma in the gravitational lensing applications, in particular the magnification and time delay effects in the strong lensing system. In case of the lens being a compact object, such as a neutron star or a black hole, the surrounded plasma can significantly affect the rays, causing complex behaviour (e.g. Rogers 2015, 2017b, 2017a; Kimpson et al. 2019; Briozzo & Gallo 2022). In the strong deflection case, the rays can make one or more revolutions around the lens, i.e. high-order images are formed (Virbhadra & Ellis 2000; Bozza et al. 2001; Perlick 2004). They were investigated for the case with surrounding plasma in Tsupko & Bisnovatyi-Kogan (2013). Such extra deflection by plasma can also change the size and the shape of black hole shadow, which has been discussed by, e.g. Perlick et al. (2015); Perlick & Tsupko (2017); Yan (2019); Bezděková et al. (2022); Briozzo et al. (2022) and review Perlick & Tsupko (2022).

The stellar wind is highly ionised plasma and can reach a high density than the average value in the interstellar medium, e.g. the electron density near Earth is around $\sim 50$ electrons per centimetre cubic (Parker 1958). The plasma near the star not only causes various magnification curves at different frequencies (Tsupko & Bisnovatyi-Kogan 2020), it also provides a potential way to study the stellar wind of different types of stars. The emission from lots of sources is weak in radio bands, which causes the plasma lensing phenomena difficult to detect. However, since the electron density near the stars can be much higher than that in ISM, the plasma microlensing effects can be significant in high radio frequency or even in sub-millimetre band.

The transformation of the microlensing curve from a single point lens to that with the presence of plasma has been first studied by Tsupko & Bisnovatyi-Kogan (2020). It has been shown that the refractive deflection in plasma can lead to a significant change in the shape of the curve, up to the disappearance of images due to strong divergence of rays. The authors called it ’hill-hole effect’, implying that with sufficient plasma density (or observing at sufficiently long wavelength), the maximum in the centre of the microlensing curve (’hill’) can be replaced by the minimum (’hole’).

In this paper, we investigate how the refraction in plasma environment changes the microlensing curve in a more general case of a binary lens. We focus mainly on the case of a star surrounded by plasma and hosting a planet. In Sect. 2, we give short summary about the plasma lensing and introduce the binary microlensing with plasma environment. We study the magnification curves for a lensing system in Sect. 3. In Sect. 4, we present the evolving of the size of the caustics. The summary of our result is given in the end. In the appendix, we present the details of how we calculate the magnification, discuss the number of solutions, some caustics and critical condition.

In this Section, we numerically calculate microlensing magnification curves for the binary system under consideration: the star surrounded by plasma, and the planet. This system is described by the lens equation (18). Normalised coordinates are used for calculation and for plotting figures: ($x_{1},x_{2}$) for the lens plane and ($y_{1},y_{2}$) for the source plane.

Numerical values of the most of lens parameters we adopt from Mao & Paczynski (1991), see Tab. 1. For the simplification of numerical calculations, we take Einstein radius of the main lens as $\theta_{E}=1$ milli-arcsec. With our choice of the lens and the source distance (Tab. 1), the corresponding lens mass is about 0.984 Solar mass. The planet is chosen as $1/1000$ of the star mass, which is about the Jupiter mass. It means that the mass ratio is $q=0.001$. Lens separation in our normalised coordinates is $\mbox{\boldmath$x_{p}$}=(1.2,0)$.

The effect of the plasma is completely determined by the parameter $B_{0}$, which depends on the plasma density profile and the frequency of observation. Increasing the parameter $B_{0}$ corresponds to considering a denser plasma or decreasing the frequency of observation (increasing the wavelength).

