If a luminous quasar has a luminosity of \(2 \times 10^{41} \mathrm{W}\), or \(\mathrm{J} / \mathrm{s}\), how many solar masses \(\left(M_{\circ}=2 \times 10^{30} \mathrm{kg}\right)\) per year does this quasar consume to maintain its average energy output?

Mass-energy equivalence is a concept from Einstein's theory of relativity. It is summed up by the famous equation: \[ E = mc^2 \]
In this equation, \(E\) stands for energy, \(m\) stands for mass, and \(c\) is the speed of light in a vacuum \(3 \times 10^8 \text{ m/s}\).
This means that mass can be converted into an equivalent amount of energy and vice versa.
When applied, this equation explains how a quasar, which emits an immense amount of energy, also implicitly involves vast amounts of mass being converted into energy.

Luminosity is the total amount of energy a celestial object emits per unit time. It is given in watts ( \(W\) ), where \( 1 \text{ W} = 1 \text{ J/s}\). For a quasar, the luminosity can be extremely high, like in our example, where it's \(2 \times 10^{41} \text{ W}\).
To find the energy output per year, you multiply the luminosity by the number of seconds in a year. Since there are 31,536,000 seconds in a year:\[ E_{\text{year}} = L \times \text{Seconds in a year} = 2 \times 10^{41} \text{ W} \times 31,536,000 \text{ s/year} = 6.3072 \times 10^{48} \text{ J/year} \]This calculation tells us how much energy the quasar outputs over an entire year.

Converting between kilograms and solar masses is essential in astrophysics, as celestial object's masses are often given in solar masses for simplicity. One solar mass (\(M_{\text{sun}}\)) is approximately equal to \(2 \times 10^{30} \text{ kg}\).
For instance, if you find that your quasar consumes \(7.008 \times 10^{31} \text{ kg/year}\), you would convert this to solar masses as follows:\[ m_{\text{solar}} = \frac{7.008 \times 10^{31} \text{ kg/year}}{2 \times 10^{30} \text{ kg/solar mass}} = 35.04 \text{ solar masses/year} \]This conversion helps simplify understanding of the mass involved, as 35.04 solar masses per year is much easier to grasp than a long number in kilograms.

A quasar is a highly luminous active galactic nucleus, powered by a supermassive black hole at its center. The luminosity of a quasar is due to the vast amounts of energy being emitted from its accretion disk, which forms from matter spiraling into the black hole.
Because quasars emit significant energy, studying their luminosity helps researchers understand the processes and dynamics within these massive objects. For example, in the given problem, the quasar's luminosity is \(2 \times 10^{41}\) watts, an exceedingly high value. By calculating energy output and relating it to mass consumption through mass-energy equivalence, we gain insight into the immense scale and power of quasars in absorbing and converting matter into energy.