INVERSE-SQUARE LAW AND MAGNITUDE FORMULAE ----------------------------------- John Pazmino NYSkies Astronomy Inc email@example.com www.nyskies.org 2015 December 3 initial 2019 April 13 current Introduction ---------- The upbringing of home astronomers includes many concepts about light and luminous output of celestial objects. Almost always they are treated as distinct subjects with little hint that some are actually mutations of a one single concept. That concept is the behavior of light radiating equally in all directions from a point source. Since celestial objects can often be treated as point sources as seen from far away and, in the absence of specific knowledge, they shine uniformly to all quarters of their world, understanding how light behaves is crucial for the astronomer. An other prime consideration is that the emission is unmolested on its way to Earth. We assume, in absence of specific information to the contrary, that the intervening space is free of diffusing, absorbing, dispersive, reflective material. A newer consideration is that the geometry of space is 'flat' over the path of the radiation. For extremely remote sources the radiation passes thru space that during the light travel is continuously expanding. ThIS expansion impresses a distortion into the way these sources illuminate Earth. I found several topics relating to luminous emission that here are discussed as facets of a one single concept, the Inverse-Square Law of radiation. I say 'radiation' because we can explore the universe in just about all wavelengths, or frequency, of electromagnetic energy, not only that in the luminous range of the spectrum. Light --- Recall that 'light' is not itself a physical substance. It can not be objectively measured without recourse to human perception. If there were no humans, the universe would be 'dark' in that it would NOT be filled with luminous sources, as some sci-fi scenarios imagine. In fact the universe would be filled with radiant energy ALL of which is INVISIBLE simply because we humans aren't there to see it. Light is the human physiological response to a limited range of wavelength of radiation incident into the eye-brain mechanism. Radiation within about 360nm and 720nm wavelength excites the eye- brain to yield the the sensation of vision. Incident radiation outside these limits do not produce vision. It took until the mid 20th century for astronomers to fully appreciate the distinction between light and general radiative emission. Photometers and light-meters ended up merely approximating the eye-brain response to radiation. Photography also was such an effort, leading to film chemistry that mimicked the human vision process. In history, only light was available for exploring the universe form Earth. Our atmosphere filtered out large segments of the incoming radiation. Certain other segments simply were never imagined to exist. The Sun was treated as a source of 'light' and 'heat', altho both were part of the one flux of radiation with different responses by humans. These were vision and warmth. Other radiations from celestial objects were hinted at in the 19th century, such as infrared studied by Herschel and radio eaves detected by Tesla. They were poorly studied for the crude instruments of the day. It wasn't until the 1930s when Jansky and Reber made deliberate investigations of radio-band noise from beyond Earth that we realized the potential range of other radiations emitted by celestial targets. The opening of astronomy to the full range of celestial radiation began in the 1960s with astrophysical satellites. By the 1980s just about the entire electromagnetic spectrum was open for exploration from space. Inverse-Square Law ---------------- One of the most lousy explanations in astronomy is for one of the most basic ideas of astronomy! We all see the diagram of rays from a point diverging as they recede outward. They pass thru screens with one, four, nine, maybe sixteen squares. The trick is to see that as the distance from the source increases, the density of the rays received on the screens decreases as the square of that distance. Altho very technicly this picture is correct, altho sometimes incorrectly drawn!, it is hardly a useful mechanism for furthering the home astronomer's career. Let's see what REALLY happens to the radiant energy. Place a sphere, of any desired radius, centered on the source. All radiation from the source must intercept this sphere, which in maths is called a Gauss sphere. The amount of radiation over the sphere is the same for all such spheres of what ever radius we choose. This is the key to understanding the inverse-square law. We define irradiation as the radiant energy, or energy, the flow or flux, passing a unit area of the sphere. If the power output of the source is P, in watts, and the radius of the sphere is r, in meters, the irradiation, I, is cited in watt/meter2 at radius r from the source. The radiant energy is captured by the whole area of the sphere, which is 4*pi**(r^2), so. I = P / (4 * pi * (r ^ 2)) And that is it! We derived the Inverse-Square Law in a simple, elegant, mature, wholesome manner. +---------------------------------+ | INVERSE-SQUARE LAW OF RADIATION | | | | I = P / (4 * pi * (r ^ 2)) | +---------------------------------+ The sphere enclosing the source does not have to a sphere, nor does it have to be centered on the source. Any closed surface, with no holes or tears, is valid. it may have folds, so the radiation enters and leaves thru them. It can be shown that in all cases the Inverse- Square law holds true. The difference is the complexity of tracing the rays thru irregular enclosing surfaces. We used a centered sphere for the simplicity