Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 36: Special Relativity

Problem 41

Chapter 36, Problem 41

A modest supernova (the explosion of a massive star at the end of its life cycle) releases 1.5 x 10⁴⁴ J of energy in a few seconds. This is enough to outshine the entire galaxy in which it occurs. Suppose a star with the mass of our sun collides with an antimatter star of equal mass, causing complete annihilation. What is the ratio of the energy released in this star-antistar collision to the energy released in the supernova?

Verified step by step guidance

Step 1: Recall the formula for energy released during matter-antimatter annihilation. The energy released is given by Einstein's mass-energy equivalence formula:

E = m c^{2}

, where

m

is the total mass annihilated and

c

is the speed of light.

Step 2: Determine the total mass involved in the annihilation. Since the star and the antimatter star each have the mass of our Sun, the total mass annihilated is twice the mass of the Sun:

m = 2 M

_☉, where

M

_☉ is the mass of the Sun (approximately

1.989 \times 10 30^{} kg

Step 3: Substitute the total mass into the energy formula. The energy released in the annihilation is:

E

_annihilation = 2 M_☉c2. Use the value of

c

, the speed of light, as

3.00 \times 10 8^{} m/s

Step 4: Calculate the ratio of the energy released in the annihilation to the energy released in the supernova. The ratio is given by:

Ratio = E

_annihilationE_supernova, where

E

_supernova is the energy released in the supernova, given as

1.5 \times 10 44^{} J

Step 5: Simplify the ratio expression by substituting the values for

E

_annihilation and

E

_supernova. This will yield the final ratio, which compares the energy released in the star-antistar collision to the energy released in the supernova.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Release in Annihilation

When matter and antimatter collide, they annihilate each other, converting their entire mass into energy according to Einstein's equation E=mc². For two stars of equal mass, such as our sun, the total energy released during their annihilation would be the sum of their masses multiplied by the speed of light squared, resulting in a significant energy output.

Supernova Energy Output

A supernova is a powerful and luminous explosion that occurs at the end of a massive star's life cycle. The energy released during a supernova can be immense, typically around 10⁴⁴ joules, which is enough to outshine entire galaxies for a brief period. Understanding this energy scale is crucial for comparing it to other astronomical events.

Energy Ratio Calculation

To compare the energy released in different astrophysical events, one can calculate the ratio of their energies. This involves dividing the total energy from one event by the total energy from another. In this case, the ratio of energy from the star-antistar annihilation to that of the supernova will provide insight into the relative magnitudes of these cosmic phenomena.

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