Imagine that you are a particle physicist, and you have come up with a theory that some gamma-ray bursts come from the decay of a new and exotic sub-atomic particle that you think might exist.
You calculated that if this particle really does exist and decay as you predict, the decaying particles would need to be about 100 astronomical units away from the Earth, to give the observed typical fluence of the gamma-ray bursts.
Unfortunately, you've just found out that you made a mistake in your prediction. The energy put out when one of your particles decays is actually a factor of 15.9 greater than you originally thought.
How far away (in astronomical units) must the typical decaying particle be now, to give the fluences observed?
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To determine the new distance at which the typical decaying particles must be from Earth to give the observed fluences, we need to consider the change in energy released during the decay.
Given that the energy released during the decay is now 15.9 times greater than originally predicted, we can write the following equation:
(Energy released originally) x (15.9) = (Energy released at new distance)
Since the fluence observed is directly related to the energy released, we can use this equation to find the new distance. However, we also need to know the relationship between distance and energy.
In general, the energy is inversely proportional to the square of the distance. Mathematically, this can be written as:
(Energy released originally) / (Distance originally)^2 = (Energy released at new distance) / (Distance at new distance)^2
Now, substituting the values we have:
(Energy released originally) / (100 AU)^2 = (Energy released originally) x (15.9) / (Distance at new distance)^2
Simplifying the equation by canceling out the common factor, we get:
1 / (100 AU)^2 = 15.9 / (Distance at new distance)^2
To solve for the distance at the new location, we can rearrange the equation as follows:
(Distance at new distance)^2 = (100 AU)^2 / 15.9
Taking the square root of both sides gives:
Distance at new distance = sqrt((100 AU)^2 / 15.9)
Evaluating the expression, we find:
Distance at new distance ≈ sqrt(10,000 AU^2 / 15.9)
Distance at new distance ≈ sqrt(628.93 AU)
Distance at new distance ≈ 25 AU
Therefore, the typical decaying particle must be approximately 25 astronomical units away from Earth to give the observed fluences, given the revised energy released during decay.
To determine the new distance of the typical decaying particle, we need to consider the change in energy output. Let's call the original predicted distance D1 and the new distance D2.
The fluence of the gamma-ray bursts is directly related to the energy emitted by the decaying particle. Since the energy released during decay is 15.9 times greater than originally predicted, the fluence will also be 15.9 times greater than what was initially expected.
Given that the original predicted distance (D1) was 100 astronomical units, and the fluence has increased by a factor of 15.9, we can set up the following equation:
Fluence_new = Fluence_original * 15.9
Since fluence is inversely proportional to the square of the distance (D), we can write:
Fluence_new = Fluence_original * (D1^2 / D2^2)
Substituting the relevant values into the equation, we have:
Fluence_original * (D1^2 / D2^2) = Fluence_original * 15.9
Cancelling out the Fluence_original from both sides of the equation, we get:
(D1^2 / D2^2) = 15.9
Taking the square root of both sides, we obtain:
D1 / D2 = sqrt(15.9)
Rearranging the equation to solve for D2, we have:
D2 = D1 / sqrt(15.9)
Plugging in the value for D1 (100 astronomical units) into the equation, we get:
D2 = 100 / sqrt(15.9)
Calculating the value, we find:
D2 ≈ 100 / 3.986 ≈ 25.07 astronomical units
Therefore, the typical decaying particle must now be approximately 25.07 astronomical units away from Earth to give the now observed fluences.