Measuring Microlensing Parallax via Simultaneous Observations from Chinese Space Station Telescope and Roman Telescope

Gravitational microlensing is powerful in detecting cold planets, including those beyond the snow line and those that are unbound to any star (e.g., Mao & Paczynski 1991; Gould & Loeb 1992; Sumi et al. 2011). Statistical studies have found that cold planets are abundant (Gould et al. 2010; Suzuki et al. 2016), and that unbound planets may be as common as main-sequence stars in the Galaxy (Mróz et al. 2017, 2019). Observational constraints on unbound planets, or free-floating planets (FFPs), can provide important constraints on the formation and evolution of planetary systems (see Zhu & Dong 2021 for a recent review).

The standard microlensing technique usually does not yield measurements of the lens mass. This is because the microlensing timescale, t_E, involves multiple physical parameters

$$\begin{eqnarray}&&{t}_{{\rm{E}}}\equiv \displaystyle \frac{{\theta }_{{\rm{E}}}}{{\mu }_{\mathrm{rel}}}.\end{eqnarray} \tag{ 1 }$$

Here μ_rel is the relative proper motion between the lens and the source, and θ_E is the angular Einstein radius

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}\equiv \sqrt{\kappa {M}_{{\rm{L}}}{\pi }_{\mathrm{rel}}},\end{eqnarray} \tag{ 2 }$$

where $\kappa \equiv 4G{({c}^{2}\mathrm{au})}^{-1}\approx 8.14\,\mathrm{mas}\,{M}_{\odot }^{-1}$ is a constant, M_L is the mass of the lens and ${\pi }_{\mathrm{rel}}=\mathrm{au}({D}_{{\rm{L}}}^{-1}-{D}_{{\rm{S}}}^{-1})$ is the lens–source relative parallax, with D_L and D_S the distances to the lens and the source, respectively (Gould 2000).

The lens mass can be measured or directly constrained if two of the three physical quantities—angular Einstein radius, microlensing parallax and lens flux—are measured (e.g., Yee 2015). Microlensing parallax is the lens–source relative parallax scaled by the angular Einstein radius (Gould 1992)

$$\begin{eqnarray}&&{\pi }_{{\rm{E}}}\equiv \displaystyle \frac{{\pi }_{\mathrm{rel}}}{{\theta }_{{\rm{E}}}}.\end{eqnarray} \tag{ 3 }$$

For faint microlenses such as FFPs, the only way to directly measure the mass is therefore the combination of θ_E and π_E. The microlensing parallax is of particular importance, as it alone can already yield strong constraints on the lens mass under general assumptions of the lens kinematics (Han & Gould 1995; Zhu et al. 2017a).

Many methods have been proposed to measure the microlensing parallax in general microlensing events (e.g., Gould 1992, 1994a; see a brief summary in Zhu et al. 2015). In the context of FFP events, the ideal way is to obtain observations from two observatories that are separated by a large fraction of the projected Einstein radius

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{r}}_{{\rm{E}}} & \equiv & \displaystyle \frac{\mathrm{au}}{{\pi }_{{\rm{E}}}}\\ & = & 0.016{\left(\displaystyle \frac{{M}_{{\rm{L}}}}{{M}_{\oplus }}\right)}^{1/2}{\left(\displaystyle \frac{{\pi }_{\mathrm{rel}}}{0.1\,\mathrm{mas}}\right)}^{-1/2}\,\mathrm{au}.\end{array}\end{eqnarray} \tag{ 4 }$$

Here the normalization has been chosen for an Earth-mass object at a typical microlensing distance (D_L ≈ 4 kpc). A combination of telescopes at Earth and the Sun–Earth L2 point, with a maximum projected separation of 0.01 au, is therefore ideal for the detection of microlensing parallax for free-floating planetary-mass objects.

Studies have looked into the feasibility of measuring microlensing parallax from Earth and L2. The Roman Space Telescope, previously known as WFIRST, is going to conduct multiple microlensing campaigns from L2 (Spergel et al. 2015). With a 2.4 m aperture and 0.3 deg² field of view, Roman is expected to detect over one thousand bound planets and hundreds of FFPs out of its 5 × 72 day microlensing campaigns (Penny et al. 2019; Johnson et al. 2020). The potential of utilizing a Roman-like telescope for microlensing parallax observations has long been realized (e.g., Gould et al. 2003; Han et al. 2004; Yee 2013; Gould et al. 2021). In particular, Zhu & Gould (2016) and Street et al. (2018) studied the use of ground-based telescopes, either the Korea Microlensing Telescope Network (KMTNet, Kim et al. 2016) or the Rubin Observatory (previously known as LSST) to augment the microlensing parallax measurements of Roman. See also Bachelet & Penny (2019) and Ban (2020) for the feasibility of using two L2 satellites (i.e., Euclid and Roman) to determine microlensing parallax parameters.

The Chinese Space Station Telescope (CSST) is a planned mission currently scheduled to launch in late 2023 and start scientific observations in 2024 (Zhan 2011, 2018; Cao et al. 2018). CSST has an aperture of 2 m and a field of view of 1.1 deg², and it will be in a ∼ 400 km low-Earth orbit (LEO) with an orbital period around 95 minutes. The primary science goal of CSST is to understand the nature of dark matter and dark energy using a number of cosmological probes, such as galaxy clusterings as well as weak and strong gravitational lensings (e.g., Gong et al. 2019; Zhang et al. 2019). The high resolution and large field of view of CSST make it also a wonderful mission to conduct space-based microlensing surveys (Gould 2009). Similar to those by Euclid and Roman (Penny et al. 2013, 2019; Johnson et al. 2020), the microlensing survey by CSST will have sensitivity to low-mass planets on both wide and unbound orbits.

