The solar energy arriving at the outer edge of Earth's atmosphere from the Sun has intensity \(1.4 \mathrm{kW} / \mathrm{m}^{2}\) (a) How much mass does the Sun lose per day? (b) What percent of the Sun's mass is this?

First, let's find the total power (energy per unit time) emitted by the Sun, using the given intensity and the surface area of a sphere. The intensity (I) of the solar energy is given as \(1.4\, \mathrm{kW/m^2}\) and the distance (r) from the center of the Sun to the Earth is approximately \(1.5 \times 10^{11}\ \mathrm{m}\). We can use the surface area of a sphere (\(4\pi r^2\)) to calculate the total power emitted by the Sun: Power (P) = Intensity (I) × Surface Area (A) P = \(1.4\,\mathrm{kW/m^2} \times 4\pi(1.5 \times 10^{11}\,\mathrm{m})^2\)

Remember that 1 kW equals 1000 W. So, we need to convert the power from kilowatts to watts. P = \((1.4 \times 10^3)\,\mathrm{W/m^2} \times 4\pi(1.5 \times 10^{11}\,\mathrm{m})^2\)

Now, solve for the power: P = \(6.366 \times 10^{26}\,\mathrm{W}\)

To find the mass loss per day, we need to convert power to energy and use the energy-mass equivalence formula (E = mc²), where E is energy, m is mass, and c is the speed of light (\(3 \times 10^8\ \mathrm{m/s}\)). Energy per day (E) = Power (P) × Time (one day in seconds) E = \(6.366 \times 10^{26}\,\mathrm{W} \times 24\,\mathrm{h/day} \times 60\,\mathrm{min/h} \times 60\,\mathrm{s/min}\) E = \(5.496 \times 10^{31}\,\mathrm{J}\) (Joules) Now, we can use the energy-mass equivalence formula to find the mass loss (m): m = \(\frac{E}{c^2}\) m = \(\frac{5.496 \times 10^{31}\,\mathrm{J}}{(3 \times 10^8\,\mathrm{m/s})^2}\) m = \(6.104 \times 10^{11}\,\mathrm{kg}\) So, the Sun loses \(6.104 \times 10^{11}\,\mathrm{kg}\) of mass per day.

Now, let's find the percentage of the Sun's mass that is lost per day. The mass (M) of the Sun is approximately \(1.989 \times 10^{30}\,\mathrm{kg}\). Mass loss percentage = \(\frac{mass\, loss\, per\, day}{total\, mass\, of\, the\, Sun} \times 100\%\) Percentage = \(\frac{6.104 \times 10^{11}\,\mathrm{kg}}{1.989 \times 10^{30}\,\mathrm{kg}} \times 100\%\) Percentage ≈ \(3.067 \times 10^{-6}\%\) Thus, the Sun loses about \(3.067 \times 10^{-6}\%\) of its mass per day.