A The energy flux carried by neutrinos from the Sun is...

A The energy flux carried by neutrinos from the Sun is estimated to be on the order of 0.4 $\mathrm{W} / \mathrm{m}^{2}$ at the Earth's surface. Estimate the fractional mass of the Sun over $10^{9}$ yr due to the emission of neutrinos. The mass of the Sun is $2 \times 10^{30} \mathrm{kg}$ . The Earth-Sun distance is $1.5 \times 10^{11} \mathrm{m}$ .

Key Concepts

Mass-Energy Equivalence

Mass-energy equivalence, encapsulated in Einstein's equation E = mc², provides the relationship between a loss of mass and the energy emitted. This concept allows one to estimate how much mass is lost from an object, such as the Sun, after it releases a certain amount of energy over time.

Energy Flux

Energy flux is the measure of energy per unit area per unit time. In astrophysics, it is used to quantify how much energy from a source, such as the Sun, reaches a given location, and is critical when translating local measurements into total energy outputs.

Inverse-Square Law

The inverse-square law explains how the intensity of energy radiated from a point source diminishes with distance. It is fundamental in calculating the total power output from energy flux measurements taken at a specific distance from the source.

Luminosity

Luminosity is the total amount of energy emitted by an object per unit time. It is derived by considering the energy flux over the entire surface of a sphere centered on the energy source and is crucial for understanding the overall energy output of stars.

Transcript

00:01 So given the following information that the flux, which is the radiated power per unit area, is 0 .4 watts per meter squared, that we're looking at a time interval of this radiative power of a billion years, that the mass of the sun is 2 times 10 to the 30 kilograms, and that the distance from the sun of the earth is 1 .5 times 10 to the 11 meters.

00:23 We're going to note a couple of things first, that the area of a sphere is 4 pi are squared.

00:29 We're going to need that, and that one year is equal to 3 .156 times 10 to the 7 seconds.

00:35 We're also going to need that to convert from years to seconds since we're working in si units.

00:41 So we're going to write the energy emitted over this billion year period as delta e.

00:49 I write it as delta e because it's associated with a change in the sun's mass.

00:54 So this delta e can be written as the flux times the area.

01:00 That the radiated power is emitted into.

01:04 In other words, we can think of this radiation or these neutrinos really as being emitted into a huge sphere that has a radius equal to the distance between the sun and the earth.

01:18 So just imagine a giant sphere that is centered on the sun and has a radius equal to the distance from the sun to the earth.

01:27 So, delta e is then equal to flux, times area, times the time interval, which is a billion years.

01:35 So flux is p over a, power per unit area.

01:39 A is that the area of that sphere we just talked about, and t is the time interval.

01:46 And so now we substitute in the area of that sphere, which is 4 pi r squared, and t is the time interval...

Please give Ace some feedback