This paper discusses the potential of measuring microlensing parallax by CSST alone and by combining CSST in LEO and a Roman-like telescope at L2. We show in Section 2 that such a joint program can measure microlensing parallax for the majority of short (and ultra-short) timescale microlensing events. Section 3 explains the potential of CSST and Roman in measuring parallax and thus presents a full lens solution of caustic-crossing binaries (including planetary events). A brief discussion is given in Section 4.

The microlensing light curve arising from a point source and a single lens is given by Paczynski (1986)

$$\begin{eqnarray}\begin{array}{rcl}F(t) & = & {F}_{{\rm{S}}}[A(t)-1]+{F}_{\mathrm{base}},\\ A(t) & = & \displaystyle \frac{{u}^{2}+2}{u\sqrt{{u}^{2}+4}}.\end{array}\end{eqnarray} \tag{ 5 }$$

Here F_S is the source flux at baseline, F_base is the total flux (source and possible blend) at baseline, and u is the distance between the source and the lens at a given time t normalized to the Einstein radius. Because of the microlensing parallax effect, telescopes at different locations (namely L2 and LEO) see different values of u

$$\begin{eqnarray}\begin{array}{rcl}{u}_{{\rm{L}}2}^{2} & = & \displaystyle \frac{{(t-{t}_{0,{\rm{L}}2})}^{2}+{t}_{\mathrm{eff},{\rm{L}}2}^{2}}{{t}_{{\rm{E}}}^{2}};\\ {u}_{\mathrm{Leo}}^{2} & = & \displaystyle \frac{{(t-{t}_{0,\mathrm{Leo}})}^{2}+{t}_{\mathrm{eff},\mathrm{Leo}}^{2}}{{t}_{{\rm{E}}}^{2}}.\end{array}\end{eqnarray} \tag{ 6 }$$

Here t₀ is the peak time and t_E is the event timescale. We use the effective timescale t_eff ≡ u₀ t_E rather than the impact parameter u₀ in describing the event evolution, because t_eff is almost always better constrained in observations than is u₀.

The microlensing parallax measured from two well separated observatories can be approximated as (Refsdal 1966; Gould 1994a)

$$\begin{eqnarray}&&{{\boldsymbol{\pi }}}_{{\boldsymbol{E}}}=({{\pi }_{{\rm{E}}}}_{\parallel },{{\pi }_{{\rm{E}}}}_{\perp })=\displaystyle \frac{\mathrm{au}}{{D}_{\perp }}\left(\displaystyle \frac{{\rm{\Delta }}{t}_{0}}{{t}_{{\rm{E}}}},{\rm{\Delta }}{u}_{0}\right),\end{eqnarray} \tag{ 7 }$$

where ${{\pi }_{{\rm{E}}}}_{\parallel }$ and ${{\pi }_{{\rm{E}}}}_{\perp }$ are the components parallel and perpendicular to the source trajectory, respectively. Here D_⊥ is the projected separation between the two observatories, t_E is the event timescale, and Δt₀ ≡ t_0,LEO − t_0,L2 and Δu₀ ≡u_0,LEO − u_0,L2 are the differences in peak times and impact parameters as seen from two observatories, respectively.

By writing the parallax vector in the form of Equation (7) we have assumed that (a) the event timescale t_E is (effectively) the same at two locations and that (b) both t_E and the projected separation D_⊥ are invariant in the process of the event. The latter is a reasonable assumption for short (less than a few days) timescale events, which are the focus of this section. Regarding the former, the difference in t_E arises from the relative motion between the two telescopes, which we separate into two parts. The relative motion between Roman and Earth ( < 1 km s⁻¹) is very small compared to the lens-source projected motion, which is typically ${\tilde{v}}_{\mathrm{hel}}\gtrsim 200\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (see Appendix B of Zhu & Gould (2016)). The relative motion between CSST and Earth varies within an orbital period of CSST (T ≈ 90 min) and caps at ∼ 8 km s⁻¹. For events with t_E ≫ T, the time-averaged relative velocity is negligible compared to the transverse velocity ${\tilde{v}}_{\mathrm{hel}}$. For events with t_E comparable to T, the relative motion between CSST and Earth is in fact useful in further breaking the standard four-fold degeneracy in two-location observations (Refsdal 1966; Gould 1994a; Zhu et al. 2017b), as in such cases CSST can be effectively treated as more than one static telescope. We will further discuss the use of this effect in Section 2.2.

While π _E is an important quantity that is frequently used in the literature, a better parameter that is more directly constrained in observations and better connected to the lens kinematics is (Dong et al. 2007; Zhu & Gould 2016)

$$\begin{eqnarray}&&{\boldsymbol{\Lambda }}\equiv \displaystyle \frac{{{\boldsymbol{\pi }}}_{{\boldsymbol{E}}}{t}_{{\rm{E}}}}{\mathrm{au}}=\displaystyle \frac{1}{{\tilde{v}}_{\mathrm{hel}}}=\displaystyle \frac{1}{{D}_{\perp }}({\rm{\Delta }}{t}_{0},{\rm{\Delta }}{t}_{\mathrm{eff}}).\end{eqnarray} \tag{ 8 }$$

We therefore rely on this parameter to quantify the detectability of the microlensing parallax effect in such an LEO+L2 configuration.

