total energy and apparent bolometric magnitude

Question

So I have been given this question:

A typical Type Ia Supernova emits $10^{44}\ \mathrm{J}$. All nuclear weapons tested by humans have released a total of about $2\times10^{6}\ \mathrm{TJ}$ of energy. Assume the Supernova energy is released at the distance of the Sun. Determine how close to your retina the nuclear weapons energy would have to be released to make both appear to have the same apparent bolometric magnitude.

The issue I'm have is normally apparent magnitudes I would have to have joules per second where this is in just total joules.

So is it that with bolometric magnitude there is an automatic assumption that when total power is mention you assume joules a second?

Could someone please advise?

I would just like to point out I am not looking for a direct answer to this question, I am just confused with why total energy has been used. I have read posts on wiki which seems to confuse me even more. I have just used the above question as an example from a book.

Answer 1

You have been given a terrible question and your confusion is totally justified.

The magnitude system is a measure of the brightness of objects which is a measure of the energy received per unit time interval.

Without knowing a characteristic timescale over which the energy is released in each case, then it is not answerable. Note that the timescale for energy release in the case of supernovae and bombs are different by many orders of magnitude - a type Ia supernova has a bolometric light curve (e.g. see Fig.6 of Scalzo et al. 2014) that lasts 2-3 weeks, peaking at around $10^{36}$ Watts.

Given this, we can compare with the Sun's bolometric luminosity of $3.8\times 10^{26}$ Watts and apparent bolometric magnitude of -26.8, to estimate that the peak apparent bolometric magnitude of a type Ia supernova at the distance of the Sun would be -50.3.

You cannot do the same thing with nuclear explosions. They last a fraction of a second (in terms of their light output). Or are you meant to assume that you take the average flux over the 70 years that mankind has had nuclear weapons?? The question is a nonsense unless you are given some sort of timescale over which this explosive energy is released. About the only sane things you can do are (a) use ~70 years (actually, most of the energy was released during the 1950s and early 1960s) (b) use the duration of a single nuclear explosion or (c) (and I imagine what was intended) use a similar duration as for a typical type Ia supernovae so that the question should have begun - "Imagine that X and Y amounts of total energy are emitted over a similar timescale..."

Answer 2

If you take a look at how the magnitude system is derived, it usually starts with an equation similar to the following

$$100^{(m_1 - m_2)/5} = \frac{F_2}{F_1}$$

where $m$ is the apparent magnitude and $F$ is the flux. The subscripts denote two different objects. This equation derives directly from the fact that, by definition, a magnitude 1 star is 100 times brighter than a magnitude 6 star.

Now if you look at the equation above, you'll see the right hand side is a ratio of fluxes, and necessarily it is unitless. What this actually means is that you can derive the magnitude system with any measure of "brightness" be it a flux which has units of $\mathrm{J\ s^{-1}\ m^{-2}}$, a luminosity which has units of $\mathrm{J\ s^{-1}}$ or power which also has units of $\mathrm{J\ s^{-1}}$. The only constraint is that you have to consider the same "type of brightness" (for want of a better term) for both sources so your ratio remains unitless. That is, you can't use flux for one star and power for another. They both have to be flux or they both have to be power. In your case, you're given the total energy output, which you can assume happens over the same time scale since they're both just explosions.

To be honest though, your problem doesn't really rely on that knowledge. In fact, your problem becomes exceedingly simple because it says to assume that $m_1=m_2$ which means the ratio of the flux from both sources is unity by the equation above. You just have to say $F_1 = F_2$, relate flux to luminosity and distance, and solve for your unknown.