Supernova SN1987A was first visible at Earth in 1987 . ( a ) How many years B.P. (before present

Supernova SN1987A was first visible at Earth in 1987 . ( a )...

Key Concepts

Light Travel Time

This concept refers to the time taken by light to traverse the distance between the event (such as a supernova explosion) and an observer on Earth. Because light travels at a finite speed, the observed time of an event can differ significantly from the actual time at which it occurred, depending on the distance involved. It is essential for understanding the 'years before present' in astronomical contexts.

Cosmic Ray Propagation

Cosmic ray propagation involves the movement of high-energy particles, such as protons, through space after being emitted by astrophysical events. These particles travel at speeds very close to the speed of light but slightly slower, which results in a small delay in their arrival compared to light. This delay depends on the particle’s energy and its relativistic speed, providing insights into the timing and dynamics of the originating astrophysical event.

Relativistic Kinematics

Relativistic kinematics deals with the behavior of objects moving at speeds close to the speed of light. When particles such as protons are accelerated to high energies (e.g., 100 GeV), their speeds approach that of light, but are still marginally lower. The resulting slight difference in travel times compared to light involves calculations of factors like the Lorentz factor and beta (v/c), which are crucial for quantifying the delay in particle arrival times relative to photons in high-energy astrophysical phenomena.

Transcript

00:01 All right, in this question, we're talking about supernova, in particular, supernova in 1987a.

00:07 And the first question asks us when did it occur.

00:12 All right, so you know that light travels at the speed of light, and the light reached us in 1987, which is an astronomical terms basically now.

00:23 So its distance in light years is going to be equal to the amount of years ago that it happened.

00:32 Distance to this star is 1 .68 times 10 to the 5 light years.

00:42 So therefore, t time ago before present at which it happened is equal to 1 .68 times 10 to the 5 years.

00:54 All right, simple enough.

00:56 Now question b says that some protons were launched at the time of a supernova with 100 giga electron volts of kinetic energy and we want to know at what time do these protons reach the earth so i'm going to call this um t arrival okay so the time which they arrive is going to be the same thing as the time it takes for the protons to travel to earth um minus the time of arrival or the minus the amount of time that takes the light to arrive added to 1987 okay so we need to figure out the velocity of the protons in order to figure out what or how long it should take the protons to arrive so we can find velocity from energy normally we use the formula k -e is equal to one -half and b squared.

02:06 But this is for non -relativistic velocities.

02:08 This formula will not work for very high velocities like you have after supernova.

02:14 So instead, we have to use a different formula, which is that kinetic energy is equal to m, which in this case is the massive proton, times c squared, times the quantity of 1 over the square root of 1 minus v squared over c squared, and then minus 1 close quantity.

02:38 All right, so in order to solve the velocity, we have to do several steps of algebra.

02:43 We can start by dividing this stuff off to the other side.

02:47 So we'll have ke over mpc squared is equal to 1 over the square root of 1 minus v squared over c squared, and then minus 1.

03:02 Now we can add one to the other side, multiply this term up, divide this term down, so that's going to give us v is equal to, so i'm doing a lot of steps at once, but hopefully you can follow the algebra.

03:17 1 minus the quantity of 1 over 1 plus energy over mp, c squared...

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