Engineering & Science
Astrophysical Virial Theorem: Galaxy Cluster Dynamical Mass & M/L Ratio
Calculate total gravitating dynamical mass M_vir = (5 * R_eff * sigma_r^2) / G (in units of 10^14 solar masses) and dynamical Mass-to-Light ratio M / L from line-of-sight velocity dispersion sigma_r.
Recreate Fritz Zwicky's 1933 Coma Cluster dark-matter discovery!
How did Fritz Zwicky (1933) discover that the Coma galaxy cluster contains ~85% invisible dark matter? By applying the Virial Theorem 2*K + U = 0! For an isotropic, gravitationally bound spherical system of effective half-mass/virial radius R_eff (in Megaparsecs, 1 Mpc = 3.0856776e22 m) whose member galaxies have 1D line-of-sight radial velocity dispersion sigma_r (in km/s, so 3D velocity squared <v^2> = 3 * sigma_r^2), balancing kinetic energy against gravitational binding energy U = -(3/5)*G*M^2/R_eff yields the isotropic virial mass estimator: M_vir = (5 * R_eff * (sigma_r ^ 2)) / G!
Look at the numbers for a Coma-like rich galaxy cluster with R_eff = 1.50 Mpc, radial dispersion sigma_r = 1,000 km/s, and total starlight luminosity L = 5.0 x 10^12 L_sun (0.05 x 10^14 L_sun): M_vir = 17.44 x 10^14 M_sun (~1.74 quadrillion suns!), giving a dynamical Mass-to-Light ratio M / L = 17.44 / 0.05 = 348.8 M_sun/L_sun—vastly exceeding normal stellar populations (M/L ~ 3 to 5)!
| Galaxy System (R_eff = 1.50 Mpc) | Line-of-Sight Dispersion sigma_r | Virial Mass M_vir (10^14 M_sun) | Dynamical M / L (at L = 0.05e14 L_sun) |
|---|---|---|---|
| Coma-Class Rich Cluster | 1,000 km/s | 17.44 x 10^14 M_sun | 348.8 M_sun / L_sun (Dark-matter dominated!) |
| Poorer Galaxy Group (Half sigma_r) | 500 km/s | 4.36 x 10^14 M_sun (1/4x mass!) | 87.2 M_sun / L_sun |
| Invalid Radius or Dispersion <= 0 | Domain Guard | Guarded Input Error | "Invalid Cluster Inputs" |
Isotropic Virial Dynamical Mass M_vir in Units of 10^14 Solar Masses
Evaluates (5 * R_Mpc * 3.0856776e22 * (sigma_kms * 1000)^2) / (6.6743e-11 * 1.98847e30 * 10^14).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Cluster Inputs", ROUND((5 * B2 * 3.0856776e22 * ((B3 * 1000) ^ 2)) / (6.6743e-11 * 1.98847e30 * 1e14), 2))Cluster Mass-to-Light Ratio M_vir / L_total (Solar Units M_sun / L_sun)
Divides M_vir (in 10^14 M_sun) by total optical luminosity B4 (also entered in 10^14 L_sun).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Cluster Inputs", ROUND(((5 * B2 * 3.0856776e22 * ((B3 * 1000) ^ 2)) / (6.6743e-11 * 1.98847e30 * 1e14)) / B4, 1))Try the example
Use the original values or change them to test your understanding.