In our calculations, we consider different values of $B_{0}$. Since the plasma density profile is specified (Tab. 1), the different values of the plasma strength $B_{0}$ correspond to different observational frequency. Generally, if we consider frequency from $0.65$ GHz to 100 GHz, this covers from low radio frequency to the milli-metre band. Such a range can give us the plasma lensing parameter $\theta_{0}$ from 1 to 0.03 milli-arcsec (Fig. 2). We take $\theta_{0}=0.1,0.7,0.8$ milli-arcsec. With $\theta_{E}=1$ milli-arcsec, this leads to $B_{0}=0.1^{3},\,0.7^{3},$ and $0.8^{3}$. With values $n_{0}$, $R_{0}$, $D_{d}$, $D_{s}$ from Tab. 1, these three values of $B_{0}$ correspond to the observational frequencies roughly $21$ GHz, $1.1$ GHz and $0.93$ GHz respectively. We show the angular scale length ($\theta_{0}$) as a function of the observational frequency in our plasma lensing model in Fig. 2. The red, orange and green horizontal line show the three $\theta_{0}$ that we mentioned above. One can see that, as long as the frequency down to a few GHz, $\theta_{0}$ will be comparable with the Einstein radius of the star, and the plasma effects can be significant.

In all our calculations, we use $n_{0}=5.6$ cm^-3, see Tab. 1. We also note that the electron density near the stars can have large uncertainties. Higher density can be found in the atmosphere of compact objects. It leads to the number density of order of $n_{e}\sim 10^{19}$ cm^-3 on the surface of compact objects (e.g. around black hole or neutron star, Wielgus et al. 2016; Rogers 2017b). In such case, one expects to see significant plasma effects in the sub-mm bands, or even in the infrared bands. Moreover, higher electron density can be found in accretion disc around a black hole (e.g. Artemova et al. 2006), about $10^{-5}$ g/cm³, see also Bisnovatyi-Kogan (2011). One thus expects some complicated microlensing effects in the strong lensed QSOs.

In order to obtain the magnification curves, we move a source behind the the lens system, and calculate the magnification caused by the lens system. It is possible that the lens can generate multiple images. However, due to the small image separations, we assume that the multiple images cannot be resolved spatially, and thus calculate the total magnification. In most cases, the magnification will be dominated by only one image. We follow the similar approach in Tsupko & Bisnovatyi-Kogan (2020) to solve the lens equation and calculate the magnification for each source position. More details about the method of calculation can be found in the Appendix A.

We begin with the simple case of a single star surrounded by plasma (i.e. no planet, $q=0$). We consider this case for comparison with the work in Tsupko & Bisnovatyi-Kogan (2020), and also as an illustration to explain the ’hill-hole’ effect. The magnification curve and caustics for the power-law plasma density profile with $h=2$ are shown in Fig. 3. With different values of $B_{0}$, one can obtain regular single peak, trident, or volcano hole shapes of the curves.

With the inclusion of a planet ($q=0.001$, $\mbox{\boldmath$x_{p}$}=(1.2,0)$), the magnification curves will be perturbed, and will be more complicated. In the figure of lensing map along $x_{1}(y_{1})-$axis (Fig. 4), we can see that when the plasma density is low (or the observational frequency is high), the plasma deflection is weak, and similar to the case in vacuum. The lensed images caused by the star will be on the both sides of the lens, while the perturbations caused by the planet will always on the same side of the planet, even when the deflection caused by plasma becomes strong. In Fig. 5, we show the critical curves (blue) and caustics (red) for all three cases of $B_{0}$ that are considered for single lens on Fig.3. In the left panel, they are almost identical to the case in vacuum. With the increasing of $B_{0}$ the plasma firstly generates another caustics. They also pushes the caustics of planet outwards, and compress the caustic generated by the planet.

Several impact parameters and inclination angles of the trace of the background source are studied, as shown by the dotted straight lines in Fig. 5. The black trace connects the star and the planet, and the green trace is parallel to the black one with an offset of $y_{2}=0.01$. The other two have an inclination to the axis of star and planet. The functions of the four traces are given by

where we write the trajectory of the source as a function of time $t$. The time $t$ is in unit of $t_{E}$ given an arbitrary tangential velocity of the source in the source plane. In the choice of our trace functions, the sources have the same tangential velocity. Then we can calculate the magnification curves $\mu(t)$ for the four traces, by inserting Eqs.(21) – (24) into Eq.(18). The total magnification is calculated and shown in Figs. 6.

