Weighing the Invisible: Gravitational Lensing Calculations Quiz

If General Relativity predicts that light is deflected by a massive object, and if that deflection is twice what Newtonian gravity would predict, then the correct formula for the angle in radians is alpha = 4GM / rc², where G is the gravitational constant and c is the speed of light.

If the deflection formula alpha = 4GM / rc² places the impact parameter 'r' in the denominator, then as the light passes further from the mass (larger 'r'), the angle 'alpha' must decrease; therefore, they are inversely proportional.

If the equation alpha = 4GM / rc² is derived from General Relativity, and if relativity links gravity to the spacetime metric and the speed of light, then 'c' must represent the universal constant 299,792,458 m/s.

If a lens, source, and observer are perfectly aligned, then the light forms a symmetrical ring; if this theoretical ring is a hallmark of Einstein's theory, then it is named the Einstein radius.

If the formula for the Einstein angle is theta_E = [ (4GM/c²) * (D_ls / (D_l * D_s)) ]^1/2, then the calculation depends on the mass, the various geometric distances, and the speed of light; if temperature does not affect gravitational pull, it is excluded.

If the equation for theta_E involves the square root of mass (M^1/2), then to solve for M, we must square both sides of the equation; if we square both sides, then the mass M becomes directly proportional to the square of the Einstein angle (theta_E²).

If the standard gravity and light equations provide results in radians, and if 1 radian equals 180/pi degrees, and if 1 degree contains 3600 arcseconds, then the conversion factor is required to reach the units used in observational astronomy.

If D_l is the observer-lens distance and D_s is the observer-source distance, then the geometric gap between the lens and source is conventionally labeled D_ls.

If the Einstein angle theta_E is proportional to the square root of the mass (sqrt(M)), and if the mass is replaced by 4M, then sqrt(4M) is equal to 2 * sqrt(M); therefore, the radius doubles.

If the universal gravitational constant G is measured in m³ / (kg * s²), then to ensure units cancel correctly in a physics formula, mass must be in kilograms and distances must be in meters.

If the deflection angle alpha is equal to 4GM / rc², and if the impact parameter 'r' is halved (0.5r), then dividing by 0.5 is mathematically equivalent to multiplying by 2; therefore, the deflection angle doubles.

If gravitational lensing measures the total mass (visible and invisible) required to bend light by a specific angle, and if that total mass is much greater than the mass of visible stars, then the difference represents the dark matter content.

If the lens is a low-mass object like a star or planet rather than a galaxy, then the angular separation of images is too small to resolve; if we can only see the magnification of the source light, the process is microlensing.

If the deflection angle is 4GM / rc², and if the Schwarzschild radius (R_s) is defined as 2GM / c², then the deflection can be written as 2 * R_s / r; therefore, the R_s term is exactly half of the numerator factor.

If the alignment and mass density are sufficient to produce non-linear effects, then the result is strong lensing; if strong lensing occurs, we observe distinct geometric features like rings, arcs, and multiple images of the same source.

If the universe is expanding and has a specific geometry, then the relationship between physical size and angular size changes over great distances; if we are calculating angles across billions of light-years, then angular diameter distances must be used.

If gravity is a property of mass regardless of its physical state, and if gas, dark matter, and plasma all possess mass, then any concentration of these materials can act as a gravitational lens.

If galaxies have a mass distribution where density drops as 1/r², then the simplest model used to calculate their lensing properties is the Singular Isothermal Sphere.

If D_s is very large, then the distance between the lens and source (D_ls = D_s - D_l) is approximately equal to D_s; if D_ls is approximately D_s, then the fraction D_ls / D_s simplifies to 1.

If extra mass exists in the foreground or background, or if the lens is elliptical rather than circular, the light path changes; if redshifts are inaccurate, the calculated distances and mass will be incorrect.