Math Exploration 12.1

Compute the Einstein radius for a solar mass object in the outskirts of our Galaxy. We assume that the object is a point mass, thus all of its mass lies within the impact parameter, b. Assume some typical values for distance: given the dimensions of our Galaxy, assume that the most likely distance to a gravitational lens will be about 10 kpc. For simplicity we will also assume that the distance between the lens and source is also 10 kpc.

• Given: Mass M = 2e30 kg, D_LS = D_LO = 10 kpc
• Find: θ_E, the Einstein radius in radians and arcseconds
• Concept(s):
  \begin{equation} \theta_E=\sqrt{\left(\frac{4GM(b)}{c^2}\right)\left(\frac{D_{LS}}{D_{LO}D_{SO}}\right)} \end{equation}
  where c = 3e8 m/s and 1 kpc = 3.09e19 m
  and D_SO = D_LS + D_LO
• Solution:
  First, get the distances in SI units (meters):
  D_LO = D_LS = 10 kpc × (3.09e19 m / 1 kpc) = 3.09e20 m
  D_SO = D_LS + D_LO = 20 kpc × (3.09e19 m / 1 kpc) = 6.18e20 m
  Now we can plug the numbers in to get θ_E:
  

  \begin{equation} \theta_E=\sqrt{\left (\frac{(4)(\rm 6.67e{-11}\ \rm{N m^2/k‌g^{2}})(\rm 2e30\ \rm{k‌g})}{(\rm 3e8\ {\rm{m/s}})^2} \right )\left(\frac{\rm (3.09e20\ \rm{m})}{(\rm 3.09e20\ \rm{m})(\rm 6.18e20\ \rm{m})} \right ) }\end{equation}

  
  = 3.1e-9 radians.
  This is a very small angle. It is better expressed in arcseconds. The conversion factor for radians to arcseconds is: 1 radian = 2.06e5 arcseconds.
  θ_E = (3.1e-9 radians)(2.06e5 arcseconds / radian) = 6.4e-4 arcseconds
  We are not currently able to measure such tiny angles using optical telescopes, either from the ground or in space. Nonetheless, this type of gravitational lensing has been used to study our Galaxy’s halo. We learn how this is done in the next section of the chapter.