What Does Absolute Magnitude in the B Filter of a Distant Galaxy Refer To?

[RingNebula57]

When referring to the absolute magnitude ,for example, in the B filter of a distant galaxy(considered point source) what are we talking about? Are we talking about the observed redshifted flux density(luminosity density) hypothetically seen from 10 pc away (1) ; or are we talking about the actual intinsic flux density hypothetically seen form 10 pc away (2)?
I encountered an exercises in which it is written that (1) is the case but I don't undesrand why. Is it because the blue (B) in (1) is diffrent from the B in (2) because of the redshift?

 

[Chalnoth]

The answer is (2). The absolute magnitude is independent of the redshift of the object. The apparent magnitude is affected by redshift.

 

[RingNebula57]

Is it okay if I show you the problem (with solution) because I don't want to have any confusion? Shall I post it here or on homework forum?

 

[PeterDonis]

If you already have a solution, there is no point in posting in the homework forum. If you have attempted a solution but you're not sure if it's correct, then you can post the problem and your attempt at the solution in the homework forum.

 

[Chalnoth]

If you already have a solution, there is no point in posting in the homework forum. If you have attempted a solution but you're not sure if it's correct, then you can post the problem and your attempt at the solution in the homework forum.

Yeah, if you don't need help with the solution, but just want help understanding what's going on, this is a good place to ask.

 

[RingNebula57]

Ok , so the problem states:
" Within the field of a galaxy cluster at a redshift of z = 0.500, a galaxy
which looks like a normal elliptical is observed, with an apparent magnitude
in the B filter mB = 20.40 mag. The luminosity distance corresponding
to a redshift of z = 0.500 is dL = 2754 Mpc. The spectral
energy distribution (SED) of elliptical galaxies in the wavelength range
250 nm to 500 nm is adequately approximated by the formula:
Lλ(λ) ∝ λ^4
(i.e., the spectral density of the object’s luminosity, known also as the
monochromatic luminosity, is proportional to λ^4.)
(a) What is the absolute magnitude of this galaxy in the B filter ?
(b) Can it be a member of this cluster?
Hint: Try to establish a relation that describe the dependence of the
spectral density of flux on distance for small wavelength interval. Normal
elliptical galaxies have maximum absolute magnitude equal to -
22 mag. For blue band, effective midpoint wavelength is 445nm and
FWHM is 94nm. "

And the solution is:

"
λemitted =λobserved/(1+z)
Let monochromatic flux of the galaxy be denoted by S(λ), then for small wavelength interval Δλ,

In case you can't see the subscripts for λ in formula 19.97 , they are "ob" and "emitted".
The reason for the question is the third line of the pictured solution. According to the solution λ=λobserved, so when the 10 pc monochromatic flux is calculated they use the observed spectral energy distribution Lλ(λ). In my opinion the first line is correct because that is the definition of the luminosity distance, but on the second line I think it should've been Lλ(λobserved/(1+z)).
Am I wrong somehow?

 

[RingNebula57]

By the way, is their solution ok?