We utilize Fisher matrix analysis to evaluate the detectability of the parallax effect. For the Roman survey, we use 15 minute cadence and the expected photometric performance out of the 53 s exposure from Penny et al. (2019). For the CSST survey, we assume a duty cycle of 40% and 8 observations per orbit (corresponding to ∼5 minute cadence). The photometric performance of the CSST microlensing survey is derived from the online CSST ETC simulator. ⁴ We have assumed high zodiacal light and average Earth shine light for the background contamination and an extinction law of E(B − V) = 0.94. A typical M0V star in the bulge is chosen as the microlensing source. With an exposure time of 60 s and a systematic floor of 0.001 mag, we can then simulate the noise curve of the CSST microlensing survey. The result is plotted in Figure 1 together with the noise curve of Roman from Penny et al. (2019).

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The parameters used to model the combined data set are

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{\theta }} & = & ({t}_{{\rm{E}}},{t}_{0,{\rm{L}}2},{t}_{\mathrm{eff},{\rm{L}}2},\\ & & {F}_{{\rm{S}},\mathrm{Leo}},{t}_{0,\mathrm{Leo}},{t}_{\mathrm{eff},\mathrm{Leo}},c).\end{array}\end{eqnarray} \tag{ 9 }$$

Here F_S,LEO is the source flux at baseline in the bandpass of CSST at LEO, and we renormalize the flux values such that F_S,LEO = 1 corresponds to a 21 magnitude star. For the other observatory, we rely on the parameter c to quantify the ratio of source fluxes in the bandpasses of the two telescopes and it is related to the observed color of the source m_LEO − m_L2

$$\begin{eqnarray}&&c\equiv \displaystyle \frac{{F}_{{\rm{S}},{\rm{L}}2}}{{F}_{{\rm{S}},\mathrm{Leo}}}={10}^{({m}_{\mathrm{Leo}}-{m}_{{\rm{L}}2})/2.5}.\end{eqnarray} \tag{ 10 }$$

We do not specify the bandpasses of the two space-based surveys, although for practical reasons we will use H- and I-band for observations from L2 and LEO, respectively. Considering the stellar colors from Pecaut & Mamajek (2013) and assuming an extinction of A_I = 1.5 and E(I − H) ≈ 1 (Gonzalez et al. 2012; Nataf et al. 2013), one finds that a Sun-like star in the bulge will have m_LEO ≈ 20, m_LEO − m_L2 ≈ 1.8 and c ≈ 5.2. Early M-dwarfs, which will probably be the primary sources of the events simultaneously observed by CSST and Roman, will have m_LEO ≈ 23, m_LEO − m_L2 ≈ 2.7 and thus c ≈ 12.

Throughout this paper we consider two types of microlensing events: a typical disk event with lens-source relative parallax π_rel = 0.12 mas (corresponding to a lens at ≈ 4 kpc) and a relative proper motion μ_rel = 7 mas yr⁻¹; and a typical bulge event with π_rel = 0.02 mas (corresponding to a lens at ≈ 7 kpc) and μ_rel = 4 mas yr⁻¹. These yield typical transverse velocities (Zhu & Gould 2016)

$$\begin{eqnarray}\tilde{v}=\frac{1}{{\rm{\Lambda }}}=\left\{\begin{array}{ll}280\,\mathrm{km}\,{{\rm{s}}}^{-1}, & \mathrm{Disk}\,\mathrm{events}\\ 1000\,\mathrm{km}\,{{\rm{s}}}^{-1}, & \mathrm{Bulge}\,\mathrm{events}\end{array}\right..\end{eqnarray} \tag{ 11 }$$

Figure 2 features example light curves of typical disk and bulge microlensing events with t_E = 0.1 d, u_0,LEO = 0.1 and a source baseline flux F_S,LEO = 1 (corresponding to a baseline magnitude of 21).

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The Fisher matrix ${{ \mathcal F }}_{{ij}}$ is then defined by

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal F }}_{{ij}} & \equiv & \sum _{\{{t}_{l}\}}\displaystyle \frac{1}{{\sigma }_{F}^{2}({t}_{l})}\displaystyle \frac{\partial {F}_{{\rm{L}}2}({t}_{l})}{\partial {\theta }_{i}}\displaystyle \frac{\partial {F}_{{\rm{L}}2}({t}_{l})}{\partial {\theta }_{j}}\\ & & +\sum _{\{{t}_{m}\}}\displaystyle \frac{1}{{\sigma }_{F}^{2}({t}_{m})}\displaystyle \frac{\partial {F}_{\mathrm{Leo}}({t}_{m})}{\partial {\theta }_{i}}\displaystyle \frac{\partial {F}_{\mathrm{Leo}}({t}_{m})}{\partial {\theta }_{j}}.\end{array}\end{eqnarray} \tag{ 12 }$$

Here {t_l } and {t_m } are the time series of the Roman telescope and CSST, respectively, and σ_F represents the uncertainty of the measured flux. We then compute the covariance matrix of the vector Λ, Σ_Λ, based on the relation between Λ and the direct observables (Equation (8)). The detectability of the parallax effect is quantified by

$$\begin{eqnarray}&&{\chi }_{{\boldsymbol{\Lambda }}}^{2}=({\boldsymbol{\Lambda }}-{\bf{0}}){{\rm{\Sigma }}}_{{\rm{\Lambda }}}^{-1}{\left({\boldsymbol{\Lambda }}-{\bf{0}}\right)}^{T},\end{eqnarray} \tag{ 13 }$$

where 0 denotes the zero-parallax case.

