THE MILKY WAY'S BLACKHOLE ----------------------- John Pazmino NYSkies Astronomy Inc www.nyskies.org nyskies@nyskies.org 2018 August 3 initial Introduction ---------- Blackholes and their behavior from Einstein's relativity theory in astronomy are an enduring attraction for home astronomers. This comes largely from the absence of easily appreciable relativity effects in everyday life and traditional astronomy. NYSkies from time to time features blackholes or relativity at its Seminars, the last being in May 2018. In July 2018 interest in blackholes spiked with news about the specimen at the center of the Milky Way galaxy. A star orbiting this blackhole exhibited redshifted radiation due to immersion in and extremely fast motion in the blackhole's strong gravity field. This event was hailed as major evidence favoring Einstein's relativity theory and our models of blackholes. As complex blackhole physics can be, much of it is surprisingly easy for home astronomers to enjoy with only high school algebra. Could this observation of this star's behavior be calculated by ordinary home astronomers? Yes, and the results are amazingly consistent with the more elaborate methods used by campus astronomers. It is easy -- exciting, fun, satisfying -- to go thru the maths. I appreciate that many readers took high school maths some long while ago. OK, have to hand an algebra review book. With the steady increase in blackhole studies the word is shifting to a one-word spelling 'blackhole'. The two-word spelling remains in wide circulation. Milky Way's blackhole ------------------- The blackhole at the center of the Milky Way galaxy was suspected in the 1990s. It was confirmed in 2008 by realizing that certain stars near it were not in random free fall but in orbit around an empty point. One star, named S2, actually completed one full orbit revolution around this point in 20088. The blackhole mass was derived from Kepler analysis of the orbits, yielding some 4.1 million Sun. While this is huge, it is nothing against the central blackholes in some other galaxies. They range into the hundred millions and billions of Suns. For many years the only way to study the Milky Way blackhole, MWBH, was via its orbiting stars. Their paths were distorted from pure Kepler paths by the spacetime gradient near the MWBH. There are no nebulae or stars close enough to interact by matter with it continuously. In this feature it is unlike a blackhole in a binary star system ro the gas-filled core of an other galaxy. In such situations gas from the regular star or enveloping nebula cascades onto the blackhole to generate X-ray and gamma-ray emission. Cygnus-X1 was discovered first as an X-ray source, then recognized as a binary star component. Early in its study the MWBH was taken as the very Sagittarius A* object, a long-known radio source near the geometric center of the galaxy. Thr MWBH is a distinct object near to, and perhaps associated with, Sagittarius A*. Some images of th galactic center label A* and MWBH separately. Other blackholes in Milky Way --------------------------- In 2014 a gas cloud bypassed the MWBH to possibly be torn apart by the steep gravity gradient. While it was swerved from its approach path and was distorted in size and shape, it survived the pass. The changes in this cloud, named G2, were an other strong evidence in favor of relativity theory. In 2017 a second massive blackhole was found near Sagittarius A*, this one within a nebula, whose circulation of material gave it away. This blackhole is only a few hundred thousand solar masses. Also in 2017 there arose the prospect of some 100 million star- mass blackholes scattered thruout the Milky Way. This was figured from the plausible number of isolated supernovae that turned into blackholes over the life of the galaxy. So far none were found probably because they are so isolated from other material to interact with. They possibly could e found gy their gravitational lensing of background objects. In May 2018, apart from star S2, suspicion sprang up of many thousands of blackholes emerged. These seem, so far as at mid 2018, to be star-size, ones to tens of solar masses. his was suggested by apparent agitated circulation in nodules of nebula in the galactic center. star S2 ----- The stars orbiting the MWBH are named 'S-number'. One, S2, attracted intense attention in the late 2010s because it was approaching its pericentron, closest point in its blackhole orbit. Will this star's radiation show a redshift due to the deeper gravity or from its extra high speed? It should but will the redshift in be severe enough to separate from the Doppler shift of a kepler orbit? One prior pericentron was observed in 2002 but results were not definitive due to use of prior-generation instruments. Pericentron occurred on 19 May 2018. Observations and conclusions were released in late July 2018. Yes, star S2 had excess