jbartlo@ouchem.chem.oakland.edu (Joseph Bartlo) sent me some extremely helpful corrections to the documents and programs in this directory. We both agreed I ought to roll his corrections into my documents but I haven't had time to do so yet; I thought I ought to set them out to avoid the propogation of errors, though. ============================================================================== Date: Wed, 2 Aug 1995 05:37:26 -0400 To: rusin@math.niu.edu From: jbartlo@ouchem.chem.oakland.edu (Joseph Bartlo) Subject: SUNCALC Hi Dave, I find your SUNCALC explanation quite interesting. Your solution to the problem of finding solar position is well thought-out; and as you say, mathematically sound. However, in some ways I think you go about solving the problem 'the hard way'. Perhaps more importantly, a few things are neglected which significantly affect solar position; which I point out below. 2 references which are extremely useful for solar position and other astronomical calculations are: Explanatory Supplement to the Ephemeris (to the Astronomical Almanac) Astronomical Algorithms The top one is published by the United States Naval Observatory, and the one below was written by Jean Meuss. The Explanatory Supplement includes detailed diagrams, geometry, and equations. Jean Meuss' book does somewhat, and is more concerned with practical aspects of getting useful answers efficiently. I have FORTRAN programs which calculate solar position and sunrise/ sunset times based on those references; and ones to calculate solar energy flux considering scattering and absorption by our atmosphere and earth. Those are both for broadband (photon frequency not considered) and spectral (photon frequency considered) solar energy. I can send you any or all of those programs, if you wish. The 2 *main* problems with your method of calculation are: In calculation of solar position, earth's ellipsoidal orbit has a greater effect than you realize. True, earth's variation in orbital speed is small over a day; but over a sequence of days, change in \phi accumulates; to the point that *time of solar noon (NOON) varies by about 30 min in local time over the course of a year*. When expressing solar energy collection, atmospheric influenece is much too great to neglect without large error. Many atmospheric constituents both absorb solar energy and scatter it in all directions, a significant amount back to outer space. Thus, solar energy flux over our entire globe at locations under cloudless skies is only about 2/3 of what it would be if no atmosphere were present. Clouds increase that efffect to the point that slightly less than half reaches ground. Because of scattering and absorption of solar energy, solar incidence angle does not have as great of an effect on its collection than it otherwise would. Below I include comments to your discussion APPARENT POSITION OF THE SUN IN THE SKY and SUNCALC program, since you asked for them. I appreciate your effort at an original solution to this problem:) Joseph Bartlo Introduction: You say "we" when you mean "I". Unfortunately, 'we' are taught to do that:) Section 1: You mention "about 23 degrees, familiar from all desk globes". I state this because I grew up thinking that obliquity of the ecliptic was 23 1/2 degrees *exactly*, because it was written as such (or 23.5 deg) 'everywhere'; while it is 23.44 deg and gradually decreasing. It is intersting that the intersection of your 2 planes with our earth's orbit *define* equinoxes. Contrary to popular usage, equinoxes *are those point of intersection*, and the days we call equinoxes are those days our earth passes by them. You state in one place "earth's revolution about the sun" when you obviously mean 'about its axis'. It is not clear to me what \alpha is, since it is not clearly defined; but I think it is obliquity of the ecliptic. Section 3: Are you sure that NOON is not the time at which our sun is highest in the sky in some places? A physical analysis states that dU/dt would be exactly 0 at noon if solar declination did not change, which is related to /phi. Still, even though solar declination changes; increasing during our winter and spring and decreasing during our summer and fall; its incremental change just before or after solar noon may be less than that in arc of the ecliptic. That is stated in my terminology, if you can understand it. At the poles, where our sun continuously rises or sets, solar noon is obviously not the time at which solar elevation angle (angle of our sun's center above the horizon) is greatest; but at lower latitudes, I think it may be. Interesting question. Section 4: Neat explanation, but I have not picked at details regarding it. I learned a few things from it:) Section 5: Your discussion regarding clock time is true except for effects due to variations in d\phi/dt. As stated above, that causes a very significant change in local clock time of NOON over the course of a year. For example, if NOON occurs at 11:58 am at a location on 6 NOV, it will occur at 12:28 pm on 9 FEB. Mean solar time is defined as you have done, such that each day is exactly 24 hours; rather than being slightly more or less, depending on season. An 'equation of time' is used to account for that. It is difference between solar longitude and right ascension at a specific time; with angles expressed as time (15 deg per hour), as you do. Equations to calcuate it are usually expressed as Fourier series based on theory