A quasar has a luminosity of \(10^{41}\) watts \((\mathrm{W}),\) or \(\mathrm{J} / \mathrm{s}\), and \(10^{8} M_{\circ}\) to feed it. Assuming constant luminosity and 20 percent conversion efficiency, what is your estimate of the quasar's lifetime?

The concept of mass-energy equivalence is one of Einstein's most famous theories, given by the equation \[ E = mc^2 \] where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) m/s.
This equation tells us that mass can be converted into a huge amount of energy.
In the context of our quasar, we can use this equation to convert the quasar's mass into energy.
The available mass is \( 10^8 M_\odot \) and each solar mass \( M_\odot \) is about \( 1.989 \times 10^{30} \) kg.
This tells us how much energy is available to power the quasar's luminosity.
The energy from mass is calculated as: \[ E = (10^8 \times 1.989 \times 10^{30} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2 = 1.789 \times 10^{54} \text { joules} \]

Luminosity is a measure of how much energy a celestial object like a quasar emits per second.
It's expressed in watts (W), which is equivalent to joules per second \((\text{J/s})\).
In simpler terms, it's the brightness of the object.
Our quasar has a luminosity of \( 10^{41} \) watts. This indicates how quickly the quasar is emitting energy. The challenge is to sustain this rate of energy emission over time.
To find the quasar's lifetime, we will divide the total available energy by this luminosity.
This will tell us how long the energy can support the quasar's brightness.

Not all the mass-energy of the quasar is actually converted into useful energy.
This is because energy conversion processes are not 100% efficient.
In our example, only 20% of the total mass-energy is converted into usable energy.
This efficiency is given as a percentage and indicates how much of the initial input gets turned into useful output. According to the given conversion efficiency of 20%, we calculate: \[ E_{\text{usable}} = 0.2 \times 1.789 \times 10^{54} \text{ J} = 3.578 \times 10^{53} \text{ J} \] Only this fraction of energy will be used to power the quasar.

How long will the quasar shine?
We need to find its lifetime.
This is done by taking the total usable energy and dividing it by the luminosity: \[ t = \frac{3.578 \times 10^{53} \text{ J}}{10^{41} \text{ W}} = 3.578 \times 10^{12} \text{ seconds} \]
But we generally prefer a more relatable unit like years. There are approximately 31,536,000 seconds in a year, so we convert the quasar's lifetime to years: \[ t_{\text{years}} = \frac{3.578 \times 10^{12} \text{ s}}{31,536,000 \text{ s/year}} = 113.4 \times 10^{6} \] Therefore, the quasar will shine for roughly 113.4 million years.