redfshift beyond that from Doppler motion. If expressed as a false motion, star S2 was travelling about 200KPS faster than in a pure kepler orbit. The news galvanized vigorous discourse at club meetings. At starviewings in summer 2018 clubs showed the galactic center where the blackhole lives, hiding behind opaque nebulae. The star itself seems to be a Main Sequence supergiant, spectral class B0-V. This was determined when the star was well away from pericentron in weaker, more Euclides-Newton, spacetime. The spectrum implies a mass of 10-15 Suns, luminosity of many thousands of Suns. It shines faint in infrared, common band used for galactic center work, because most of its radiation is emitted in the blue and shorter wavelengths. The other stars in the galactic center seem to be supergiants. This could be an instrumental bias forcing less luminous stars to miss detection. Peri-what? -------- In classical celestial mechanics the closest point in an orbit to the central attrahent is 'peeri-' attached to the neuter-nominative form of the Greek name of the body. For an orbit around Jupiter, this point is the perizeon. When binary stars were discovered around 1800 the closest point was named priastron. Beyond that, in the 20th century , there was no proper way to coin terms for the closest point around newly orbited bodies. Sadly for classicists, bastard terms are in circulation from the Space Age. A spacecraft around Jupiter passes thru a perijov or perijup[!]. The situation now is that we rad many terms for the closest point in a blackhole's orbit. Some are: ---------------------------------------------- * periastron, Greek for 'star', 'celestial body' * pericentron, Greek for 'center' * perinigricon, Lain for 'black thing' * peribothron, Greek for 'pit', 'pothole' ----------------------------------------- I stick with prericontron in this piece. Do note that for the mathematical figure of an ellipse the term is periapsis, 'nearby apse'. Observing the MWBH ---------------- We do not opticly see the Milky Way center, with its blackhole, because it is obscured by opaque nebulae in spiral arms between us and the center. The hotspot on the Milky way band in the sky is the feeble showthru of the intensely luminous center. In the clear it would create a full Moon level of ground lighting. We first examined the center via radio wavelengths in the late 1940s. The intervening nebulae are transparent to certain radio bands, allowing us to map out the spiral arms of our galaxy and determine the location of the galactic center. We were confident of a good fix by the mid 1950s that we redimensioned the galactic coordinate grid to put its origin at the galactic center. The radio source Sagittarius A* is almost smack at the center, with the blackhole nearby. That the blackhole has a small nonzero galactic coordinate doesn't bother us, When astrophysical spacecraft were fielded in the 1960s we found many X-ray and gamma-ray sources in the direction of the Milky Way center. In the 198os we found that certain infrared wavebands penetrate the Milky Way nebulae to unveil the galactic center. We found many new sources sending out radiation in IR bands. It was impossible to secure a parallax on objects in the Milky Way center for instrumental constraints. At their distance of some 8,000 parsecs, based on models of the galaxy, the parallax was well over an order smaller than the best optical parallaxes we could capture. The GAIA spaceprobe can measure parallaxes of microarcseconds but it works only in the optical bands. It can not see into the center and, in fact, is limited in range by the 'fog' of interstellar medium. Infrared observations are important also because relativity effects near the MWBH increase the wavelength of emitted radiation. The increase in extreme situations may, as we observe from a safe distance away, push optical emission into the infrared region. Parameters of MWBH and S2 ----------------------- Here I collect some parameters for the star and blackhole for ready reference. ------------------------------------- parameter | value | remarks ----------------+-------------+-------- S2 orbit period | 16.05 year | 1st full lap in 2008 S2 semimajor ax| 1,030 AU | = (parsec dist)*(ang dim) S2 excentricity | 0.884 | storng elliptical orbit S2 spectrrum | B0 V | upper end of Main Sequence S2 Sun mass | 10-15 | from mass-luuminosity rule pericentron time | 2018 May 19 | 2nd pericentron, 1st in 2002 event horizon | 12.3e6 km | = 0.0822 AU pericentron dist | 120 AU | = (SMA)*(1-excenty)| BH distance | 8,000 pc | close to Sagittarius A* BH sun mass | 4.1 million | small for central BH ------------------------------------------- All values are subject to revision as vigorous examination of the MWBH continues. Excess redshift of S2 ------------------- Star s2's orbital motion is monitored by the radial speed Doppler method and astrometry. The speed obtained