of the maths. Watts and lumens -------------- We could have stopped here if this was an article within ordinary physics. The radiant power of the source is in watts. The distance, radius of the Gauss sphere, is in meters. The irradiation is then is watt/meter2. As a separate branch of science we played with 'light' which we treated as a luminous flux from the source. We dimensioned this flux in lumans and the incidence of this flux on the Gauss sphere was in lumens.meter2. Lumen is the analogy in light to the watt in total radiation. The Inverse-Square law for light is identical to the general one, except for the units of measure. The lumems received per unit area is the illumination of that area. There still seems to be no definite word for the luminous output of a source. Lumwnpower is a common one. On and off physicists tried to determine the equivalence of watts and lumen. A lamp bulb consumes electric in watts and emits light in lumens of light. The effort was frustrated by the weak knowledge of the hums vision mechanism and the tenet that light was really a physical substance. Since watt is a measure of mechanical poert, the equivalence was sometimes called the mechanical equivalent of light. Try this experiment ----------------- You need a light-meter or other device that measures incident illumination in lux or foot-cadle. Recall that for all intents and purooses one foot-candle equals ten luc. Or, in essences, the 'metric foot-candle' is 10 lux. In any rectangle room at home place a bare-bulb lamp in any convenient spot, like an end table. it does not have to centered in the room but in a place you can work all around it. Measure the dimensions of the room and work out its surface area, all in square meters. All other lighting in the room are now turned off, leaving only the test lamp turned on. it may be easiest to do this experiment at night with window shades pulled down. With the light-meter take illumination readings within each square meter of the room, including floor and ceiling. You may need a step- stool to reach the ceiling and work around immobile furniture to get at certain parts of the wall. tabulate the readings, m2-by-m2, and strike their sum. You in principle must multiply each by the area associated with it, because you already did the measurements in square-meter cells, the multiply is '1m2' for each reading. You may may have to do the explicit multiply for fractional square-meters sections of the room. The sum should be equal to the lumen rating of the lamp bulb, which should be marked on the bulb or its package. A tolerance of 10- 15 percent would be quite a good success. You actually surrounded the lamp with a Gauss surface, the room, to capture all of the lumens sent out by the lamp. You collected the illumination all over the room, summed them, to [more or less] equal that lumen output. You had to work in sections because the illumination over the room is very uneven. If the room was a sphere centered on the lamp, you would need only one reading, knowing it was the same all over the room. This assumes, as is largely true, the bulb radiates uniformly in all directions. Apparent magnitude --------------- By history astronomers did not care about the physical measure of illumination. They used, as invented by Hipparchus, a scale of 'brightness' for the stars. From the Greek era until the mid 1800s this scale was informal, based on eyeball assessments of the stars. Hipparchus himself assigned rank 1 to the brightest stars, about 15 of them above the horizon of Greece. The fainter stars, in 'steps' or 'shades', earned ranks 2 thru 6. Since the brilliance of a star was its greatness, the ranking was called 'magnitude'.A star of Hipparchus rank 3 was a star of magnitude 3. Until the 19th century magnitude assigned to star was sometimes erratic among astronomers. For telescopic stars, with no prior history of magnitude ranking, the scale diverged severely. Pogson in the mid 1800s formalized the scale with a logarithmic sequence. He measured by crude photometers that stars five magnitude apart were almost 100 time different in illumination. He set the ratio to exactly 100 and made each magnitude step equal to 5root(100), about 2.512. This scale preserved the bulk of ratings of stars already catalogued and provided a rational way to rate telescopic stars. This eyeball ranking is still used by home astronomers for monitoring variable stars, comets, meteors. The target is compared to field stars having designated magnitude ratings. The logarithm scale of magnitude is +-----------------------------+ | HIPPARCHUS-POGSON MAGNITUDE | | | | M - M0 = -2.5 * log(I/I0) | +-----------------------------+ M0 and I0 are the zeropoint values for magnitude and illumination. The logarithm is on base 10, the common or Briggs scale. The minus signum forces the ranks to INCREASE in value for fainter stars, matching the trend of the Hipparchus scale. newcomers sometimes find this relation confusing because a brighter star 'should' jave a greater value. Tink of magnitude as a rank or class in a company or camp or club. The lower-level members have a hifher rank number. The importnat members have the lower number, such as 'fist vice-chair'. As a member progresses and improves his status he moves to a higher class, with a lower numaaber. Newer developments ------ ----- At first we banked off of Polaris, assigning it magnitude 2.0. In the 1890s electrophotometry made magnitude assessments more objective, less personalized, and allowed for a link to the physics photometry system. The I0 for Polaris, with M0 = 2.0, was that electric current generated by its starlight. Effort was spent to filter the incoming radiation to accept that range