Question

Ultra-high-energy gamma rays are found to come from the Equator of our galaxy, with energies up to \(3.5 \cdot 10^{12} \mathrm{eV}\). What is the wavelength of this light? How does the energy of this light compare to the rest mass of a proton?

Short answer

Answer: The wavelength of the ultra-high-energy gamma ray is \(3.56 \cdot 10^{-17} \mathrm{m}\). The energy of the gamma ray is 3733 times the rest mass energy of a proton.

Step-by-step solution

Convert the energy to SI units

Given energy, \(E = 3.5 \cdot 10^{12} \mathrm{eV}\). To convert it into SI units (joules), we need to multiply it by the charge of an electron. \(1 \mathrm{eV} = 1.6 \cdot 10^{-19} \mathrm{J}\) So, \(E = 3.5 \cdot 10^{12} \mathrm{eV} \times 1.6 \cdot 10^{-19} \frac{\mathrm{J}}{\mathrm{eV}} = 5.6 \cdot 10^{-7} \mathrm{J}\)

Calculate the wavelength of the gamma ray

We will use the energy-wavelength relationship given by Planck's constant (\(h\)) and the speed of light (\(c\)). For a photon, the energy-wavelength relationship is given by \(E = h \cdot c / \lambda\) where \(E\) is the energy, and \(\lambda\) is the wavelength. Rearranging for \(\lambda\), we get \(\lambda = h \cdot c / E\) Using the values for Planck's constant (\(h = 6.63 \cdot 10^{-34} \mathrm{Js}\)) and the speed of light (\(c = 3.0 \cdot 10^{8} \mathrm{m/s}\)), we can compute the wavelength as follows: \(\lambda = \frac{6.63 \cdot 10^{-34} \mathrm{Js} \times 3.0 \cdot 10^{8} \mathrm{m/s}}{5.6 \cdot 10^{-7} \mathrm{J}} = 3.56 \cdot 10^{-17} \mathrm{m}\)

Compare the energy with the rest mass of a proton

To compare the energy of this light with the rest mass of a proton, we need to convert the proton mass into energy using the mass-energy equivalence formula. Einstein's mass-energy equivalence formula is given by \(E = m \cdot c^2\) where \(m\) is the rest mass of a proton, which is approximately \(1.67 \cdot 10^{-27} \mathrm{kg}\), and \(c\) is the speed of light, \(3.0 \cdot 10^{8} \mathrm{m/s}\). The energy of the proton is given by \(E_\mathrm{proton} = 1.67 \cdot 10^{-27} \mathrm{kg} \times (3.0 \cdot 10^{8} \mathrm{m/s})^2 = 1.50 \cdot 10^{-10} \mathrm{J}\) Now, we will compare this energy with the energy of the gamma ray by taking their ratio \(\frac{E_{\mathrm{gamma}}}{E_\mathrm{proton}} = \frac{5.6 \cdot 10^{-7} \mathrm{J}}{1.50 \cdot 10^{-10} \mathrm{J}} = 3733\) The energy of the ultra-high-energy gamma ray is 3733 times the rest mass energy of a proton.

Key concepts

Energy-Wavelength Relationship

Gamma rays, like all electromagnetic waves, have a unique relationship between their energy and wavelength. This relationship is governed by Planck's equation:

• The equation is given by: \[ E = \frac{h \cdot c}{\lambda} \]where:
  • \( E \) is the energy of the photon,
  • \( h \) is Planck's constant \((6.63 \cdot 10^{-34} \mathrm{Js}) \),
  • \( c \) is the speed of light \((3.0 \cdot 10^{8} \mathrm{m/s}) \), and
  • \( \lambda \) is the wavelength.

The formula reveals an inverse relationship between energy and wavelength. As energy increases, wavelength decreases. This is why gamma rays, which have very high energies, have extremely short wavelengths. They typically measure in the range of fractions of nanometers or even smaller. Understanding this relationship helps us measure and compare different forms of electromagnetic radiation and underscores why gamma rays are such potent forms of energy.

Mass-Energy Equivalence

The concept of mass-energy equivalence is beautifully encapsulated in Einstein's famous equation:

• \[ E = mc^2 \], where:
  • \( E \) is energy,
  • \( m \) is mass,
  • \( c \) is the speed of light.

This principle highlights that mass can be converted into energy, and vice versa. It's pivotal in fields like nuclear physics and cosmology.
In practical terms, this means that the rest mass of particles, such as a proton, can be translated into an energy equivalent. For a proton, with a rest mass of approximately \(1.67 \cdot 10^{-27} \mathrm{kg}\), the energy equivalent is \(1.50 \cdot 10^{-10} \mathrm{J}\).
Comparing this with gamma ray energy helps demonstrate the sheer power of high-energy photons. While a proton's rest mass energy seems significantly large, an ultra-high-energy gamma ray can surpass this, demonstrating the staggering nature of energy at quantum levels.

High-Energy Photons

Gamma rays are the most energetic form of electromagnetic radiation. Unlike visible light, they penetrate deep into materials and have enough energy to affect atomic nuclei. This level of interaction is due to their status as high-energy photons, making them invaluable in scientific fields like astrophysics and medicine (for imaging and cancer treatment).
The energy of these high-energy photons, such as the ones originating from the galaxy's equator, can exceed billions of electron volts (eV). In cosmic contexts, they help astronomers learn about processes occurring in stars and black holes.
Understanding high-energy photons necessitates grasping their unique behavior, as they define both the limitations and possibilities within quantum mechanics. Working with such energies allows scientists to innovate ways to harness these photons to both decipher and influence the universe around us.