Wouldn't the 1054 supernova have temporarily enlightened the Earth like the Sun?

Question

The peak luminosity of all type Ia supernovae is 1.60×10³⁶ Watts. If we see a star go supernova from earth, and we observe its intensity to be 1.0 W/m2. The intensity of the Sun is about 6.33x10⁷ W/m2.

Now, I have read that a supernova explosion intensity can exceed that of the whole Milky Way (or even 50 times!. The 1054 supernova, of which the Crab Nebula is a remnant (see pictures below), was seen to be about four times the intensity of Venus. But shouldn't it at 6700 lightyears distance, have looked brighter than the Sun for a moment?

The Crab Nebula supernova remnant:

The neutron star in the nebula, spinning about 30 times per second!

Answer 1

The peak absolute magnitude of a Type Ia supernova is -19.5. The relation between absolute and apparent magnitude for nearby objects ($z \approx 0)$ is $$m = M + 5\,(\log_{10}(d) - 1)$$ where $d$ is the distance to the object in parsecs, $M$ is the absolute magnitude, and $m$ is the apparent magnitude. The -1 on the right hand side is because absolute magnitude is defined as being magnitude at a distance of 10 parsecs. The factor of 5 and the log base 10, that's also a part of the definition. The definition is more or less consistent with how human eyes work.

Difference sources provide different distances to the Crab Nebula, ranging from 6500 light years to 6700 light years, or 1993 to 2053 parsecs. I'll use 2000 parsecs as a nice round number. Doing the math, $-19.5 + 5\,(\log_{10}(2000) - 1) \approx -8$. So even a Type Ia supernova at 2000 parsecs distance would not appear anywhere as close to bright as would a full moon. It would however appear to be brighter than Venus at its brightest (maximum apparent magnitude = -4.6).

The Crab Nebula was not a Type Ia supernova. It was instead a Type II supernova, with various sources stating its peak absolute luminosity magnitude being -18 rather than -19.5. That still means that SN 1054 was brighter than is Venus at its brightest.