of wavelengths of human vision. To at least some degree the incoming watt/meter2 was approximately proportional to lumens/meter2. An immediate benefit of electrophotometry was the greaater resolution of magnitude assessment in an objectrtive verfiable process. By eyeball, stars were assessed only to a whole or half unit, which could not be objectively verified. This objective nonhuman-based method moved the study of variable stars to a solid scientific level. By end of the 19thcentury a complete electrophotmetry of the sky's naked eye stars was done, issued as the Harvard Photometry. A refined edition came out in the 19-ohs, the Harvard Revised Photometry. This catalog would be the standard for magnitude values until the 1960, when the Johnson photometry came into use. Electrophotometry showed that some of the first magnitude stars were too bright for that rank. They were given the prolongated ranks of 0 and -1. In addition, planets usually are so bright that their magnitudes were in the negative range, too. Venus is typicly -4 magnitude; Jupiter, -2. In the1920s-30s we realized that the physical photometry system was out of whack. We were, without fully appreciating it, using laboratory illuminants ultimately based on combustion of hydrocarbon fuel. Until we learned more about blackbody radiation and the spectrosensitivity of human vision, we practiced photometry under false premises. New illluminants were devised employing noncombustive processes, such as fluorescent and neon lamps, and phosphorescence. Their luminous output in no way resembled the spectral profile of combustive sources. It took decades to shift physicists to a more fundamental scheme of photometry. We dogged on with 'light' as some real physical feature that would yield to objective instrumental study. What saved us from catastrophe was the resilience of human vision to errors of measurement and application of photometry. We really didn't get out of 19th century photometric mindset until the millennium crossing. On the astronomy side we in the 1960s were establishing major new observatories in the southern hemisphere, where Polaris is out of the local sky. We moved the magnitude zeropoint to Vega, visible from most reasonable southern latitudes. Vega was defined as magnitude 0.0. For narrow-band photometry the illumination, or irradiation, from Vega within the photometer's bandwidth is defined as magnitude 0.0. An other cause to let go of Polaris was its discovery as a delta Cephei variable star! Its illumination altered over a several day cycle by 2/10 magnitude. This was a totally intolerable swing of some 23% in illumination. Polaris just ws no longer a stable reference standard. Vega itself in the 1980s came under question when we found it had a circumstellar dust ring. Could the dust pass over the star to vary the light it sends us? So far it appears that the dust ring is too inclined to our line of sight. We do keep careful watch on it. Linked at last! ------------- What is the zero-point to initialize the magnitude scale? This number is maddingly trough to find in astronomy litterature! It is vaguely mentioned as a special application of photometry. When the International System of metric units was built in the 1960s, we finally linked the photometries of physics to astronomy. In addition, within a few more years we developed electronic and digital means of recording incident radiation, enabling us to manipulate data about light in ways never before possible. One significant result was the equation of magnitude and irradiation within the visual band of the spectrum. +-----------------------------------+ | MAGNITUDE-ILLUMINATION | | | | Mapp(0.0) = 2.56e-6 lumen/meter2 | +--------------------------------- \ The initial measurements of this equivalence were done at stations under clean dry air, as best as could be found, to minimize distortion of the incoming starlight by haze and moisture. In the Space Age the equivalence is assessed from above the atmosphere. Values wander a bit among authors because the photometer spectrosensitivity profile may not match that of human vision. Vision varies widely across astronomers, as much as with any other human faculty In as much as vision is entirely an internal processing of incident irradiation by the eye-brain system, it is impossible to know just 'how bright' a given person see a target star. It is equally impossible to teach a person for 'how bright' he should see that target star. Please keep in mind that 'light' as equated to irradiation is the STIMULUS entering the human eye-brain. It is NOT -- ad can not be -- the RESPONSE of the eye-grain. Angular magnitude --------------- In all of astronomy, until the late 20th century, the magnitude system was applied only to point sources, like stars, as observed by eye. It was applied occasionally to extended sources provided they remained angularly small, like the Moon and planets. Large areas such as nebulae, comets, aurora, were not treated to the magnitude ranking. Since the magnitude system is ganged to photometric units, it should be feasible to collect light from the entire angular extent of a large target and divide it by the target's angular area. The figure is lumnen/(meter2.arcmin2) or lumnen/(meter2.arcsec2). This is next converted into magn/arcmin2 ot magn/arcsec2, the very angular magnitude of the target. Be careful. The total illumination is NOT first turned to magnutude and THEN divided by the area! This would yield a ridiculously lw dim brightness per unit area. A crucial point to mind is that angular area applies to sources either nebular in texture or not resolved into stars, like a star cluster seen by bare-eye. If the target resolves into stars, under