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Use the supplied cosmological cosmic baryon fraction $f_b={\Omega }_{\mathrm{bar}}/{\Omega }_{\mathrm{mat}}=0.2$, and assume the halo initially has that fraction. Its total baryon mass is then $M_{b,\mathrm{tot}}=6\times 10^{11}{M}_{\odot }$. The selected baryons are the lowest-angular-momentum fractionHere the given cumulative specific angular momentum distribution must be normalized separately for the baryon component: equal baryon and dark-matter distributions mean equal normalized fractions, not equal absolute masses.

The virial velocity and the largest specific angular momentum in the selected inner baryon population areEstimate the outer radius of the settled low-angular-momentum component using circular rotational support and an enclosed-mass approximation to its self-gravity. With negligible dark matter inside the component, $v_d^2\simeq G{M}_b/r_d$ and $j_d=r_d{v}_d$, soThe given rounded gravitational constant produces the same estimate. The very small radius follows from selecting a small low-$j$ fraction, rather than assigning all central baryons the halo-edge angular momentum.

These estimates assume that gas radiates energy, preserves each parcel's specific angular momentum, and settles with negligible pressure support and no strong redistribution or cancellation of its angular-momentum vectors. They also assume that the phrase “innermost baryons” selects the lowest-$j$ material, the original cosmic baryon supply is retained, and the central gas supplies the dominant gravity. A possible central black hole or an exact flattened disk potential changes the numerical coefficient; the stated baryonic mass is used for this estimate.

The low-angular-momentum baryonic disk estimate uses a cumulative distribution uniform in $j$ from zero to $j_d$, so its mean is $j_d/2$. The quoted $r_d$ is the outer radius based on the cutoff angular momentum, not a one-zone radius based on the mean. Within the same enclosed-mass approximation, $M(<j)=M_{b}j/j_d$ and circular balance give $r(j)=j{j}_d/(G{M}_b)$: the rotation curve is approximately flat and the half-mass radius is $r_d/2$. Treating every baryon as one shell with the mean $j$ would instead give $r_d/4$ and twice the velocity, a different radius convention rather than the outer edge of the supplied distribution.