A quasar has the same brightness as a galaxy that is seen in the foreground 2 Mpc distant. If the quasar is 1 million times more luminous than the galaxy, what is the distance of the quasar?

To understand how we determine the distance of a quasar using its brightness and luminosity, let's first define these two terms. Brightness is how much light from a celestial object reaches us on Earth. It's the perceived intensity of the light. Luminosity, on the other hand, is the total amount of energy emitted by the object per second in all directions.

Imagine holding a flashlight. The brightness you see depends on how far you are from it. The farther away, the dimmer it appears. Luminosity would be like knowing the bulb strength in the flashlight. A 100-watt bulb will always be a 100-watt bulb regardless of where you are in the room.

In astronomy, the relationship between brightness (\text{B}) and luminosity (\text{L}) is given by: \[ B \rightarrow \frac{L}{D^2} \] Here, \text{D} is the distance to the object. This equation tells us that brightness decreases as the square of the distance from the source increases. So, if you double the distance, the brightness decreases by a factor of four.

The inverse square law is a crucial concept in the calculation of astronomical distances. It states that a physical quantity (in this case, brightness) is inversely proportional to the square of the distance from the source.

Consider a light bulb: if you move twice as far away, the area over which its light spreads increases by four times, making it look four times dimmer. This is mathematically expressed as:

\[ B \rightarrow \frac{L}{D^2} \]

This principle applies to all sources of light in the universe, including quasars and galaxies. When comparing the brightness and luminosity of different astronomical objects, this law helps you determine how the observed brightness changes with distance.

Let's tie this back to our quasar and the foreground galaxy. The quasar is way more luminous (one million times more, in fact), but if both objects appear equally bright to us, it means the quasar must be much farther away. The inverse square law allows us to equate the brightness and adjust the luminosity to solve for the quasar's distance.

Determining the distance of astronomical objects like quasars is essential for understanding the scale of the universe. Various methods are used, but using brightness and luminosity relationship is one of the fundamental techniques.

In our exercise, we have a quasar and a galaxy with these key details:

• The galaxy is 2 Megaparsecs (Mpc) away.
• The quasar is one million times more luminous than the galaxy.
• Their apparent brightness to us is the same.

Using the inverse square law, we know:

• \( B_\text{gal} = B_\text{quasar} \)
• \( \frac{L_\text{gal}}{(2 \text{ Mpc})^2} = \frac{L_\text{quasar}}{D_\text{quasar}^2} \)

By substituting \( L_\text{quasar} = 10^6 \times L_\text{gal} \) in the above equation, we can then solve for \( D_\text{quasar} \). Simplifying, we find:

\[ D_\text{quasar}^2 = 4 \times 10^6 \text{ Mpc}^2 \]

Taking the square root to find \( D_\text{quasar} \), we get:

\[ D_\text{quasar} = 2,000 \text{ Mpc} \]

Thus, the quasar is 2,000 Megaparsecs away, illustrating how astronomers use these concepts to measure cosmic distances.