magnification, there is no 'area' sending illumination to us. The light comes from separate point sources and the total magnitude procedure is required. Extravisual magnitude ------------------- Magnitude is specificly defined for luminous radiation within the optical spectral band. For the most part home astronomers observe only by light but equipment is entering the market to observe beyond the optical range. Astronomers like to continue visualizing targets by a magnitude rating, even when the radiation is outside of the visual spectrum. They cite the 'magnitude of an ultraviolet or infrared source. The illusion may be to imagine its brightness if somehow human vision is sensitive to such radiation. A common basis of extravisual magnitude ratings is to compare the radiation to that from Vega in the same bandwidth. The irradiation from Vega is set at magnitude 0.0. Since Vega is a blackbody radiator and most nonoptical radiation is not, the comparison can prodice insane values for the 'magnitude' of a given target. Vega, for example raidates weakly in the ar infrared, making a target's infrared radiation seem enormous, with a humongous magnitude rating. In actuality the concept of magnitude as a measure of visual brightness is nonsense outside of the visual range of wavelengths. The entire premise of magnitude is to rank illumination as perceived by human vision. Radiation beyond the bandwidth of human vision produces no illumination or sensation of brightness. Astronomers working outside the optical band use magnitude as a convenient logarithmic scale of relative irradiations. The one is compared against an other. In this sense magnitude is like decibels. In no way would a radio technician claim that a signal of a given decibel strength sounds as loud as a note of the same decibel value if human hearing could hear the signal. Under the assumption that a star is a pure blackbody emitter we tried to account or the limited range of irradiation we captured as light. From spectrometry we figure out the temperature of the star and generate a blackbody radiation curve for it, using standard procedures rom thermdynamics. of the entire spectral range of this curve e work out the portion falling within the optical range. The remainder, outside the optical band, is expressed as an increment of magnitude applied to the optical magnitude. The sum, always brighter than the optical magnitude, is the bolometric magnitude. The increment, bolometric correction, is a function only of temperature and can be tabulated for handy reference. The Hertzsprung-Russell Diagram is plotted by the optical magnitude, since it is only in recent times we could measure other regions of radiation from stars. Some astronomers suggest to revise the HRD to plot bolometric magnitude Such a HRD can easily e plotted because the normal one embeds the star temperature, none caught on. Parallax ------ Parallax is the swing of our sightline from Earth to the target as Earth orbits the Sun. It is also the angular radius of Earth's orbit as seen from the target. Since stars are awfully far away, their light taking years and decades and centuries to reach us, the parallax angle is incredibly tiny. No known star has a parallax so large as one full arcsecond. Most stars making up constellations have angles in the hundredths of an arcsecond. The star distances can not be reasonably cited in terrestrial units like kilometers or Earth radii. Even the Earth-Sun distance, the astronomical unit, is way too small a unit. For example, the first star to yield a positive parallax, in the 1830s, was 61 Cygni with a distance of some 650,000 AU. Since this is one of the closer stars, other distances will be in millions of AU, a quantity that can not be easily visualized. The lightyear came into use in the mid 1800s for popular astronomy litterature and it mainstreamed in the profession by about 1900. But the lightyear has to be calculated from an other measure of distance. It is NOT a timing of the light as it travels from the target. No kidding, I see authors making this claim! With such minuscule angles, the parallax angle is inversely proportional to the distance by applying the small-angle rule of maths. The parallax angle is at the apex of the long slender triangle with the Earth orbit radius as base. For a given parallax the long sides, both essentially equal , have a definite length which can be be calculated. This length is 206,265 AU for a parallax of one arcsecond. With this length, distance to the star, inversely proportional to parallax, the one-arcsec length is a new, the 'parsec' from 'PARallax- SECond'. For quick work we can round this to 200,000 AU and 5 parsec = 1 million AU. In terms of lightyears, one parsec is 3.26 lightyears. Many astronomers simply use parsecs without switching to lightyears. The parsec has the simplicity of being identicly the reciprocal of the measured parallax. A star of 0.01 arcsecond parallax stands 1/0.01 = 100 parsec away. If you insist, that's 326 lightyears. +--------------------------------------+ | PARALLAX-PARSEC RELATION | | | | parsec distance = 1/(parallax) | +--------------------------------------+ Comparing illuminations --------------------- The Inverse-Square Law can compare illuminations from different sources to find either the distance or the radiation output. We have two sources of equal power P. One is at a given distance r0; the other, at unknown distance r. We write out the illumination received from both. I0 = P / (4 * pi * (r0 ^ 2)) I = P / (4 * pi * (r ^ 2)) Divide the lower one by the upper: I / I0 = (P / P) / (r ^ 2) / (r0 ^ 2) = 1 / (r ^ 2) / (r0 ^ 2) = (r0 ^ 2) / (r ^ 2) The received illumination from the equal sources is the