1858 Drawing of the Eclipsed Sun as Seen from Brazil

We have been talking about the sun. With the European Space Agency’s and NASA’s Solar Orbiter mission and its suite of ten instruments daily gathering untold quantities of images and data, it’s hard to imagine the time when, a century or so ago, it took years of effort – usually by single investigators – to wring out even the smallest bits of dependable data from it. Close though it is, the star for decades mocked our feeble attempts to understand it.

It was difficult enough to obtain trustworthy observational data through our watery atmosphere with invented equipment. But to make any sense of it in the early days of the 20th century, one also had to know about thermal radiation. That part of physics was still quite new yet evolving rapidly. With breakthrough equations on radiation and temperature developed in the late 19th century, the Smithsonian Astrophysical Observatory’s C. G. Abbot, about whom you are by now familiar from our earlier posts, theoretically could take his newly minted solar constant, apply some geometry, and measure the temperature of the sun’s surface from the earth’s surface. Sounds simple, right? Well, maybe. But let’s look what lay ahead of him and understand a little more of the historical context.

*The Challenge*

Despite remarkable advances in spectroscopy throughout the 19th century, beginning with Joseph von Fraunhofer’s 1814 mapping the forest of dark lines of the solar spectrum, almost nothing was known about the heat of the sun, how it is generated and sustained, whether it is stable or variable, or how long it will survive. No one had any clear idea about what causes sunspots, faculae, and prominences, or about its interior constitution. Even the most obvious question was a hair-pulling puzzle: *how hot is it*? Especially, how hot is the surface of it, the part that we see that radiates heat away from the sun and warms our globe?

To Smithsonian Astrophysical Observatory scientists Samuel Langley and his collaborator and successor, Charles Greely Abbot, the prize offered in 1876 by French Academy of Sciences for successfully determining the temperature of the sun may have been a tempting inducement. Certainly, it signaled the question’s importance to the scientific community. But both men had broad interests in understanding the entire radiation spectrum of the sun and its effects on earth – finding the sun’s temperature was but one consequence of their larger programs of solar study. Langley, for example, had long been interested in mapping the unseen infrared portion of the solar spectrum. He invented the first ‘bolometer’ for measuring the sun’s heat at different wavelengths. Abbot was fascinated by the possibility of solar variability and potential effects on the earth’s climate; for years his Grail was an ever-more precisely measured solar constant.

Yet, as they also knew, the spread of the sun’s thermal energy across all wavelengths is determined by its *temperature*. In that measure are the keys to the kingdom. Indeed, it turns out that radiation and temperature are two sides of the same coin. In fact, with some qualifying assumptions that we’ll discuss in a minute, any radiator’s temperature exhibits a *uniquely shaped radiation curve*. And the peak of that curve will reveal the effective temperature of the radiator. These were hugely important insights in the latter part of the 19th and early 20th centuries, emerging mostly from the work of Kirchoff, Štefan, Boltzmann, Wien, Planck, and others.

In short, *if* we know the amount of thermal energy emitted by the sun at earth’s distance from the sun (i.e., the solar constant), *then* we can (with the merest bit of math) determine it at the sun’s surface. And *if* we can determine the thermal energy emitted at the sun’s surface, *then *we can know its temperature *if* we know the law that relates energy to temperature. And we can *check* that result against the temperature predicted by the peak of the solar radiation curve. That is the approach that Charles Greely Abbot would take to find the sun’s surface temperature.

But others had found the task immensely confounding.

Earlier efforts to estimate the sun’s effective temperature had produced wildly disparate results. Agnes Clerke, the redoubtable historian of 19th century astronomy, reported these all-over-the-map values for the sun’s surface temperature in her 1903 book, *Problems in Astrophysics*: 4,036,000° (Ericsson, 1872); 20,000° (Francesco Rossetti, 1879; later about half that); 7,600° (Le Chatelier, 1892); 8,700° (Wilson and Gray, 1894); 5,130° (Paschen, 1895); and 6,590° (Langley, 1901). But with ever-improving values of the solar constant in hand (his 1915 value being 1.93, compared to the modern value of 1.95 cal./cm2/min or 1361 watts/m2/sec), Abbot could confine estimates of the sun’s temperature to a much narrower range.

