Question

Neutrinos fly from the Sun's center to its surface in about \(2.3 \mathrm{~s}\) because they travel at the speed of light \((c \approx 3 \times\) \(10^{10} \mathrm{~cm} \mathrm{~s}^{-1}\) ) and they undergo little interaction with matter. But a photon of light takes much longer to travel from the Sun's center to its surface because it collides with protons and free electrons and undergoes a random walk to reach the Sun's surface. A photon's step length is approximately \(1 \mathrm{~cm}\) per step [9]. The Sun's radius is \(7 \times 10^{10} \mathrm{~cm}\). How long does it take a photon to travel from the center to the surface of the Sun?

Short answer

It takes a photon approximately \(1.63 \times 10^{11} \text{ seconds}\) to travel from the center to the surface of the Sun.

Step-by-step solution

- Understanding Random Walk

A photon's travel can be modeled as a random walk. In a random walk, the mean squared displacement after N steps, where each step has length l, is given by: \[ \text{Mean Squared Displacement} = N l^2 \]

- Setting Up the Equation

To find the number of steps (N) it takes for a photon to travel from the center to the surface of the Sun, set the mean squared displacement equal to the square of the Sun's radius (R) since the displacement should roughly equal the Sun's radius. Therefore, \[ R^2 = N l^2 \]

- Solving for N

Rearrange the equation to solve for the number of steps (N): \[ N = \frac{R^2}{l^2} \]

- Plugging in the Values

Given that the Sun's radius (R) is \(7 \times 10^{10} \text{ cm}\) and the photon's step length (l) is 1 cm, plug these values into the equation: \[ N = \frac{(7 \times 10^{10} \text{ cm})^2}{(1 \text{ cm})^2} = 49 \times 10^{20} \]

- Determining the Total Time

The total time (T) taken is the product of the number of steps (N) and the time taken for each step. Given that light travels at approximately \(c \approx 3 \times 10^{10} \text{ cm/s}\), the time taken for each step of 1 cm is: \[ \text{Time per step} = \frac{1 \text{ cm}}{3 \times 10^{10} \text{ cm/s}} = \frac{1}{3 \times 10^{10}} \text{ s} \]

- Calculating the Total Time

Multiply the number of steps (N) by the time per step to find the total time: \[ T = N \times \frac{1}{3 \times 10^{10}} \text{ s} = 49 \times 10^{20} \times \frac{1}{3 \times 10^{10}} \text{ s} \] \[ T = \frac{49 \times 10^{20}}{3 \times 10^{10}} \text{ s} = \frac{49}{3} \times 10^{10} \text{ s} = 16.33 \times 10^{10} \text{ s} \]

Key concepts

random walk

When photons travel from the Sun's center to its surface, their journey is not straightforward. Instead, it follows a phenomenon known as a 'random walk'. Imagine walking in a crowded room, bumping into people and changing direction randomly each time you collide with someone. This process significantly slows down the photon's journey.

In scientific terms, a random walk is described by how the mean squared displacement (the average distance squared) relates to the number of steps (N) taken and the length of each step (l). The formula for mean squared displacement is: \[ \text{Mean Squared Displacement} = N l^2 \] This concept helps us understand and calculate the complex path photons take.

When applied to photons in the Sun, each step they take is roughly 1 cm long before they bump into another particle, and the total number of such steps defines how far they have traveled.

mean squared displacement

To understand how far a photon travels in the Sun due to its random walk, we use the concept of mean squared displacement (MSD). The MSD indicates the average squared distance a photon covers after a certain number of steps. This becomes particularly useful for particles like photons that move in random directions due to numerous collisions.

The mean squared displacement is mathematically represented as: \[ \text{MSD} = N l^2 \]

Here, N is the number of steps, and l is the step length. By setting the MSD equal to the Sun's radius squared \[ R^2 = N l^2 \], we can solve for N, the total number of steps needed for a photon to cover the distance from the center to the surface of the Sun. In this case:

\[ N = \frac{R^2}{l^2} \]

Given the Sun's radius and the step length, we find that photons require an immense number of steps, 49 x 10^20, to get to the Sun's surface.

photon step length

Photon step length is a crucial component in calculating how long it takes for photons to travel from the Sun's center to its surface. Step length denotes the average distance a photon moves in a straight line before it collides with another particle and changes direction.

In the context of the Sun, the average step length for a photon, due to the dense environment filled with particles, is about 1 cm. Even though each step is incredibly tiny compared to the Sun's radius of 7 x 10^10 cm, the photon's journey consists of an astronomical number of these steps.

This continuous sequence of small steps underlies the concept of a random walk and explains why the transport time for photons is vastly longer compared to neutrinos, which travel nearly unobstructed through the Sun's matter.

time calculation for light travel

Calculating the total time it takes for a photon to travel from the Sun’s center to its surface involves summing the time taken for each tiny step over the vast number of steps involved. Given a step length of 1 cm and the speed of light (c) approximately 3 x 10^10 cm/s, we can calculate the time per step: \[ \text{Time per step} = \frac{1 \text{ cm}}{3 \times 10^{10} \text{ cm/s}} = \frac{1}{3 \times 10^{10}} \text{ s} \]

Next, we know the number of steps (N) from the random walk calculation: \[ N = 49 \times 10^{20} \]

Therefore, the total time (T) is: \[ T = N \times \frac{1}{3 \times 10^{10}} \text{ s} = \frac{49 \times 10^{20}}{3 \times 10^{10}} \text{ s} \] This results in:

\[ T = \frac{49}{3} \times 10^{10} \text{ s} \simeq 16.33 \times 10^{10} \text{ s} \]

This enormous time frame illustrates just how slow the photon's random walk is compared to the almost instantaneous journey of less interactive particles like neutrinos.