How is the difference of bolometric magnitudes not dependent on the stars' radii?

The difference of 2 bolometric magnitudes is given by:

$$M_{bol, ★} - M_{bol, ☉} = -2.5 \cdot \log \left( \frac{L_★}{L_☉} \right)$$

But Pogson's equation is:

$$M_{bol, ★} - M_{bol, ☉} = -2.5 \cdot \log \left( \frac{F_★}{F_☉} \right)$$

where $$F_★=\frac{L_★}{4\pi R^2}$$, so how come the first equation isn't dependent on the radius?

• Welcome to astronomy SE! Would you mind adding a link to Pogson's equation, just as a service for those who do want to learn (more) about it, please? May 12 at 7:09
• May 12 at 7:23

Therefore, the second relation for the two fluxes is about the apparent magnitudes (which describe the brightness of an astronomical object observed from Earth), $$m-m_\odot = -2.5 \log F/F_\odot$$