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 redshift, we need to make use of the formula:
Δλ/λ = (GM/c²) * Δr / r
Where:
- Δλ is the change in wavelength
- λ is the initial wavelength
- G is the gravitational constant (6.67430 × 10^-11 N m^2/kg^2)
- M is the mass of the star (equivalent to the mass of the Sun, approximately 1.989 × 10^30 kg)
- c is the speed of light in a vacuum (approximately 3.00 × 10^8 m/s)
- Δr is the change in radius of the star
- r is the initial radius
In this case, the initial wavelength (λ) is 500 nm (or 500 × 10^-9 meters) and the initial radius (r) is 10 km (or 10,000 meters). We want to calculate the gravitational redshift caused by a star with the mass equivalent to the Sun.
Let's plug in the given values into the formula:
Δλ/λ = (6.67430 × 10^-11 N m^2/kg^2 * 1.989 × 10^30 kg / (3.00 × 10^8 m/s)^2) * Δr / 10,000 meters
Since the change in radius (Δr) is not given, we assume that the change is negligible compared to the initial radius.
Δλ/λ = (6.67430 × 10^-11 N m^2/kg^2 * 1.989 × 10^30 kg / (3.00 × 10^8 m/s)^2) * 0 / 10,000 meters
Simplifying further:
Δλ/λ ≈ 0
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 negligible.