Find the approximate gravitational red shift in 500 nm light emitted by a compact star whose mass is that of sun but whose radius is 10 km.
To find the approximate gravitational red shift in 500 nm light emitted by a compact star (with the same mass as the Sun but a different radius), we can use the formula for gravitational red shift:
Δλ/λ = GM / (c^2R)
where:
Δλ is the change in wavelength,
λ is the original wavelength of light,
G is the gravitational constant (approximately 6.674 × 10^-11 m^3/kg/s^2),
M is the mass of the star,
c is the speed of light in a vacuum (approximately 3 × 10^8 m/s),
R is the radius of the star.
Now, let's plug in the known values into the formula:
Δλ/λ = (6.674 × 10^-11 m^3/kg/s^2) × (mass of the Sun) / [(3 × 10^8 m/s)^2 × (10,000 m)]
The mass of the Sun is approximately 1.989 × 10^30 kg.
Δλ/λ = (6.674 × 10^-11 m^3/kg/s^2) × (1.989 × 10^30 kg) / [(3 × 10^8 m/s)^2 × 10,000 m]
Simplifying the equation:
Δλ/λ = (6.674 × 1.989 × 10^30) / [(3 × 10^8)^2 × 10,000]
Δλ/λ ≈ 8.845 × 10^-7
The approximate gravitational red shift in 500 nm light emitted by this compact star is 8.845 × 10^-7. This means that the wavelength of the light is stretched or shifted to longer wavelengths due to the gravitational effects.