inverse square ratio of their distances. We know the standard distance r0 and we solve for the unknown r I / I0 = (r0 ^ 2) / (r ^ 2) r ^ 2 = (I / I0 ) / (r0 ^ 2) We can also have two sources of equal distance r but one has a known power P0 and the other has unknown power P I / I0 = (P / P0) / (r ^ 2) / (r ^ 2) = (P / P0) / (1) = P / P0 Solve for unknown P P = P0 / (I0 / I) P = P0 * I / I0 Absolute magnitude ---------------- This is an attempt to standardize the illumination system of stars by artificially setting the stars at a one distance from us. The distance is 10 parsecs, chosen for a good maths reason. For that distance, and the actual distance and illumination, or magnitude, of the star, a new fake magnitude is computed. The usual explanation is that it's the apparent magnitude the star would shine if it somehow was placed 10 parsecs away. This computed magnitude is the absolute magnitude, a misleading name which we may for ever more be stuck with. Better names would have been 'normalized magnitude' or 'reduced magnitude'. There is nothing 'absolute' about absolute magnitude and it is not even a property of the star. To obtain the absolute magnitude we must have in hand the apparent magnitude and the distance to the star. Both are observed data from the star. Absolute magnitude was first used in the 1910s when we accumulated databases of parallax and apparent magnitude of stars. The data were captured by the then-new incorporation of photographic astrometry and electrophotometry. The absolute magnitude comes recta mente from the Inverse-Square Law and definition of magnitude. We compare the same star, with output P, at two distances. One is the actual distance r; the other, the standard one of 10pc, r0. I / I0 = (P / P0) / (r ^ 2 / r0 ^ 2) = (P / P) / (r ^ 2 / r0 ^ 2) I left out the 4*pi factor since it immediately cancels out in the denominator: ((4*pi*r^2)/(4*pi*r0^2)) -> *r^2/r0^2). Also P0 is P because we are working with one source moved between tow distances. I / I0 = 1 / (r ^ 2 / r0 ^ 2) = r0 ^ 2 / r ^ 2 Because we be astronomers we work with the magnitude scale, not the raw photometric scale. We first take the log of both sides log(I / I0) = log(r0 ^ 2 / r ^ 2) We now apply the definition of magnitude. log(I) - log(I0) = log(r0 ^ 2) - log(r ^ 2) = 2 * (log(r0) - 2 * log(r)) We apply the magnitude definition to both sides. -2.5 * (log(I) - log(I0)) = -2.5 * 2 * (log(r0) - log(r)) Mapp - Mabs = -2.5 * 2 * (log(r0) - log(r)) Mapp is the apparent magnitude recorded for the star at its real distance r. Mabs is the artificial, absolute, magnitude of the star if it was 10pc away. Mapp - Mabs = -5 * log(r0) - (-5 * log(r)) = -5 * log(r0) + 5 * log(r) Now comes the trick. We purposely picked 10pc as the distance for absolute magnitude BECAUSE the log of 10 is one! Astronomers hate maths as much as any one else. Mapp - Mabs = -5 * log(10) + (5 * log(r)) = -5 + 5 * log(r)) = 5 * log(r) - 5 Typicly we know the distance and Mapp and solve for Mabs Mapp - Mabs = 5 * log(r) - 5 -Mabs = 5 * log(r) - 5 - Mapp Mabs = -5 * log(r) + 5 + Mapp = Mapp - (5 * log(r)) + 5 Recall that the distance in parsec is merely 1/parallax, where parallax is the actual observed parameter of the star. Mabs = Mapp - (5 * log(r)) + 5 = Mapp - (5 * log(1/pi)) + 5 = Mapp + (5 * log(pi)) + 5 +---------------------------------+ | ABSOLUTE MAGNITUDE FORMULA | | | | | Mabs = Mapp + (5 * log(pi)) + 5 | | | | Mabs = Mapp - (5 * log(r)) + 5 | + ----------------------------------+ where I give both versions, for distance and for parallax. By this formula and stating the Earth-Sun distance in parsecs, a very small number, we find that the Sun's absolute magnitude is +4.8. If our Sun was removed to 10pc, a modest remoteness for a star, it would be among the mediocre tars in the sky. This helps to show how insignificant our Sun is on the scale of even nearby stellar reealm. Because most stars are beyond 10 parsecs from Earth, their Mabs is much brighter, algebraicly smaller, than their Mapp. Their is no special meaning attached to the numerical value of Mabs since it drives from the arbitrary choice of the 10pc standard distance. The derivation here joining the Inverse Square law to magnitude is almost neglected in the normal astronomy tuition. The two are treated as unrelated / separate topics. Luminosity -------- Altho 'absolute magnitude' is a poor choice of words for the normalized magnitude on a 10-parsec distance, it is a handy way to compare the luminous output of stars. With the Sun as a unit emitter of light, the relative output of any other star, in solar units, is the star's luminosity. This is NOT the star's full radiant output because luminosity ignores radiation beyond the visual spectrum. Stars in general emit the bulk of their radiation in the visual range, by the blackbody mechanism. In the era when we could not observe beyond the optical band, we had no confident accounting for the extravisual radiation. We let luminosity equal radiopower and live with ay discrepancy. Stars placed the same distance away shine with magnitudes consonant with their luminous emission. That is From the magnitude definition m - m0 = -2.5 * log(L / L0) where L is the luminosity, luminous output, in place of P, the full radiation output. Set m0 to the absolute magnitude of the Sun, +4.8, and I0 to the solar unit of luminosity m - +4.8 = -2.5 * log(L / 1) = -2.5 * log(L) When m is set to the absolute magnitude of a star, the luminosity ratio falls out Mabs - +4.8 = -2.5 * log(L) log(L) = (Mabs - +4.8)) / -2.5 = -0.4 * (Mabs - +4.8) L = 10 ^ (-0.4 * (Mabs - +4.8)) +----------------------------------------+ | ABSOLUTE MAGNITUDE-LUMINOSITY RELATION | | | | L = 10 ^ (-0.4 * ( Mabs - +4.8)) | | | | Mabs = -2.5 * log(I) + 4.8 | +----------------------------------------+ Star catalogs generally list either absolute magnitude or luminosity for its stars, It happens that ou may need the other figure. The Mabs or I formulae can be put into computr code for easier passage between the two. Star Deneb, alpha Cygni, is 1.3 magnitude and about 900 parsecs away. This is uncertain due to possible filtering by interstellar medium along the Milky Way. How much more luminous than Sun is Deneb? Mabs = Mapp - (5 * log(r)) + 5 = +1.3 - (5 * log(900)) + 5 = +6.3 - 5 * log(900) = +6.3 - 5 * 2.9542 = +6.3 - 14.7712 = -8.4710 This is brilliant! it approximates the brightness of a half Moon. L = 10 ^ (-0.4 * ( Mabs - +4.8)) = 10 ^ (-0.4 * (-8.4710 - +4.8)) = 10 ^ (-0.4 * -13.2710) = 10 ^ (5.3085) = 203,400 r, rounded because of the uncertainty of distance, 200,000 time more luminous than the Sun. Deneb is, in fact, among the most luminous stars visible in our sky. Mind well the distinction between Mabs and I. Mabs is an artificial parameter while luminosity is a part of the radiation output of the star. We see this distinction by imagining we are at a abase on an exoplanet. Our catalog with absolute magnitude is worthless while that with luminosities remains valid. (Mapp-Mabs) by itself is the 'distance modulus', a function only of the target's distance. It is routinely employed for galactic studies. Under 2010s methods we can not measure the parallax of extragalactic objects. We wok only with distaances in parsec. Distance modulus is rough, with values commonly cite to only the whole Extrasolar planets ---------------- With the incandescent interest in extrasolar planets and there being, in 2015, some 90 bare-eye planetary stars over the whole celestial sphere, a new use for the absolute magnitude equation sprang up. When we see in out sky a planetary star, we can ask: 'How bright is out Sun in that star's sky?' This amounts to figuring out the apparent magnitude of the Sun at the star's distance, given the absolute magnitude of the Sun. That's +4.8. We shuffle the distance modulus formula into an apparent magnitude form Mapp - Mabs = - 5 + 5 * log(r) Mapp = Mabs - 5 + 5 * log(r) +------------------------------+ | APPARENT MAGNITUDE FORMULA | | | | Mapp = Mabs - 5 + 5 * log(r) | +------------------------------+ For the specific case of the Sun seen from a planetary star this collapses to a very simple form Mapp = Mabs - 5 + 5 * log(r) = +4.8 - 5 + 5 * log(r) = -0.2 + 5 * log(r) = 5 * log(r) - 0.2 +--------------------------+ | SUN'S APPARENT MAGNITUDE | | | | Mapp = 5 * log(r) - 0.2 | +--------------------------+ This is a very simple formula! Remember that r is in parsec, not lightyear. By applying his formula to a few planetary stars, we find that our Sun would be among the dimmer stars in the planet's sky. This is based on human vision, of course. That's because stars in our sky tend more to be luminous than the Sun. As example to a star just 20 parsecs away the Sun would be a 6.3 magnitude star. This is the utter threshold of detection for good eyesight in a dark sky. For example, planetary star Hamal, alpha Arietis, is 20 parsecs away. How bright is the Sun in its planet's sky? Mapp = 5 * log(r) - 0.2 = 5 * log(20) - 0.2 = 5 * (1.3010) - 0.2 = 6.5050 - 0.2 = 6.3050 From Hamal's planet our Sun is a 6.3 magnitude star, at the threshold of human vision, in the local sky. In Earth's sky Hamal is a 2.0 magnitude star. The disparity of brightness, 4.3 magnitudes, translates into hamal being some 50 times more luminous than the Sun. Total magnitude ------------- When two or more stars are angularly so close that they blend into a single point, their separate illuminations add to a total single value. This total illumination yields a total magnitude for the set of stars. This total is always brighter than the group's brightest star. Total magnitude is almost always treated only for double stars, where the illuminations of two stars are added. In the old days, before calculettes, textbooks commonly had tables of two-star magnitude. A variation was a table of magnitude difference between the stars versus magnitude increment for the brighter one. The summation method applies to any set of close stars, like an open cluster, tight conjunction, compact asterism. If by bare eye or low power the group merges into a single source, the method works. First, the magnitude of each star is converted into illumination. The illuminations are summed. The sum is converted into the total magnitude. That may be: +---------------------------------------------------+ | TOTAL MAGNITUDE OF A CLOSE SET OF STARS | | | | (tot magn) = -2.5 * log(sum(10 ^ (-0.4 * magnX))) | +---------------------------------------------------+ Where magnX is the magnitude of each star in the group. In the Pazmino CLuster the trapezium of four brightest stars gives the bulk of the cluster's illumination. Little more is added by the decorative dim stars. The trapezium stars are about 7.5, 7.6, 7.7, and 7.8 magnitude, varying slightly among authors. What is the total magnitude of the Pazmino Cluster? (tot magn) = -2.5 * log((10 ^ (-0.4*7.5)) + ... + 10 ^ (-0.4*7.8)) = -2.5 * log(1.000e-3 + 9.210e-4 + 8.318e-4 + 7.586e-4) = -2.5 * log(3.511e-3) = -2.5 * (-2.455) = 6.136 This is within the casual estimates from deepsky observers. The magnitude in observing litterature is 6 to 6-1/2. More than four stars may be best handled by a computer program. It asks for the number of stars and then for the magnitude of each in turn. It outputs the total magnitude of the group. In the two-star case, many instances can be passed up when the magnitude difference is more than 2-1/2 magnitude. The contribution