*A Voyage to the Sun*

Abbot’s solar constant measured the mean amount of solar radiation streaming onto an imaginary sphere outside earth’s atmosphere; to derive the sun’s temperature he’d have to know its value at the sun’s surface.

As with gravitational force, radiation from the sun in all directions changes exponentially with the change in distance from it; in mathematical lingo, it varies inversely as the square of the distance from it. Think of two enormous spheres centered on the sun. The first encompasses the outer visible surface of the sun, a radius of almost 700 thousand kilometers. The other, vastly larger sphere extends about 150 million kilometers to earth’s orbit. Now let’s imagine we are on a voyage from earth to the sun, from the outer to the inner sphere. As we approach the sun, the detector aboard our spaceship records this power-of-two increase of solar radiation as our distance from it diminishes. At half the distance the radiation will have increased fourfold. We can take the trip safely and cheaply with the vehicle of mathematics, and compute how the solar constant will exponentially increase all the way to the solar photosphere, that inner sphere. In the early 20th century, the distance between the earth’s orbit and the sun’s photosphere was well enough known for Abott to put his solar constant on the metaphorical spaceship, travel inward, and estimate the radiation emitted at the sun’s surface. By virtue of this inverse-square relationship and our great distance from the sun, solar radiation per square meter at the sun’s surface is over 46,000 times greater than its value at earth’s orbital distance.

The value of the solar constant at the sun’s outer surface or photosphere is usually called the *solar flux* and is typically stated in watts per square meter. It’s not to be confused with temperature of the sun surface, which we’ll talk about in a minute. If we use the modern value of 1361 watts per square meter for the solar constant outside of the earth’s atmosphere, we’d find a solar flux at the sun’s photosphere to be about 63 million watts per square meter of the sun’s vast surface. How do we grasp that number? With difficulty! Go to your hardware store and buy a million sixty-watt bulbs, put them in a one-meter square frame and you’ll almost have the idea…

At this point Abbot could compute the sun’s surface temperature if he knew the physical law by which to translate solar radiation at the sun’s surface into temperature. If you felt a little light-headed on our exponential ride toward the sun involving powers of two, brace yourself: we’re elevating now to the *fourth* power as we get into the critical and historically interesting relationship between temperature and heat

*The Intimate Relationship between Heat and Temperature*

If I feel and measure heat from a candle a foot away, how can I tell the temperature of the candle? For much of the 19th Century, this relationship was unknown, and led to the wildly varying guesses of the sun’s temperature. Fortunately, Slovenian physicist (and poet) Jožef Stefan (1835 – 1893) in 1879 solved this problem by showing that, for an ideal (''blackbody'') radiator – one that emits and absorbs radiation in equal amounts, in perfect equilibrium – the radiation is proportional to the *fourth power* of its temperature. Stars are not perfect radiators. Selective absorption of certain wavelengths of photons in their outer atmospheres, different for stars of different temperatures, for example, make their radiation emissions rather less than would be the case of a perfect blackbody. Nevertheless, they are not far from being blackbody radiators, and the radiation law can be applied with reasonable confidence to the stars, including the sun, to give a tolerably good estimate of temperature, which in these circumstances is called the *effective *temperature. (Stefan's equation relating radiation and temperature came to be called the *Stefan-Boltzmann law* in honor of his student, Ludwig Boltzman, who contributed to the theoretical foundations for the law.) The effect of the fourth-power temperature relation is dramatic. The slightest increase in temperature of a star (or any other blackbody radiator) translates to an enormous growth in the body's radiative output: a doubled temperature implies a sixteen-fold increase in energy output.

With this powerful tool, Abbot could use his solar constant to mathematically reach across 150 million kilometers of space and determine the temperature of our closest neighboring star that warms our earth.

*How Hot is the Sun's Photosphere?*

According to the Stefan-Boltzmann law, the radiant energy flux per square meter i.e., the solar flux of a perfect (blackbody) radiator is equal to the fourth power of the temperature (in degrees Kelvin) times an extremely small number known as the Stefan-Boltzmann constant (usually denoted by σ). How small? Well, its .0000000567. The sun as noted is not a perfect blackbody radiator, and there are areas of absorption in the solar spectrum which make its radiation curve (intensity plotted against wavelength) differ somewhat from the ideal blackbody curve, especially at shorter wavelengths. But these differences are not huge, and Abbot could reasonably estimate the sun’s effective temperature using the Stefan-Boltzmann law. With all this, from Abbot’s 1908 value of the solar constant (2.01 calories), he found the sun’s effective temperature to be 5,962° Kelvin, the closest value yet to the currently accepted value of 5,772° K.

