Worked exercise — Sun, Milan-Naples and mass loss

Problem

On Earth sunlight arrives with intensity $I_\oplus = 1000\;\text{W/m}^2$; the Earth is $r_\oplus = 1{,}5\cdot 10^{11}\;\text{m}$ from the Sun. (a) What is the Sun’s total luminosity $P_\odot$ (power radiated in all directions)? (b) At what distance from the Sun’s centre does the radiation pressure equal atmospheric pressure $P_\text{atm} = 10^5\;\text{Pa}$? (c) At what rate (in kg/s) does the Sun lose mass due to radiation?

Solution

(a) Luminosity of the Sun. Energy spreads over concentric spheres: the intensity at distance $r$ is $I = P_\odot/(4\pi r^2)$. At the Earth’s distance it equals $I_\oplus$, so

$P_\odot = I_\oplus\,4\pi r_\oplus^2 = 1000\cdot 4\pi\cdot (1{,}5\cdot 10^{11})^2 \approx 2{,}8\cdot 10^{26}\;\text{W}$

$\ev{P_\odot \approx 2{,}8\cdot 10^{26}\;\text{W}}$

(b) Distance at which the radiation pressure equals $P_\text{atm}$. For complete absorption the radiation pressure is $P_\text{rad} = I/c$. Setting $P_\text{rad} = P_\text{atm}$ requires an intensity

$I = c\,P_\text{atm} = 3\cdot 10^8\cdot 10^5 = 3\cdot 10^{13}\;\text{W/m}^2$

From $I = P_\odot/(4\pi r^2)$ we obtain the distance from the centre:

$r = \sqrt{\frac{P_\odot}{4\pi I}} = \sqrt{\frac{2{,}8\cdot 10^{26}}{4\pi\cdot 3\cdot 10^{13}}} \approx 8{,}6\cdot 10^5\;\text{m}$

$\ev{r \approx 8{,}6\cdot 10^5\;\text{m} \approx 860\;\text{km}}$

This distance is still well inside the Sun (the solar radius is about $7\cdot 10^8\;\text{m}$): the radiation pressure only becomes “atmospheric” at $860\;\text{km}$ from the centre, roughly the same as the distance between Milan and Naples ($\sim 770\;\text{km}$).

(c) Mass loss due to radiation. By mass-energy equivalence $E = mc^2$, every unit of radiated energy corresponds to a mass loss. The power $P_\odot$ corresponds to a mass-loss rate

$\dot m = \frac{P_\odot}{c^2} = \frac{2{,}8\cdot 10^{26}}{(3\cdot 10^8)^2} \approx 4{,}4\cdot 10^9\;\text{kg/s}$

$\ev{\dot m \approx 4{,}4\cdot 10^9\;\text{kg/s}}$

About $4$ million tonnes per second turned into light: enormous in absolute terms, but negligible compared with the Sun’s total mass ($\sim 2\cdot 10^{30}\;\text{kg}$).

Connections

Topics: Electromagnetic waves Concepts: Intensity of a wave · Radiation pressure · Mass-energy equivalence Skills: Fermi estimates