of illumination by the dimmer star is less than 0.1 magnitude and the total magnitude is substantially that of the brighter component. delta Cephei star --------------- If we had some way to independently know the absolute magnitude of a source and we record its apparent magnitude we can immediately find the distance to the source. delta Cephei, also Cepheid, stars were first applied to star distances in about 1910. We when we found that the absolute magnitude of a Cepheid star is a monotonic function of its period of oscillations of brightness. The magnitude is the mean between maximum and minimum luminous emission. his is expressed in the Period-Luminosity Relation. The name reflects the use of absolute magnitude as luminosity. discovered in delta Cephei stars are very luminous, letting us see them in other galaxies. Measuring their period is a matter of monitoring them for a few complete cycles of oscillation. Cepheids have a unique profile of light variation, not shared by any other kind of variable star. This makes it feasible to pick them out from a crowd of other variable stars. With the period in hand we read out the absolute magnitude for the star. The monitoring also captures the apparent magnitude, also of the mean between max and min illumination. We solve the absolute magnitude equation for r and insert the known values of Mapp and Mabs Mapp - Mabs = 5 * log(r) - 5 Mapp - Mabs + 5 = 5 * log(r) (Mapp - Mabs + 5) / 5 = log(r) +-----------------------------------+ | DELTA CEPHEI DISTANCE FORMULA | | | | log(r) = (Mapp - Mabs + 5) / 5 | +-----------------------------------+ ` zeta Geminorum is a delta Cephei star varying between +3.6 and +4.2 magnitude in a 10.148 day period. How far away is the star? The absolute magnitude of a delta Cephei star is either read off of a Period-Luminosity graph or calculated from a formula fitted to that graph. We use here the formula, which is one of several variations Mabs = -2.78 * log(period) - 1.43 = -2.78 * log(10.148) - 1.43 = -2.78 * (1.0064) - 1.43 = -2.7978 - 1.43 = -4.2278 This formula is a curve-fit against a plotted P-L graph and is valid only for 'classical', delta Cephei, stars. It does not apply to W Virginis or RR Lyrae stars. The Gaia astrometric spaceprobe in 2018 determined of distance to Polaris, the nearest Cepheid, as 131.1 parsecs. This is within the range previously assessed for calibrating the relation. The Mabs is the average between maximum and minimum brilliance. The Mapp of zeta Geminorum is the average of its maximum and minimum illumination, O3.6+4.2)/2 = 3.9. Then log(r) = (Mapp - Mabs + 5) / 5 = (+3.9 - -4.2278 + 5) / 5 = 13.1218 / 5 = 2.6256 r = 422.2706 -> 422 parsec Main-sequence fitting ------------------ In the Hertzsprung-Russell Diagram stars that shine by the hydrogen-helium energy process align along a narrow band, the Main Sequence. A star on the MS has a specific absolute magnitude and spectral class corresponding to its mass. The laws of nature work the same every where such that in an aggregate of stars, like a cluster or galaxy, the MS is the same as that for nearby stars. Unless we previously know the distance to the cluster we can not plot its stars on an HRD by their absolute magnitude. We plot the stars by their apparent magnitude. The cluster's MS is displaced vericly relative to the MS of a standard HRD. This displacement is measured as (Mapp - Mabs) a;ong a given spectral class. Several values are taken off from several points on the two Main Sequences and an average is worked up. This magnitude displacement is a distance modulus. It gives directly the distance of the cluster. +--------------------------------+ | MAIN-SEQUENCE FITTING FORMULa | | | | log(r) = (Mapp - Mabs + 5) / 5 | +--------------------------------+ For an example, the sigma Orionis cluster has a MS shifted 8.2 magnitude fainter than the standard MS. (Mapp - Mabs) = +8.2. The separate Mapp and Mabs aren't needed because the shift is scaled direcctly off of the HRD in magnitude units. Note that the target's MS is always faintr than. below, under the standard MS. If the MS is above, over, brighter than the standard MS, the target is close enough to yield a regular parallax. log(r) = (Mapp - Mabs + 5) / 5 = (+8.2 + 5) / 5 = 13.2 / 5 = 2.6400 r = 436.5 parsec A collateral method is applied to single stars, not part of a cohaerent group. If the star can be spectrmetricly placed on the standard Main Sequence, its absolute magnitude is read out. This is subtracted from the star's apparent magnitude to get (Mapp - Mabs). The above formula yields the star's distance. This method is commonly called spectrometric distance0 or spectrometric parallax. spectrometric distance is weak for stars off of the Main Sequence. Such stars do not have unique plots on the HRD, and a distance modulus can not be confidently calculated. Type-Ia supernova --------------- Certain supernvae are members of a binary star. In the course of the star's life it siphons from its companion to gradually increase in mass. Eventually the mass crosses the Chandrasekar limit, the largest mass a star can have before it by its own gravity collapses into a supernova. Because the increase is gradual, it seems that all such stars trip into supernova at about the same limiting mass and erupt into about the same luminance or absolute magnitude. There are many kinds on supernova but only the Type-Ia has this unique property of a uniform peak magnitude. other supernova processes generate unpredictable peak brilliance. In addition, the Type-Ia star has a unique spectrum and light output profile, This lets us recognize a Type-Ia star if we miss catching it at peak emission. We fit the observed profile to ones from previous