Greenhouse effect and strong stellar winds (due 2025 Oct 2)

A major worry of our time is global heating. Here, we make overly simplistic estimates of the greenhouse effect, to get a sense of numbers. On average, the Earth should emit as much flux as it receives for the Sun but the actual ground temperature is higher than what you would expect from that because the atmosphere is optically thin to (mostly optical) sunlight reaching the Earth, but optically thick to (mostly infrared) radiation from Earth to space.

1. Show that the equilibrium temperature the Earth would have in the absence of an atmosphere is \(T_{\rm p}\simeq255\) K. How does this compare with the actual mean surface temperature on Earth?
2. As a very crude approximation, imagine the atmosphere as a single opaque layer with a uniform temperature that receives heat from the ground (at temperature \(T_{\rm g}\)) and radiates as much energy towards the ground as it radiates towards space. What is \(T_{\rm g}\)?
3. Now treat our atmosphere as a grey one (using the two-stream approximation, RL 1.8; see also wiki:Grey_atmosphere) to derive \(T(\tau)\).
   • What is \(\tau\) on Earth?
   • What would it be on Venus?
   • Can one make the ground temperature arbitrarily high? If not, what assumption breaks down?
   • Given that Earth's surface temperature has increased by about 1.5 degrees C since the pre-industrial era, can CO₂ be the dominant opacity source? If not, what would be? (You don't have to look them up, but the IPCC reports are remarkably readable.)

In hot stellar winds, free-free absorption is the main opacity source in the continuum. Its dependendies are discussed in RL 5.3. For this problem set, we use that Wolf-Rayet stars are hot, around 20000 K, while we are dealing with near-infrared radiation at a few microns, so \(h\nu/kT\ll1\).

1. Show that for a wind with mass-loss rate \(\dot{M}\) and (constant) velocity \(v_{\rm w}\), the radius at which the radial optical depth to infinity reaches unity scales as \(R_{1}\propto\lambda^{2/3}(\dot{M}/v_{\rm w})^{2/3}T^{-1/2}\).
2. How does the flux \(F_{\nu}\) received on Earth scale? In principle, as pointed out by Wright & Barlow 1975MNRAS.170...41W, given the constants, one could estimate \(\dot{M}\) from \(F_{\nu}\), a distance and the wind velocity (how would you estimate the latter? do you reproduce that we do not have to worry about \(T\)?). If the wind were clumped (i.e., had significant density variations around the mean), would the estimate be an over-, still good, or under-estimate of \(\dot{M}\)?
3. Calculate \(F_{\nu}\) by integration of the radiative transfer equation (continuing to assume thermal emission), appropriately scaling so that \(\tau=1\) towards \(R_{1}\). We can define an effective radius \(R_{\rm eff}\), the radius at which \((R_{\rm eff}/d)^{2}\pi{}B_{\nu}=F_{\nu}\) (use units of \(R_{1}\)). At which \(\tau\) does this occur?
4. Hillier at. 1983ApJ...271..221H found that one could calculate line profiles using that the line absorption coefficient, like that for free-free, scales as \(n^{2}\) (we briefly discussed in class why, and will return to that). Derive the line profile by again integrating the radiative transfer equation, writing the line absorption coefficient as \(\alpha_{\rm line}(\nu)=\alpha_{0}(n)\phi(\nu-\nu_{0})\) and using that the width of \(\phi(\Delta\nu)\) is much smaller than \(v_{\rm w}/c\) (like done in class). What observed line profile do you get? Is it in between the completely optically thin and thick cases discussed in class? Does it depend on the relative strength of the line? (For concreteness, define relative strength as \(\tau_{\rm line}(v=0)/\tau_{\rm cont}\).)

Note that the above can fairly easily be extended to the case where the wind is accelerating, the temperature changes, or the stellar photosphere is visible. I used the ideas in my thesis (1993PhDT........76V) to analyze infrared spectra of the X-ray source Cygnus X-3, which I found showed Wolf-Rayet features proving the companion was a helium star (1992Natur.355..703V). In later observations, the features were much weaker and varied with orbital phase. I attributed this to the wind being strongly ionized by the X-ray source, except in the star's shadow (1993A&A...276L...9V, 1996A&A...314..521V). The second paper has most of the analysis, and includes a figure (Fig. 9) with a fit to a regular Wolf-Rayet star; the line profiles fit remarkably well! The paper omits the appendix included in the thesis (chapter 4), which goes through a derivation similar to that asked for her, but keeping constants in place (and introducing a two-temperature wind). If you're lost, feel free to read that.