Abbot was nevertheless skeptical of his temperature results because he didn’t know the extent to which the sun deviated from being a perfect blackbody radiator, the theoretical assumption upon which these equations were constructed. Abbot’s values for the equation constants, too, differed modestly from modern values. Though reflecting the knowledge of the day, his data for the earth-sun distance, solar radius, and the Stefan-Boltzmann constant were all slightly smaller than modern values. Using Abbot’s 1915 value for the solar constant of 1.93, with modern physical values (not then available to him), we end up with 5,757° K for the solar temperature, only 15 degrees off from the modern value.

*Another Way: Climbing the Radiation Curve*

Abbot, using mathematical tools in the new science of astrophysics, also had the idea of using Wein’s law to estimate the temperature of the sun (after German physicist Wilhelm Wien, 1864 – 1928) from the *radiation curve* he had deduced from his data. To do this, one must examine the radiation data from the sun, as Abbot did, and determine where its intensity peaks. Since at each temperature the distribution of blackbody radiation will be different (according to an equation discovered by Max Planck, and called the *Planck curve*), each temperature will have its own defined curve, and its unique summit: the peak intensity will be different for each curve. The curves I created below (from Planck's equation) illustrate how dramatically temperature affects their shape:

It is the peak intensity that chiefly determines the colors of stars (and other hot objects) as they appear to us. The Wien relation between the peak wavelength and temperature is shown by this simple relation:

*T* = 2,898/*λmax*

where the temperature *T* is at the peak wavelength λmax. This may be one of the simplest equations in physics, so savor it! Our task is to look at the solar energy curve, find its peak wavelength, divide it into that big number, and when you’ve done that, out will pop the temperature. But finding the peak isn’t always easy. Abbot estimated the maximum radiation from the Sun to be at about .433μ (microns). This estimate was difficult, in Abbot’s words, because “[t]he position of maximum energy in the solar spectrum as represented by the table and chart just given is rather indefinite, because there are in fact two maxima separated by the region of G lines in the violet.” Below is Abbot’s solar radiation curve, drawn from Mt. Wilson observations of the Sun in 1905 and 1906. You can see by looking at the valleys in the graph the problem in estimating the solar peak in the violet end of the spectrum:

Abbot also used values for the constant in the equation of 2,921 (from Paschen) and 2,940 (from Lummer) and obtained high values for the solar temperature of 6,750° K and 6,790°K, respectively. Abbot attributed the discrepancy between the temperature he found by the Stefan-Boltzmann law and the Wien law to be attributable to the fact that the sun is not a true blackbody radiator. While the difference between the temperatures deduced from Stefan’s and Wien’s laws is an indication that the radiation of the source is not exactly that of a blackbody, the main cause of Abbot’s discrepancy was that his wavelength for the peak intensity was too low – toward the violet end. It therefore suggested a slightly higher energy spectrum than is actually the case for the sun, and a correspondingly higher temperature. If we turn the Wien equation around it can tell us what that peak emission ought to be. Using the solar temperature of 5,772° K, the peak of the solar curve is .502 (i.e., half a micron), a bit higher than Abbot’s 1908 estimate of .433 microns.

*Postscript on the Sun’s Luminosity*

From knowledge of the solar flux, Abbot could also have determined the *luminosity* of the sun, although I did not see a reference to his doing so in the *Annals* or later papers; it really wasn’t part of his dedicated task to do. But we can do it. To find it, we just need to multiply the per-square-meter solar flux by the total surface area of the sun. Using the formula for the surface area of a sphere and the above-mentioned radius of the sun, the calculation yields a luminosity for the sun of about 3.85 × E26 watts. For comparison, the pale blue supergiant star Deneb in the constellation Cygnus has a luminosity of about 9.9 × E31 watts, almost 260,000 times more luminous than our own modest star.

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