For any chosen set of microlensing parameters we apply the above Fisher matrix analysis to numerically evaluate the detectability of the parallax effect. One example output is featured in Figure 3, for which we have assumed a maximum projected separation between LEO and L2 D_⊥ = 0.01 au.

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To gain theoretical insights, we also derive the following analytical scaling relation in the high-magnification (u₀ → 0) regime (see Appendix for the detailed derivation)

$$\begin{eqnarray}&&{\chi }_{{\boldsymbol{\Lambda }}}^{2}\propto \frac{{D}_{\perp }^{2}{F}_{{\rm{S}},\mathrm{Leo}}}{{t}_{{\rm{E}}}{u}_{0,\mathrm{Leo}}^{2}}f(c).\end{eqnarray} \tag{ 14 }$$

Here $f(c)\equiv {\chi }_{{\rm{\Lambda }}}^{2}(c)/{\chi }_{{\rm{\Lambda }}}^{2}(c=1)$ captures the dependence of ${\chi }_{{\rm{\Lambda }}}^{2}$ on the color parameter c and is normalized by the value of ${\chi }_{{\rm{\Lambda }}}^{2}$ at c = 1 (see Figure 4). As affirmed in Figure 3, the above scaling relation matches the numerical results reasonably well and breaks down at both large (u₀ ≳ 1) and small u₀ values (u₀ ≲ 0.01). The former is due to the breakdown of the high-magnification regime. The latter is due to the fact that we have assumed u_0,LEO = u_0,L2 in the analytical derivation. According to Equation (7), Δu₀ = π_E,⊥ D_⊥. For typical disk and bulge events, Δu₀ ≈ 0.01 and 0.001, respectively, and thus u_0,L2 = u_0,LEO + Δu₀ → u_0,LEO as long as u_0,LEO ≫ 0.01. Additionally, the above scaling typically breaks down at t_E ≲ 0.1 days when the event timescale is comparable to the orbital period of an LEO satellite around Earth.

With Equation (14) and the numerical results from Figure 3, we can rewrite the detection significance of the parallax effect in ${\chi }_{{\boldsymbol{\Lambda }}}^{2}$ as the following expressions

$$\begin{eqnarray}{\chi }_{{\boldsymbol{\Lambda }}}^{2}=\left\{\begin{array}{ll}37\left(\frac{{F}_{{\rm{S}},\mathrm{Leo}}}{1}\right){\left(\frac{{t}_{{\rm{E}}}}{10\,\mathrm{days}}\right)}^{-1}{\left(\frac{{u}_{0,\mathrm{Leo}}}{0.3}\right)}^{-2}{\left(\frac{{D}_{\perp }}{0.01\,\mathrm{au}}\right)}^{2}f(c), & \mathrm{Bulge}\,\mathrm{event}\\ 274\left(\frac{{F}_{{\rm{S}},\mathrm{Leo}}}{1}\right){\left(\frac{{t}_{{\rm{E}}}}{10\,\mathrm{days}}\right)}^{-1}{\left(\frac{{u}_{0,\mathrm{Leo}}}{0.3}\right)}^{-2}{\left(\frac{{D}_{\perp }}{0.01\,\mathrm{au}}\right)}^{2}f(c), & \mathrm{Disk}\,\mathrm{event}\end{array}\right..\end{eqnarray} \tag{ 15 }$$

The above expressions can be readily used to evaluate the detectability of the parallax effect in any given single-lens event. Specifically, with the maximum projected separation of D_⊥ = 0.01 au between Earth and L2, the combination of CSST and Roman should be able to detect the microlensing parallax effect for typical bulge (disk) events with 0.1 ≲ t_E/days ≲ 10 (0.1 ≲ t_E/days ≲ 80), if the source is 23 mag at baseline and the impact parameter is ∼0.3. These values enclose the bulk of the expected FFPs from Roman, especially those with low-mass (and thus preferentially terrestrial) lenses (e.g., Johnson et al. 2020).

2.1. Lens Mass Determinations

Besides the microlensing parallax, another critical parameter toward a lens mass measurement is the angular Einstein radius θ_E, which is usually determined from the finite-source effect (Yoo et al. 2004). For single-lens events, the finite-source effect becomes prominent when the impact parameter u₀ becomes comparable to or less than the scaled source size ρ. The fraction of microlensing events with complete lens mass measurements out of all events with parallax detections can be estimated as

$$\begin{eqnarray}&&{P}_{\mathrm{mass}}\sim \displaystyle \frac{\rho }{{u}_{0}^{\mathrm{crit}}}=\displaystyle \frac{{\theta }_{\star }}{{\mu }_{\mathrm{rel}}{t}_{{\rm{E}}}{u}_{0}^{\mathrm{crit}}}.\end{eqnarray} \tag{ 16 }$$

Here θ_⋆ = 0.3 μas is the typical value for the source angular size, which corresponds to an early M dwarf with a radius of 0.5 R_⊙ in the Galactic Bulge. The quantity ${u}_{0}^{\mathrm{crit}}$ is the critical impact parameter above which the parallax effect is no longer detectable. For given source and lens properties, it can be derived from the scaling relations in Equation (15) with a certain ${\chi }_{{\boldsymbol{\Lambda }}}^{2}$ threshold for the parallax detection. We use a ${\chi }_{{\boldsymbol{\Lambda }}}^{2}$ threshold of 10 and set an upper limit on ${u}_{0}^{\mathrm{crit}}$ at 0.7. The latter takes into account the fact that the scaling relations are no longer valid approximations when u₀ approaches unity.

