Shift in wavelength of photons from the Sun using energy argument

[etotheipi]

I came across a question recently which involved calculating the change in wavelength of a photon between being emitted from the surface of the sun and arriving at the Earth.

The method that was implied involved calculating the GPE's of the photon (assuming the photon to have a mass h/[c lambda]) at the start and end points in order to calculate the overall decrease in its GPE, with M_ph, M_e and M_s being the photon, Earth and Sun masses respectively and r_s, r_e and r_se being the Sun's radius, Earth's radius and Earth-Sun distance respectively. This turns out to be: $$\Delta U = GM_{ph}[-\frac{M_{e}}{r_{e}}-\frac{M_{s}}{r_{se}}+\frac{M_{s}}{r_{s}}+\frac{M_{e}}{r_{se}}]$$ Since the term M_s/R_s is substantially larger than the others, we omit all of the other terms and find the following approximate expression for change in GPE: $$\Delta U = GM_{ph}\frac{M_{s}}{r_{s}}$$The last step is to equate this change in GPE to the change in energy of the photon from which we can approximate the change in wavelength of that photon:$$\Delta E_{ph} \approx hc \frac{\Delta \lambda}{\lambda^{2}} = \frac{GM_{s}}{r_{s}}\frac{h}{\lambda c}$$ This yields the result $$\frac{\Delta \lambda}{\lambda} = 2.1\cdot10^{-6}$$Whilst I can understand the mathematical steps, I have trouble understanding the intuition for this last part. Why can we equate the increase in the photon's GPE to the decrease in the energy associated with its wavelength? Thanks a bunch.

 

[phyzguy]

Energy is conserved. So if the photon gains GPE as it climbs out of the sun's gravitational well, it must lose energy from somewhere. Since the photon's energy is hc/λ, and h and c are constants, λ must increase.

 

[etotheipi]

Energy is conserved. So if the photon gains GPE as it climbs out of the sun's gravitational well, it must lose energy from somewhere. Since the photon's energy is hc/λ, and h and c are constants, λ must increase.

Thanks for the really quick reply! That makes sense. I was also wondering how rigorous is this argument considering that the photon only really has an effective GPE?

 

[phyzguy]

I think what you have outlined is a heuristic argument which is not completely rigorous. To do it properly, you need to use General Relativity. This Wikipedia article has a description of how you do that, as well as experimental tests. I find the Pound-Rebka experiment really fascinating. They were able to measure the increase in wavelength of light propagating from the basement of their lab up to the roof, as it climbed out of the Earth's gravitational field.