Chandrasekhar Mass Limit & White-Dwarf Radius in a Spreadsheet

Use a consistent mass–radius approximation

For an idealized cold, electron-degenerate white dwarf, use M_Ch/M_sun = 5.816/mu_e^2 and R/R_sun = (0.0225/mu_e)×SQRT((M_Ch/M)^(2/3)−(M/M_Ch)^(2/3)). These paired coefficients follow the Nauenberg expression reproduced by Smith and Vande Putte (2006), equation 3.

mu_e is the mean mass per free electron in atomic-mass units; for fully ionized carbon-12 or oxygen-16, A/Z=2. This convention gives M_Ch=1.454 solar masses when mu_e=2. Do not silently replace it with 1.44 while keeping other coefficients fixed.

This is an idealized zero-temperature approximation, not a detailed evolutionary model. Its radius tends to zero as M approaches M_Ch from below; that mathematical limit is not a physical zero-radius star or a universal Type Ia explosion criterion. The formula does not decide a real star’s fate.

Set up the inputs and derived limiting mass

B2 holds stellar mass in solar masses, initially 0.6. B4 holds mu_e, initially 2. B3 is a formula derived from B4, not a separate editable limiting mass. This keeps a composition change consistent in both calculations.

B5 and B6 are the adopted conversion radii in kilometres: 695700 for the Sun and 6371 for Earth. Their ratio converts solar radii into Earth radii; changing these conversion constants changes the reported units, not the dimensionless stellar model.

Place the following formula in B3. Keep its unrounded value for the radius calculation; round only the displayed result.

=IF(NOT(ISNUMBER(B4)),"Enter numeric mu_e",IF(B4<=0,"mu_e must be positive",5.816/B4^2))

Validate the domain before evaluating the square root

C2 contains the audit below. Mass, mu_e and conversion radii must be numeric and positive. The model requires 0<M<M_Ch. At or above the limiting mass, the result is a model-domain message rather than an explosion or collapse classification.

The mathematical requirement mu_e>0 does not make every positive value a realistic stellar composition. Choose mu_e from the fully ionized composition being modelled; this example does not infer composition from observed data.

=IF(COUNT(B2,B4,B5,B6)<>4,"Enter numeric mass, mu_e and radii",IF(MIN(B2,B4,B5,B6)<=0,"Inputs must be positive",IF(B2>=B3,"Outside model: M must be below M_Ch","OK")))

Compute the radius in solar units

C3 contains the following dimensionless radius. Both mass ratios are dimensionless because B2 and B3 are in the same solar-mass units.

The old formula used a fixed Earth-radius coefficient times M^(-1/3); it did not preserve the normalization of this mass-ratio expression. With the stated constants, the corrected 0.6-solar-mass example is approximately 1.3734 Earth radii. At 1.2 solar masses it is approximately 0.6224 Earth radii.

=IF(C2<>"OK",C2,(0.0225/B4)*SQRT((B3/B2)^(2/3)-(B2/B3)^(2/3)))

Convert to Earth radii and kilometres

C4 contains the Earth-radius result below. C5 contains =IF(C2<>"OK",C2,C3*B5), the radius in kilometres. These are radii, not diameters.

To display four decimals, use =IF(ISNUMBER(C4),ROUND(C4,4),C4) in a separate cell or apply a number format. The practice outputs keep full precision.

Try increasing B2 from 0.6 to 1.2, then to the derived B3 value. The radius shrinks before the domain guard takes over. Changing B4 updates B3 and therefore changes the valid domain automatically.

=IF(C2<>"OK",C2,C3*B5/B6)

Further reading

• Smith & Vande Putte (2006), Testing the criteria for stable mass transfer in cataclysmic variables — equation 3
• Even & Tohline (2009), Constructing Synchronously Rotating Double White Dwarf Binaries — zero-temperature model context