In the high observed frequency case (small $B_{0}$), the magnification curve is almost identical to that in vacuum. One can see that from the black and red curves in Fig. 6 panel 2. The central peak generated by the star behaviours like that without the planet. The plasma deflection weakens the peaks in the magnification curves. With the increasing of $B_{0}$ the single central peak becomes trident, and then volcano hole. The caustic generated by the planet shrinks. The double peaks generated by the planet move outwards of the star. In the intermediate case ($B_{0}=(0.7)^{3}$), both plasma and gravity cause significant deflection. Thus we can see all the peaks. Such signatures are distinguished from that without plasma or single star with plasma environment.

In this Section, we discuss the structure of the caustics of binary system in the presence of plasma and their evolution with changes in parameters.

First, let us comment on the caustics for the case of a single lens surrounded by plasma (Fig. 3). For $h=2$ power law profile, there are two caustics: central as a point (associated with gravity) and a circular (associated with plasma). As long as the plasma influence is small, the plasma caustic has a large radius. E.g., in Fig. 3 for $B_{0}=0.1^{3}$, the radius of the plasma caustics extends beyond the figure. For central panels in Fig. 3, where the parameter $B_{0}$ is larger ($0.7^{3}$), we see both caustics: the red dot in the centre of the figure and the red circle around it. With the further increasing of the $B_{0}$, there will be an transition, i.e., the hole in the magnification curve forms and the central caustic disappears at the same time. We use a horizontal width $\Delta_{cc}$ to indicate the size of the caustics, which is the diameter of the caustics in an axis-symmetric case. In Fig.  7, we show the evolving of the size of caustics as a function of $B_{0}$. Since the central caustic is a point, its $\Delta_{cc}$ is zero. With the plasma density increasing further, the plasma caustic shrinks and finally merges with the central caustic. One can see that there is only one (plasma) caustic from the right panel of Fig. 3. After merging, the caustic will increase with the influence of the plasma, $B_{0}$. The transition without planet has been given analytically (Tsupko & Bisnovatyi-Kogan 2020), see Appendix D for more detail. The critical value is $B_{0}=2/3^{3/2}\approx 0.3849$ ($B_{0}^{1/3}\approx 0.7274$, and the corresponding frequency in our density model is around 1.07 GHz).

Now let us consider a binary system where the star is surrounded by plasma. Note that even for the vacuum case, the existence of the planet will generate extra caustics and change the central caustic as well (e.g. Bozza 1999). It has been approximately studied in an analytic way or numerically in vacuum (e.g. An 2005; Chung et al. 2005; An 2021). However, with the extra deflection by the plasma, the analytic form is difficult to obtain.

In presence of plasma in binary system, more caustics will be generated by the lens, e.g. more than one central caustics, the one associated with the planetary and the one associated with plasma. Here we only consider a simple case: a large separation between the star and planet, the two caustics generated by star and planet do not merge together. One interesting difference exists from the case of single star with plasma environment. In case of single star, when the plasma becomes sufficient strong, all the light rays will be diverged, i.e. no solution of image at all. However, when there is a planet, the gravity of the planet will weaken the diverging caused by the plasma. One solution of image to the lens equation exists, although the magnification of this image can be extremely small. Only when the refraction deflection by plasma is strong enough ($B_{0}^{1/3}\sim 5$), there will be no solution at all. Thus the definition of hole in the magnification curve is different from that in the single star case. We consider the hole forming when there is only one image with small magnification, $<10^{-5}$. In our lensing system, it will happen when $B_{0}^{1/3}\approx 0.7273$ (Appendix D). We can estimate the number of caustics by two ways: first we can use the number and the positions of peaks in the magnification curve; on the other hand, we can see from the pattern of the caustics. We find that the peak number drops from six to four when $B_{0}^{1/3}\approx 0.7278$, two peaks related to the central caustics disappear, which imply that the central caustics disappear. The caustics can have either diamond-like shapes (four cusps) or triangle-like shapes (three cusps) (e.g. Bozza 1999; Zhang 2022). One or two of the cusps are located on the star-planet axis, i.e. $y_{2}=0$. We thus define the width $\Delta_{cc}$ as the distance between the points where caustic crosses an axis connecting the star and planet. In Figs. 8-9, we show the evolving of $\Delta_{cc}$ with $B_{0}^{1/3}$ and $x_{p}$. From left panel of Fig. 8 we can see that both caustics shrink with the increasing of the plasma effects. While when $B_{0}^{1/3}>0.93$, the size of planetary caustic will increase again with $B_{0}$ (we do not consider a case with such low frequency of observation, thus not show in the figure). When $B_{0}^{1/3}\sim 0.7275$, the caustics caused by plasma and part of central caustics merge (green curves in Fig. 13). The other caustic which is not merge with the plasma caustic is regarded as the central one. We can see the transition, i.e. a drastic decreasing of the size of the central caustics. The size of the caustic generated by plasma have almost the identical behaviour as that without planet (Fig. 7). Only the transition point is slightly different, $B_{0}^{1/3}\sim 0.7275$. The caustic generated by the planet is larger than that by the central one. In all cases, $\Delta_{cc}$ of planet decreases with $B_{0}^{1/3}$ and $x_{p}$. The position of the planet caustics will be changed by plasma as well, but no transition has been found.