and observation, in which only a few terms are significant. That is the largest source of error in your calculation of solar position. Section 6: By "Dawn and Dusk", you obviously mean 'sunrise and sunset'. This is mainly a question of terminology, but I address it because dawn and dusk are often thought of as twilight times before sunrise and after sunset. You may consider the following definitions expressed in the Astronomical Almanac (beginning and ending times): Sunrise/sunset - solar zenith angle of 90 deg 50 min Civil twilight - solar zenith angle of 96 deg Nautical twilight - solar zenith angle of 102 deg Astronomical twilight - solar zenith angle of 108 deg One can note that all occur when solar zenith angle (angle between Up-axis and a ray to the apparent position of our sun) is > 90 deg. Sunrise/sunset occur at such an angle due to atmospheric refraction; bending of light by our atmosphere. Thus, when our sun's center is on the horizon, it would be about 50 arc min below the horizon if no atmosphere were present. Sunrise and sunset times are typically defined according to the position of our sun's center, to be consistent with other stellar observations. However, our sun's disc occupies about 32 arc min in our sky. I define sunrise and sunset times to be those at which our sun comes in or goes out of view. At low and middle latitudes, atmospheric refarction increases day length by 5-10 min, but by much more at higher latitudes, at which location our sun's apparent position remains very close to the horizon for greater time periods. At arctic locations, even calculations based on our sun's center and upper limb differ by tens of minutes in some circumstances; but only are 2-3 min different at low and middle latitudes. Section 7: In discussing solar energy, you state kWh in a place where you obviously mean kW. You may consider the following definition: solar constant - solar energy flux perpendicular to the solar beam at our earth's mean distance to our sun, in abscence of atmospheric influence The 'solar constant' is not constant, but slightly varies due to sunspots, solar flares, other solar acivity; and to a very small extent, interplanetary matter. However, it is about 1370 W/m^2; and typically varies by only a few W/m^2. Of course, the amount reaching our earth's atmosphere (extra-terrestrial) varies as you stated; from about 1330 W/m^2 at perihelion to 1415 W/m^2 at aphelion. Solar energy is greatly scattered and absorbed, even by a 'clear' atmosphere. Gas molecules scatter it approximately proprtional to frecuency^4. Thus, colors toward the blue end of the visible spectrum are preferentially scattered, and our sky appears blue. Aerosols scatter proportional solar energy to frequency. They also scatter more toward the blue end of the visible spectrum, but also significant amounts in other colors. Thus, on a day with many aerosol particles (dust, smoke, pollen, salt, etc.) in our atmosphere, a milky yellow-white appearance may is seen around the solar disc. Clouds scatter solar energy fairly equally in all frequencies, and thus appear whitish. Thier bases often appear dark beacuse very little solar energy is reflected up toward them and scttered back toward ground. You may consider typical summertime magnitude of solar energy flux (in W/m^2) near NOON in Illinois: Extra-terrestrial solar energy flux: 1335 cos(40-22) = 1270 CLOUDLESS SKY Global (all) solar energy flux incident to horizontal: 1000 Direct solar energy flux (solar beam) incident to horizontal: 850 Diffuse solar energy flux (scattered) incident to horizontal: 150 CLOUDY SKIES (averaged over a sufficient time period) About 20% cloud-covered : Global 900 Direct 720 Diffuse 180 About 50% cloud-covered : Global 700 Direct 450 Diffuse 250 About 80% cloud-covered : Global 500 Direct 280 Diffuse 220 Overcast : Global 200 Direct 2 Diffuse 198 Cloudy sky amounts are those typical, and depend very much on cloud types. Section 8: Parallax (difference between observations from earth center and surface) makes a difference in solar elevation angle of up to a little more than 8 arc min. That is small compared to some other inaccuracies. SUNCALC: I notice that you mistakenly put "June equinox" and "December equinox" in your output. It seems that "energy collection per day" includes nighttime periods as well as daytime, since that is much less than percentages at specific times. ============================================================================== Date: Fri, 11 Aug 1995 20:14:50 -0400 To: rusin@math.niu.edu (Dave Rusin) From: jbartlo@ouchem.chem.oakland.edu (Joseph Bartlo) Subject: Re: SUNCALC Hi Dave, Thank you for your kind comments. I think it would be best if you did rewrite my comments, so that they fit in better with the rest of your text (not seeming akward because of grammatical differences). It is fine with me if you do so, so long as you mention that I made a few suggestions. I have 3 corrections to make on what I wrote to you: I refered to earth's orbit as ellipsoidal, whereas now I recall that an ellpsoid is a 3-D trace of an ellipse about one of its axes. Quasi-elliptical is a better term for it. I got aphelion and perihelion reeversed, as I do too often. Jean Meeus' name is as spelled here. His is an excellent book to reference! I am happy that my comments are helpful to you. Please do not hesitate to send a messgae to me if you have further questions, comments, or ideas you wish to discuss. Joseph