## 2 Microlensing Parallax  
In gravitational lensing, the surface brightness, which is the flux per area, is conserved. The total flux increases or decreases, since the area increases or decreases. In microlensing, distinct images, due to the gravitational effects of the lensing system, are not seen, but, instead, magnification or demagnification of the source star is observed; the images are not resolved. Since the Jacobian matrix gives the amount of change in the source star flux in each direction, the transformation of the original source to the stretched source can be mapped by the Jacobian. The absolute value of the inverse of determinant gives the amount of magnification. [PERSON]'s ring forms when there is an exact alignment of the source, lens and observer and is an important parameter for the basis of gravitational microlensing equations. [PERSON]'s ring radius, \(\theta_{E}\), can be defined by Equation (2). \[\theta_{E}=\sqrt{\frac{4 GMD_{LS}}{c^{2}D_{L}D_{S}}} \tag{2}\]  
where \(M\) is the total mass of the lensing system, \(D_{LS}\) is the distance from the lens to the source, \(D_{L}\) is the distance from the observer to the lensing system, and \(D_{S}\) is the distance from the observer to the source [5]. From the formalization from [6], rewriting this in terms of relative lens-source parallax, \(\pi_{rel}\), where \(\pi_{rel}=\frac{AUD_{LS}}{D_{s}D_{L}}\), we obtain  
\[\theta_{E}=\sqrt{kM\pi_{rel}} \tag{3}\]Here \(k=\dfrac{4G}{c^{2}AU}\) and AU is 1 Astronomical unit or \(1.5\times 10^{8}\) km. If we define microlensing parallax in terms of the relative lens-source parallax, we obtain \(\pi_{E}=\dfrac{\pi_{rel}}{\theta_{E}}\)[7, 8],  
\[M =\dfrac{\theta_{E}^{2}}{k\pi_{rel}} \tag{4}\] \[=\dfrac{\theta_{E}}{k\pi_{E}} \tag{5}\]  
The amplification of a single lensed microlensing event light curve with time dependency is given by Equation (6) [5]. \[A(t)=\dfrac{u_{0}^{2}+\left(\dfrac{t-t_{0}}{t_{E}}\right)^{2}+2}{\left[u_{0}^{ 2}+\left(\dfrac{t-t_{0}}{t_{E}}^{2}\right)\right]^{1/2}\left[u_{0}^{2}+\left( \dfrac{t-t_{0}}{t_{E}}\right)^{2}+4\right)\right]^{1/2}} \tag{6}\]  
The flux at each time sample, \(t\), is given by Equation (7) [5]. \[F(t)=F_{s}A(t)+F_{b} \tag{7}\]  
Thus, from a photometric curve, for a single microlensing event, we can obtain the parameters: \(t_{0}\), \(t_{E}\), \(u_{0}\), \(F_{s}\) and \(F_{b}\) from a microlensing photometric curve. All of the parameters are defined in Table 1.  
In Table 1, \(\mu_{rel}\) is the relative lens-source proper motion. Here, obtaining the lens mass remains unresolved, as we have two unknown parameters: \(\theta_{E}\) and \(\pi_{rel}\). In order to break the degeneracy to obtain specific microlensing parameters, measuring the parallax, \(\pi_{E}\), offers once such solution [9, 10]. If we obtain \(u_{0}\), we can solve for \(M\), given \(\pi_{E}\). According to [8], microlensing parallax can be measured in three ways:  
* Motion of the Earth around the sun causing an annual parallax;
* Two or more space based observatories, separated by a significant baseline;
* Terrestrial parallax measured using a ground and space based observatory. In our work, we focus on a constellation of a space based observatory to create simultaneous parallax measurements. We can define the parallel and perpendicular shifts due to a microlensing parallax as in [6, 8]. Let us assume **o** as the vector for the motion of the observatory. In a telescope constellation, from [8], **o** = \(<o_{1},o_{2}>\), where:  
\[o_{1}=\epsilon_{\parallel}cos\Omega \tag{8}\]  
and  
\[o_{2}=\epsilon_{\parallel}sin\Lambda sin\Omega \tag{9}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \(t_{0}\) & Time of peak magnification \\ \hline \(t_{E}\) & [PERSON] ring crossing time: \(\dfrac{\theta_{E}}{\mu_{rel}}\) \\ \hline \(\mu_{0}\) & Impact parameter in units of \(\theta_{E}\) \\ \hline \(F_{s}\) & Microlensing source star flux \\ \hline \(F_{b}\) & Microlensing source star blended flux \\ \hline \end{tabular}
\end{table}