In Fig. 9 (left), we present the evolving of the size of caustics as a function of the star-planet separation $x_{p}$. As one would expect, the larger the separation the smaller the caustics. Additional tests with different $B_{0}$ have been performed, and similar trend has been found. For the size of the caustics generated by plasma, the planet has little effect. As shown in Fig. 9 (right), $\Delta_{cc}$ increases with $x_{p}$ slowly, for the dependence on $x_{p}$ of plasma caustic is much weaker compared with that of other caustics on $x_{p}$.

Microlensing provides a powerful tool in searching and studying the exoplanets. Taking into account of the stellar wind (plasma), the magnification curves generated by the system can be different at low frequencies. This happens due to refraction in the plasma, which leads to chromatic deflection of the light ray in addition to gravitational deflection. We present some magnification curves with different observational frequency and different trajectories of the background source for the system of a star surrounded by plasma and a planet (Figs. 5 and 6). When the observational frequency is high, the magnification curves are almost indistinguishable from those without plasma. But as the observational frequency decreases to a few GHz, the plasma effects will be significant. For example, the peak in the magnification curve generated by the star becomes trident or even volcano hole. The position of the peak generated by the planet changes with frequency as well.

We also present the evolving of the size of the caustics with plasma density ($B_{0}$) and the separation of star and planet ($x_{p}$) in Figs. 8 and 9. With the increasing of $B_{0}$, the size of the central caustics and planetary caustic decrease gradually, and the central caustics disappear after $B_{0}$ reaching a critical value. For the size of the caustic caused by plasma, it decreases at the beginning and then increases. With $x_{p}$ increases, the size of the central caustics and planetary caustic decrease, but the size of the caustic caused by plasma almost unchanged.

In our study, we find some difference from the case without planet. When the lens is consisted of star and plasma only, the hole forms and the central caustic disappears at the same time, and the caustics have the regular circular shapes. While with the inclusion of a planet, the transition value of the plasma is different. More caustics will be generated, and the shapes of caustics can be diamond-like, triangle-like or even more complex shape. The hole in the magnification curve will form before the central caustics disappear. The possible reason is that the existence of the planet slightly shifts the central caustics to the right, thus the disappearance of the central caustics requires a stronger plasma.

The plasma density that we adopted in this study is from the Solar wind. The possible observational frequency is around $\sim$GHz. There is possibility of higher plasma density or density gradient, and thus even in higher frequency, e.g. infrared, we can distinguish and study the stellar wind. There are several simplification in this work. First of all, we only study the system of one star and one Jupiter size planet. Binary stars with planet(s) or single star with multiple planets has also been found and will show complicate situations. Even for the system of single star and single planet, there is large parameter space to be explored, e.g. mass ratio, separation etc. Both the lens and the background source has been considered as point source, i.e. the finite size of the star or planet has not taken into account. Such effect needs to be included for accurate prediction in the future study. The magnification curve in different bands can provide rich information on the lens system.

XE would like to thank Chun Xia for the help on the plasma model of the Solar wind. XE is supported by the NSFC Grant No. 11873006, 11933002 and the China Manned Space Project with No.CMS-CSST-2021-A07, No.CMS-CSST-2021-A12. The work of OYuT was partially supported by the Russian Foundation for Basic Research, project No. 20-52-12053. OYuT is thankful to G.S. Bisnovatyi-Kogan for useful discussions.

The data underlying this article will be shared on reasonable request to the corresponding author.