supernovae and read out the apparent magnitude it had at peak luminance. This Mabs is -19.3, as best we know in the 2010s. Some astronomers suggest there is a leeway, maybe +/- 0.4 magnitude, due to chemical composition of the star and binary orbit dynamics. We massage the magnitude-distance formula: (Mapp - Mabs + 5) / 5 = log(r) (Mapp - (-19.3) + 5) / 5 = log(r) (Mapp + 19.3 + 5) / 5 = log(r) (Mapp + 24.3) / 5 = log(r) +------------------------------------+ | TYPE-Ia SUPERNOVA DISTANCE FORMULA | | | | log(r) = (Mapp + 24.3) / 5 | +------------------------------------+ So brilliant are these stars that home astronomers can spot them in the closer galaxies. It happens commonly that we can not see the diffuse patch of the galaxy but only the pinpoint of the star itself. With several scores of galaxies within reach of small scopes in New York City, we may observe a Type-Ia star once per decade or so. We must apply two major corrections to Mapp. Firs, the star may be dimmed by the interstellar medium of the host galaxy. The other is that beyond around 500 million parsecs the spacetime distortion effects of Hubble expansion must be considered. For galaxies which home astronomers can observe, the formula given here may be used as is. Hubble expansion is negligible and we usually have no data for interstellar dimming. In 2011 a Type I-a supernova erupted in galaxy M101. Its maximum apparent magnitude was +10.0. How far off is M82? log(r) = (Mapp + 24.3) / 5 = (10.0 + 24.3) / 5 = 34.3 / 5 = 6.8600 r = 7,244,360 -> 7,200,000 parsec Comet magnitude ------------- We now come to an obsolescent case, predicting the apparent magnitude of a comet. It was developed in the 1930s, so far as I know, when home astronomers took over much of the comet finding and observing work from campus astronomers. In the old days we knew far too little about how comets shine. We did suss out that a comet shines by reflected sunlight and self- luminance induced by solar radiation. The portion for reflected light was handled by the Inverse-Square law to factor in the Earth-comet and Sun-comet distances. We had no decent model for the induced luminance. In spite of this lack, we include the solar-induced light into the comet magnitude equation in a simple recognition factor. We had no good theory or model for the extent of a comet's tail or coma. We didn't try to add them into the comet's brightness. The equation applies only to the comet's head, ignoring extended coma and tail. When a comet is discovered it is assigned an absolute magnitude and a magnitude gradient, or slope, factor. Both are often guesses taken from similarity of the instant comet to previous ones. The absolute magnitude of a comet is the apparent magnitude when the comet is in equilateral triangle with Earth and Sun. Each side is 1 AU. Phase effect is neglected because head radiates in all directions with no shadowed side. As the Earth-comet side varies, the comet's magnitude changes according as the Inverse-Square Law. When the comet-Sun side varies the comet changes its reflected light by the Inverse-Square Law AND ALSO its luminous output induced by interaction with solar radiation. This additional light is governed by an other power law, not in general an inverse square. What is this other law? With no decent comet model until the explorations by spacecraft we could only cut-&-try a value for this other power. Choice was taken from experience with prior comets which behaved similarly to the instant one. Crashing together these factors and jumping directly into logarithms, we have the comet magnitude formula: +---------------------------------------------+ | COMET MAGNITUDE FORMULA | | | | Mapp = Mabs + 5 * log(r) + 2.5 * K * log(R) | +---------------------------------------------+ r is the comet-Earth distance in AU; R,comet-Sun distance in AU. K is the slope or gradient parameter. It is composed of 2 for the reflected light and some other number for the solar-induced light. The symbols differ widely among authors with no longer a strong effort for standardize them. One other common statement is m1 = m0 + 5 * log(DELTA) + 2.5 * n * log(r) The symbols here line up with those in the boxed equation. The equation's circumstant text should explain the symbols in each instance. The value of K can range from 2 for a dead worn-out comet to 4 or 5 for a vigorous active comet jived by the Sun. One practice for brand-new comets is to assign K = 4 and hope for the best downrange. Some astronomers combine K and the 2.5 factor into a one parameter. In such situations the slope parameter may range from 5 to 8 or 10. As the comet does its round thru the solar system, its behavior may depart form the current equation. News for the comet may contain revised values of Mabs and K to keep pace with the comet's current activity. In such news the 2.5 and K are commonly combined into a single number, ranging from 5 to 10 or 12.5. Since the mid 1990s as we learned more about comets from spacecraft visits and better comet models, the comet magnitude equation is in a steady decline. Astronomers who today hear abut it often must examine old comet litterature for information. Conclusion -------- Astronomers, on and off campus, learn of these various magnitude formula as separate topics with almost no attempt to correlate them. Here we see that the one Inverse-Square Law and the definition of 'magnitude' are the root of all these formulae. They are simple permutations of each other. Because the formulae are part of topics that can be scattered along the tuition of astronomy, it may be clumsy to collect them into a single lesson. Perhaps near the end of a course, when the separate forms are in hand, a summation can be offered..