Problem 57 - Fundamentals of Physics

Question

Quasars are thought to be the nuclei of active galaxies in the early stages of their formation. A typical quasar radiates energy at the rate of \(10^{41} \mathrm{~W}\). At what rate is the mass of this quasar being reduced to supply this energy? Express your answer in solar mass units per year, where one solar mass unit (1 smu \(=2.0 \times 10^{30} \mathrm{~kg}\) ) is the mass of our Sun.

Step-by-Step Solution

Answer

The quasar's mass is reduced at approximately 174.8 solar mass units per year.

Step 1: Understand the Problem

We need to find the rate at which the mass of a quasar is being converted into energy given that it radiates at a power of \(10^{41} \text{ W}\). We are tasked with expressing this rate in solar mass units per year.

Step 2: Recall the Energy-Mass Conversion Formula

Use Einstein’s mass-energy equivalence formula \( E = mc^2 \) to understand that energy \(E\) is related to mass \(m\) and the speed of light \(c\). We need the rate of mass change, so we use the formula for power \( P = \frac{dE}{dt} = \frac{dm}{dt} c^2 \).

Step 3: Solve for Mass Rate Change

Rearrange the formula for power to find \( \frac{dm}{dt} = \frac{P}{c^2} \). We substitute \( P = 10^{41} \text{ W} \) and \( c = 3 \times 10^8 \text{ m/s} \). Thus, \( \frac{dm}{dt} = \frac{10^{41}}{(3 \times 10^8)^2} \).

Step 4: Simplify the Mass Rate Change Calculation

Calculate \( \frac{10^{41}}{9 \times 10^{16}} = \frac{10^{41}}{9} \times 10^{-16} = \frac{1.11 \times 10^{25}}{9} \). Thus, \( \frac{dm}{dt} \approx 1.11 \times 10^{25} \text{ kg/s} \).

Step 5: Convert to Solar Mass Units per Year

We need to convert \( \frac{dm}{dt} = 1.11 \times 10^{25} \text{ kg/s} \) to solar mass units per year. 1 year \( = 365 \times 24 \times 3600 = 3.15 \times 10^7 \text{ s} \). Thus, \( 1.11 \times 10^{25} \text{ kg/s} \times 3.15 \times 10^7 \text{ s/year} \).

Step 6: Final Calculation and Conversion

Compute the mass per year in kg: \( 3.4965 \times 10^{32} \text{ kg/year} \). Since 1 solar mass unit = \( 2.0 \times 10^{30} \text{ kg} \), we divide \( \frac{3.4965 \times 10^{32}}{2.0 \times 10^{30}} \approx 174.8 \text{ smu/year} \).

Key Concepts

Energy-Mass Conversion

Energy-mass conversion is a fascinating concept that underpins many phenomena in the universe. At its heart, this concept tells us that mass can be transformed into energy and vice versa. When an object's mass is converted into energy, or when energy is transformed back into mass, the total amount remains constant due to the law of conservation of energy. This transformation is core to understanding how stars, like quasars, emit such immense energy.

For instance, in the case of a typical quasar, the energy radiated is drawn from the mass of the materials surrounding its nucleus. This process occurs at a rate proportional to the amount of mass being converted into energy. The formula to calculate this exchange is essential in physics and provides a deeper glance into the dynamics of celestial bodies and even nuclear reactions here on Earth.

Einstein’s Mass-Energy Equivalence

Einstein’s mass-energy equivalence principle, encapsulated in the famous equation \( E = mc^2 \), is perhaps one of the most renowned results in physics. This formula explains the direct relationship between mass \( m \) and energy \( E \), linked by the speed of light squared \( c^2 \).

The symbol \( c \) represents the speed of light in a vacuum, which is approximately \( 3 \times 10^8 \) meters per second. Since this number is so large, even a small amount of mass can be converted into a tremendous amount of energy. This relationship becomes particularly significant when analyzing cosmic phenomena, such as quasars or nuclear processes. By utilizing this equation, scientists can understand how astronomical objects emit energy over vast distances, helping decode some of the universe’s mysteries.

Through Einstein’s insight, the bridge between mass and energy is clearly depicted, showing that matter can be seen as a potential storehouse of energy. This principle is fundamentally why phenomena like quasars can shine so brightly across the cosmos.

Active Galaxies

Active galaxies are some of the most energetic and luminous objects in the universe. They host supermassive black holes at their centers, which are the engines driving their immense power output. Quasars are considered a class of active galaxies, characterized by their extraordinarily bright centers.

The energy produced by active galaxies, such as quasars, largely stems from the accretion of matter around the central black hole. As matter spirals into the black hole, it becomes superheated and emits vast quantities of energy across the electromagnetic spectrum. This process makes quasars visible even from distances of billions of light-years away.

Understanding active galaxies is crucial to studying the universe’s structure and evolution. These galaxies offer insights into the lifecycle of matter and energy, showcasing how mass concentration events can spur radiant luminosity, shaping the cosmos and impacting surrounding interstellar mediums.

Solar Mass Units

Solar mass units are a convenient way for astronomers and astrophysicists to quantify astronomical masses, using the mass of the Sun as a reference point. One solar mass unit (smu) is defined as \( 2.0 \times 10^{30} \) kilograms, making it a straightforward and practical metric for discussing astronomical entities.

When examining phenomena such as quasars, expressing mass rates of change or loss in solar mass units helps scientists convey immense size and scale in more comprehensible terms. For instance, the rate of mass conversion in quasars, expressed in solar mass units, simplifies comparison with other cosmic objects.

Beyond simplifying complex measurements, using solar mass units aligns with convention in the space community, aiding in the standardization of astronomical data. From planetary masses to whole galaxies, solar mass units provide a scale that makes processing and understanding the universe's vastness much more practical.