Figure 5 shows the estimated fraction of events with complete mass measurements out of those with parallax detections. Here we have included the results with typical sources (F_S = 0.16 and c = 12, corresponding to m_LEO ≈ 23 and m_LEO − m_L2 ≈ 2.7) in addition to the results with the default F_S = 1 and c = 1 combination. For events with timescales in the range of 0.1–10 days, for which the joint observations of CSST and Roman are most valuable, ≳ 0.3% ( ≳ 2%) of disk (bulge) events with typical sources and parallax detections should have complete lens mass determinations. Up to ∼40% of ultra-short-timescale events will show the finite-source effect and thus have complete lens mass determinations. This fraction is largely independent of the source properties.

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For events that do not show the finite-source effect, the lens mass can be inferred from the microlensing parallax alone (Han & Gould 1995). Such mass inferences are statistically accurate at the ∼ 30% level for typical disk events (e.g., Yee 2015; Zhu et al. 2017a), whereas the uncertainty increases to ≳ 40% for bulge events (Gould et al. 2021). Such statistically inferred lens properties are also valuable in constraining the mass distribution of isolated microlenses, especially those with low probabilities of showing the finite-source effect.

2.2. CSST Alone

The orbital motion of CSST around Earth will be useful in further resolving the four-fold degeneracy in the single-lens case (Refsdal 1966; Gould 1994a; Zhu et al. 2017b) as well as detecting the parallax effect by CSST alone. An example light curve to demonstrate the use of the orbital motion effect of CSST is displayed in Figure 6.

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For simplicity, we will treat the orbital motion effect of CSST as if two satellites were simultaneously observing with a separation of an Earth diameter (i.e., R_⊥ = 2R_⊕). With this approximation, the result from the CSST+Roman analysis (Equation (15)) is directly applicable

$$\begin{eqnarray}{\chi }_{{\boldsymbol{\Lambda }},\mathrm{CSST}}^{2}\approx \left\{\begin{array}{ll}3.7\left(\frac{{F}_{{\rm{S}},\mathrm{Leo}}}{1}\right){\left(\frac{{t}_{{\rm{E}}}}{0.1\,\mathrm{days}}\right)}^{-1}{\left(\frac{{u}_{0,\mathrm{Leo}}}{0.1}\right)}^{-2}{\left(\frac{{D}_{\perp }}{2\,{R}_{\oplus }}\right)}^{2}, & \mathrm{Bulge}\,\mathrm{event}\\ 27\left(\frac{{F}_{{\rm{S}},\mathrm{Leo}}}{1}\right){\left(\frac{{t}_{{\rm{E}}}}{0.1\,\mathrm{days}}\right)}^{-1}{\left(\frac{{u}_{0,\mathrm{Leo}}}{0.1}\right)}^{-2}{\left(\frac{{D}_{\perp }}{2\,{R}_{\oplus }}\right)}^{2}, & \mathrm{Disk}\,\mathrm{event}\end{array}\right..\end{eqnarray} \tag{ 17 }$$

Note that these scaling relations do not take into account the finite-source effect. The point-source example featured in Figure 6 has ${\chi }_{{\boldsymbol{\Lambda }}}^{2}=141$, which for the chosen parameters is a factor of ∼2 smaller than what the above scaling relation yields. This difference probably comes from the fact that the real orbit of CSST has an effective projected separation smaller than 2 R_⊕.

Ultra-short-timescale events with relatively small impact parameters almost always show the finite-source effect, so it is necessary to take into account the impact of this additional effect on the detectability of the parallax effect. The finite-source effect works in two different ways: on the one hand, it reduces the magnification when the source–lens relative distance u ≪ ρ; on the other hand, the finite-source effect increases the magnification at u ∼ ρ (Gould 1994b, see also Yoo et al. 2004; Chung et al. 2017). Whether the finite-source magnification is less or more sensitive to the variation of distance u than is the point-source magnification depends on the ratio between u and ρ. Consequently, the detectability of the parallax effect can be either enhanced (when there are more data points with u ∼ ρ) or reduced (when there are more data points with u ≪ ρ). For example, the finite-source example displayed in Figure 6 has ${\chi }_{{\boldsymbol{\Lambda }}}^{2}=161$, slightly higher than the value of the point-source case. Considering that the source should be relatively bright, here we have assumed a Sun-like star in the bulge as the microlensing source.

A more precise and comprehensive evaluation of the parallax detectability in the case of CSST alone will require additional knowledge like the position of Earth and the inclination of CSST orbit, and thus we do not attempt to derive it in the present work. In short, CSST alone should be able to detect the parallax effect of ultra-short-timescale events which have relatively high magnifications and bright sources (see also Mogavero & Beaulieu 2016). For such events, the joint observations of CSST and Roman can also be used to break the four-fold degeneracy that is common in single-lens microlensing events.

