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.
The formula to estimate the gravitational redshift is given by Δλ/λ = GM/rc^2, where Δλ is the change in wavelength, λ is the original wavelength, G is the gravitational constant, M is the mass of the star, r is the radius of the star, and c is the speed of light.
Let's plug in the values:
Δλ/λ = (6.67430 × 10^-11 N m^2/kg^2) * (1.988 × 10^30 kg) / ((10,000 m) * (3.00 × 10^8 m/s)^2)
Simplifying the equation:
Δλ/λ = (6.67430 × 10^-11 N m^2/kg^2) * (1.988 × 10^30 kg) / (10,000 m * 9.00 × 10^16 m^2/s^2)
Δλ/λ ≈ 1.483 × 10^-11
Therefore, the approximate gravitational redshift in 500 nm light emitted by a compact star with the mass of the Sun and a radius of 10 km is approximately 1.483 × 10^-11.
To find the approximate gravitational redshift, we need to apply the formula for gravitational redshift, which is given by:
Δλ/λ = GM/(Rc^2)
Where:
Δλ = Change in wavelength
λ = Original wavelength
G = Gravitational constant
M = Mass of the compact star
R = Radius of the compact star
c = Speed of light
Let's calculate step by step:
1. Convert the given wavelength to meters:
λ = 500 nm = 500 × 10^(-9) m = 5 × 10^(-7) m
2. Plug in the values into the formula:
Δλ/λ = (G × M)/(R × c^2)
G = 6.674 × 10^(-11) m^3 kg^(-1) s^(-2) (gravitational constant)
M = 1 solar mass = 1.989 × 10^30 kg
R = 10 km = 10 × 10^3 m
c = 3 × 10^8 m/s (speed of light)
Δλ/λ = (6.674 × 10^(-11) × 1.989 × 10^30)/(10 × 10^3 × (3 × 10^8)^2)
3. Calculate the value:
Δλ/λ = 4.43 × 10^(-6)
The approximate gravitational redshift in 500 nm light emitted by a compact star with a mass of the Sun and a radius of 10 km is approximately 4.43 × 10^(-6).