For binary and planetary events with caustic-crossing features, which constitute roughly half of all binary and planetary events in such well sampled observing campaigns (Zhu et al. 2014), the joint observations of CSST and Roman can also measure the microlensing parallax and thus fully determine the mass and distance of the lens system. The caustic-crossing features are smoothed on the scales of the source size, and thus two observers separated by roughly the (projected) source size should see caustic-crossing features with measurable time offsets. For typical microlensing events in the CSST and Roman campaigns, the source stars are sub-solar in size (e.g., Penny et al. 2019), and thus the LEO–L2 separation, ∼ 0.01 au = 2 R_⊙, is ideal for the microlensing parallax observations regardless of the event timescale (see also Gould et al. 2021). Figure 7 illustrates two example light curves, one of planetary nature with mass ratio q = 10⁻⁵ and the other of binary nature with q = 1. In both examples, the caustic-crossing features are significantly offset in the light curves of CSST and Roman.

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Besides the LEO–L2 separation, other factors also affect the detectability of the microlensing parallax. The determination of the full two-dimensional microlensing parallax requires that each satellite should capture (at least) two caustic crossing features. Otherwise there will be a continuous degeneracy in the parallax vector (Hardy & Walker 1995; Graff & Gould 2002). This is not rare in reality. For example, some caustic-crossing events have effectively only one caustic-crossing feature (e.g., Shvartzvald et al. 2015), or the observations of one or both satellites may fail to capture some of the caustic-crossing features (e.g., Zhu et al. 2015). This latter scenario is probably going to be the dominant factor, as CSST at LEO has a duty cycle of ∼30%–40% due to (mostly) Earth's umbra.

Caustic-crossing events may have their parallax further constrained by the orbital motions of CSST or Roman (Honma 1999). Such a complementary constraint will be especially useful for caustic-crossing events with continuous degeneracies. We refer to Zhu & Gould (2016) for more discussion.

This work studies the potential of simultaneous observations by CSST and Roman in detecting the microlensing parallax effect. We have derived the scaling relations that quantify the detectability of the parallax effect by the two satellites together and by CSST alone. Our calculations confirm that CSST and Roman together can measure the microlensing parallax for short and ultra-short timescale (∼0.1–10 days) microlensing events, which roughly correspond to brown dwarf and lower-mass lenses. CSST alone has the capability to detect the microlensing parallax for ultra-short timescale events with relatively high magnifications and bright sources. Furthermore, the separation between CSST and Roman is ideal for microlensing parallax measurements of caustic-crossing events. With the angular Einstein ring radius measurements from the caustic-crossing features (Yoo et al. 2004), these parallax measurements will lead to direct determinations of the mass and distance of the lens system.

Although the combination of LEO and L2 is not ideal to detect the parallax effect of typical Galactic microlensing events, which usually require a projected separation between observatories of several au (Refsdal 1966; Gould 1994a), it is perhaps the best approach to complete the mass measurements (or inferences) of Galactic microlenses. A space-based telescope with high spatial resolution like CSST and Roman can resolve individual bulge stars and thus constrain the properties of luminous lenses through the lens flux method (e.g., Bennett et al. 2007). The microlensing campaign by CSST or Roman alone can therefore measure/constrain the mass function of (most) stellar lenses. However, as the lens flux method does not work for sub-stellar microlenses such as brown dwarfs and FFPs, the microlensing parallax method becomes the only way to constrain the lens properties. Therefore, a joint program that can detect the parallax effect of short and ultra-short timescale microlensing events can complement the lens mass measurements/constraints of the microlensing campaigns of individual missions.

Finally, it is also worth pointing out that although our results have assumed the configuration of CSST and Roman, they are generally applicable to any combination of satellites at LEO and L2.

We would like to thank the anonymous referee for comments and suggestions that have improved the paper. We acknowledge the science research grants from the China Manned Space Project with No. CMS-CSST-2021-A11.

The scaling relation (Equation (15)) can be derived analytically in the high-magnification regime, where the light curve can be modeled with only three parameters (t₀, t_eff, F_peak) (Gould 1996)

$$\begin{eqnarray}\begin{array}{rcl}F(t) & = & {F}_{\mathrm{peak}}Q(t);\\ Q(t) & = & {\left({\tau }_{\mathrm{eff}}^{2}+1\right)}^{-1/2}.\end{array}\end{eqnarray} \tag{ A.1 }$$

Here F_peak = F_S/u₀ is the source flux at event peak and τ_eff ≡ (t − t₀)/t_eff. The joint data set of LEO and L2 can then be modeled with the parameter set

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{a}} & = & ({t}_{0,{\rm{L}}2},{t}_{\mathrm{eff},{\rm{L}}2},{F}_{\mathrm{peak},{\rm{L}}2},\\ & & {t}_{0,\mathrm{Leo}},{t}_{\mathrm{eff},\mathrm{Leo}},{F}_{\mathrm{peak},\mathrm{Leo}}),\end{array}\end{eqnarray} \tag{ A.2 }$$

and the corresponding derivatives are

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {F}_{{\rm{L}}2}}{\partial {\boldsymbol{a}}} & = & \left({F}_{\mathrm{peak},{\rm{L}}2}{Q}_{{\rm{L}}2}^{3}\displaystyle \frac{{\tau }_{\mathrm{eff},{\rm{L}}2}}{{t}_{\mathrm{eff},{\rm{L}}2}},\right.\\ & & \left.{F}_{\mathrm{peak},{\rm{L}}2}{Q}_{{\rm{L}}2}^{3}\displaystyle \frac{{\tau }_{\mathrm{eff},{\rm{L}}2}^{2}}{{t}_{\mathrm{eff},{\rm{L}}2}},{Q}_{{\rm{L}}2},0,0,0\right);\end{array}\end{eqnarray} \tag{ A.3 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {F}_{\mathrm{Leo}}}{\partial {\boldsymbol{a}}} & \to & \left(0,0,0,{F}_{\mathrm{peak},\mathrm{Leo}}{Q}_{\mathrm{Leo}}^{3}\displaystyle \frac{{\tau }_{\mathrm{eff},\mathrm{Leo}}}{{t}_{\mathrm{eff},\mathrm{Leo}}},\right.\\ & & \left.{F}_{\mathrm{peak},\mathrm{Leo}}{Q}_{\mathrm{Leo}}^{3}\displaystyle \frac{{\tau }_{\mathrm{eff},\mathrm{Leo}}^{2}}{{t}_{\mathrm{eff},\mathrm{Leo}}},{Q}_{\mathrm{Leo}}\right).\end{array}\end{eqnarray} \tag{ A.4 }$$

The Fisher matrix in Equation (12) can then be written as

$$\begin{eqnarray}{ \mathcal F }=\left[\begin{array}{cc}{{ \mathcal F }}_{{\rm{L}}2} & {\bf{0}}\\ {\bf{0}} & {{ \mathcal F }}_{\mathrm{Leo}}\end{array}\right],\end{eqnarray} \tag{ A.5 }$$

where ${{ \mathcal F }}_{{\rm{L}}2}$ and ${{ \mathcal F }}_{\mathrm{Leo}}$ are both 3 × 3 matrices.

Penny et al. (2019) performed numerical simulations for the Roman microlensing campaign and found that most of the Roman microlensing events, especially those with planetary signals, should have H ≲ 23, for which the photometric noises are dominated by the source flux (roughly 16–23 mag; see the dashed curve in Figure 1). We can therefore assume that the magnitude uncertainty σ_m ∝ F^−1/2. The Fisher matrix given in Equation (A.5) is then integrable, with

$$\begin{eqnarray}{{ \mathcal F }}_{{\rm{L}}2}={{\rm{\Gamma }}}_{{\rm{L}}2}\left[\begin{array}{ccc}\displaystyle \frac{{C}_{1}{F}_{\mathrm{peak},{\rm{L}}2}}{{t}_{\mathrm{eff},{\rm{L}}2}} & 0 & 0\\ 0 & \displaystyle \frac{{C}_{2}{F}_{\mathrm{peak},{\rm{L}}2}}{{t}_{\mathrm{eff},{\rm{L}}2}} & {C}_{3}\\ 0 & {C}_{3} & \displaystyle \frac{{C}_{4}{t}_{\mathrm{eff},{\rm{L}}2}}{{F}_{\mathrm{peak},{\rm{L}}2}}\end{array}\right]\end{eqnarray} \tag{ A.6 }$$

and

$$\begin{eqnarray}{{ \mathcal F }}_{\mathrm{Leo}}={{\rm{\Gamma }}}_{\mathrm{Leo}}\left[\begin{array}{ccc}\displaystyle \frac{{C}_{5}{F}_{\mathrm{peak},\mathrm{Leo}}}{{t}_{\mathrm{eff},\mathrm{Leo}}} & 0 & 0\\ 0 & \displaystyle \frac{{C}_{6}{F}_{\mathrm{peak},\mathrm{Leo}}}{{t}_{\mathrm{eff},\mathrm{Leo}}} & {C}_{7}\\ 0 & {C}_{7} & \displaystyle \frac{{C}_{8}{t}_{\mathrm{eff},\mathrm{Leo}}}{{F}_{\mathrm{peak},\mathrm{Leo}}}\end{array}\right].\end{eqnarray} \tag{ A.7 }$$

Here Γ_L2 and Γ_LEO are the observation cadences at L2 and LEO, respectively, and C_k (k = 1, 2, 3, 4, 5, 6, 7, 8) are constants that do not depend on the lensing parameters. Then the covariance matrix ${{ \mathcal C }}_{{ij}}$ can be given by

$$\begin{eqnarray}{ \mathcal C }=\left[\begin{array}{cc}{{ \mathcal F }}_{{\rm{L}}2}^{-1} & 0\\ 0 & {{ \mathcal F }}_{\mathrm{Leo}}^{-1}\end{array}\right],\end{eqnarray} \tag{ A.8 }$$

with

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal F }}_{{\rm{L}}2}^{-1}=\displaystyle \frac{1}{{{\rm{\Gamma }}}_{{\rm{L}}2}}\\ \times \ \left[\begin{array}{ccc}\displaystyle \frac{{t}_{\mathrm{eff},{\rm{L}}2}}{{C}_{1}{F}_{\mathrm{peak},{\rm{L}}2}} & 0 & 0\\ 0 & \displaystyle \frac{{C}_{4}{t}_{\mathrm{eff},{\rm{L}}2}}{({C}_{2}{C}_{4}-{C}_{3}^{2}){F}_{\mathrm{peak},{\rm{L}}2}} & \displaystyle \frac{{C}_{3}}{{C}_{3}^{2}-{C}_{2}{C}_{4}}\\ 0 & \displaystyle \frac{{C}_{3}}{{C}_{3}^{2}-{C}_{2}{C}_{4}} & \displaystyle \frac{{C}_{2}{F}_{\mathrm{peak},{\rm{L}}2}}{({C}_{2}{C}_{4}-{C}_{3}^{2}){t}_{\mathrm{eff},{\rm{L}}2}}\end{array}\right];\end{array}\end{eqnarray} \tag{ A.9 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal F }}_{\mathrm{Leo}}^{-1}=\displaystyle \frac{1}{{{\rm{\Gamma }}}_{\mathrm{Leo}}}\\ \times \ \left[\begin{array}{ccc}\displaystyle \frac{{t}_{\mathrm{eff},\mathrm{Leo}}}{{C}_{5}{F}_{\mathrm{peak},\mathrm{Leo}}} & 0 & 0\\ 0 & \displaystyle \frac{{C}_{8}{t}_{\mathrm{eff},\mathrm{Leo}}}{({C}_{6}{C}_{8}-{C}_{7}^{2}){F}_{\mathrm{peak},\mathrm{Leo}}} & \displaystyle \frac{{C}_{7}}{{C}_{7}^{2}-{C}_{6}{C}_{8}}\\ 0 & \displaystyle \frac{{C}_{7}}{{C}_{7}^{2}-{C}_{6}{C}_{8}} & \displaystyle \frac{{C}_{6}{F}_{\mathrm{peak},\mathrm{Leo}}}{({C}_{6}{C}_{8}-{C}_{7}^{2}){t}_{\mathrm{eff},\mathrm{Leo}}}\end{array}\right].\end{array}\end{eqnarray} \tag{ A.10 }$$

The covariance matrix of the vector Λ can be expressed as

$$\begin{eqnarray}{{\boldsymbol{\Sigma }}}_{{\boldsymbol{\Lambda }}}=\displaystyle \frac{1}{{D}_{\perp }^{2}}\left[\begin{array}{cc}{\sigma }^{2}({{\rm{\Lambda }}}_{\parallel }) & 0\\ 0 & {\sigma }^{2}({{\rm{\Lambda }}}_{\perp })\end{array}\right],\end{eqnarray} \tag{ A.11 }$$

and

$$\begin{eqnarray}\left\{\begin{array}{ll}{\sigma }^{2}({{\rm{\Lambda }}}_{\parallel })=\displaystyle \frac{1}{{C}_{1}{{\rm{\Gamma }}}_{{\rm{L}}2}}\displaystyle \frac{{t}_{\mathrm{eff},{\rm{L}}2}}{{F}_{\mathrm{peak},{\rm{L}}2}}+\displaystyle \frac{1}{{C}_{5}{{\rm{\Gamma }}}_{\mathrm{Leo}}}\displaystyle \frac{{t}_{\mathrm{eff},\mathrm{Leo}}}{{F}_{\mathrm{peak},\mathrm{Leo}}}\approx \displaystyle \frac{{t}_{{\rm{E}}}{u}_{0,\mathrm{Leo}}^{2}}{{F}_{{\rm{S}},\mathrm{Leo}}}\left(\displaystyle \frac{1}{{C}_{1}{{\rm{\Gamma }}}_{{\rm{L}}2}b}+\displaystyle \frac{1}{{C}_{5}{{\rm{\Gamma }}}_{\mathrm{Leo}}}\right) & \\ {\sigma }^{2}({{\rm{\Lambda }}}_{\perp })=\displaystyle \frac{{C}_{4}}{({C}_{2}{C}_{4}-{C}_{3}^{2}){{\rm{\Gamma }}}_{{\rm{L}}2}}\displaystyle \frac{{t}_{\mathrm{eff},{\rm{L}}2}}{{F}_{\mathrm{peak},{\rm{L}}2}}+\displaystyle \frac{{C}_{8}}{({C}_{6}{C}_{8}-{C}_{7}^{2}){{\rm{\Gamma }}}_{\mathrm{Leo}}}\displaystyle \frac{{t}_{\mathrm{eff},\mathrm{Leo}}}{{F}_{\mathrm{peak},\mathrm{Leo}}}\approx \displaystyle \frac{{t}_{{\rm{E}}}{u}_{0,\mathrm{Leo}}^{2}}{{F}_{{\rm{S}},\mathrm{Leo}}}\left(\displaystyle \frac{{C}_{4}}{({C}_{2}{C}_{4}-{C}_{3}^{2}){{\rm{\Gamma }}}_{{\rm{L}}2}b}+\displaystyle \frac{{C}_{8}}{{C}_{6}{C}_{8}-{C}_{7}^{2}){{\rm{\Gamma }}}_{\mathrm{Leo}}}\right) & \end{array}\right..\end{eqnarray} \tag{ A.12 }$$

Here we have assumed that u_0,L2 ≈ u_0,LEO. With the definition of χ² given by Equation (13), we thus have

$$\begin{eqnarray}&&{\chi }_{{\boldsymbol{\Lambda }}}^{2}=\displaystyle \frac{{{\rm{\Lambda }}}_{\parallel }^{2}}{{\sigma }^{2}({{\rm{\Lambda }}}_{\parallel })}+\displaystyle \frac{{{\rm{\Lambda }}}_{\perp }^{2}}{{\sigma }^{2}({{\rm{\Lambda }}}_{\perp })}\propto \displaystyle \frac{{D}_{\perp }^{2}{F}_{{\rm{S}},\mathrm{Leo}}}{{t}_{{\rm{E}}}{u}_{0,\mathrm{Leo}}^{2}}.\end{eqnarray} \tag{ A.13 }$$