while s2 is far from pericentron matches closely that expected from a Kepler orbit. Near pericentron there appeared in the spectrum of S2 a displacement of wavelength that exceded the Kepler amount. Some of this was detected at the 2002 pericentron but the earlier vintage instruments gave only persuasive results. The extra spectral displacement comes from Einstein's time dilation feature of rapid motion and strong gravity. The two causes are distinct. The motion dilation effect is part of special relativity; gravity dilation, general. Both effects were observed successfully else where in astronomy and space flights, but mostly were considered too obtuse for home astronomers to understand. . One early trip-up for many readers was the way the news media presented the observations of star S2. S2 displayed the extra redshift came only from gravitational redshift, missing out attention to the motional redshift.. When readers versed in Einstein physics tried to replicate the gravitational redshift they came up with quite one half of the claimed amount. They applied the correct formula, inserted correct parameter values, checked the arithmetic, and still fell short. What went wrong? The news was incomplete. The discovery article presenting the observations and conclusion stated clearly that the excess was the sum of the both relativity effects:: gravity and speed. The time dilation redshift from each cause just happened to be about equal, roughly 100KPS. Readers working ou only the gravity effect came up with an amount quite one half the 200KPS noted in the litterature and they went crazy looking for their mistake.. Mass of MWBH ---------- Orbital motion can determine the mass of the attrahent body. Most home astronomers know Kepler's Law of Periods for the solar system: (period)2 = (semimajpr axis)3. The units are years and AU. Newton elaborated the law as (period)^2 = (semimajor axis)^3 / (mass1 + mass2) mass1 for the solar system is the Sun; mass2, the planet, both in solar units. It turns out that the mass of a planet is very small against the Sun, no more than 1/1,000 for Jupiter. (mass1+mass2) is nearly one for any planet, fooling Kepler to miss out this mass term in his law. The Newton modification is routinely applied to binary stars, where mass1 and mass2 are of the same order, the both bodies being stars. Since the observed parameters are period and semimajor axis, the binary star formula is (mass1 + mass2) = (semimajor axis)^3 / (period)^2 Note well that only the SUM of the wo masses is obtained. In binary star work other information, like from spectrometry, allocates mass1 and mass2 between the stars. This formula was used to find the mass of the MWBH. Star S2 over most of its orbit is in gravity too weak to significantly distort its geometry off of a straight Kepler path. Observation of S2 for some 20 years revealed its period as 16.05 years and its semimajor axis as 1,030AU. Mass1 is the blackhole; mass2, S2. Even as a supergiant S2'd mass is vanishingly small against the blackhole. We have (mass1 + mass2) = (semimajor axis)^3 / (period)^2 (mass1 + 0) = (semimajor axis)^3 / (period)^2 (mass1) = (semimajor axis)^3 / (period)^2 = (1,030AU)^3 / (16.05yr)^2 ) / (256) = 4.242e6 -> 4.2 million Sun This is agreeable with the generally cited 4.1 million Suns. Event horizon ----------- Typical astronomy instruction explain that the gravity field of a point mass is an inverse square function of the distance from the point. The field strength decreases with increasing distance. This is a straight reading of Newton's law of gravity. Passed over for most of astronomy is the INCREASE of gravity field strength with with PROXIMITY to the point. The strength grows without limit, toward infinity, for extreme proximity. Newton recognized this as some horrible flaw in his theory. The gravity strength rises to infinity at zero distance from the point. No, he did not discover the blackhole effect. He explained away the problem by noting that celestial bodies have solid surfaces. These prevent extreme approach. stopping it at a finite distance from the point. The gravity field pins at a maximum finite value. An other feature of the gravity field is the energy needed to escape it fro a given distance away. This is usually cited as a upward, vertical, velocity. This is based on the only reasonable way to give an object enough kinetic energy to leave the gravity field. This is by impulse, like from a rocket launch. The escape speed increases toward infinity with proximity to the attrahent body. At some distance the escape speed equals lightspeed, a. The escape speed continues to increase closer to the body but the faster-than-light motion is still beyond today's physics to properly understand. The distance out from the gravity source where the escape speed is lightspeed is the event horizon or Schwarzchild radius. The latter honors Schwarzchild's description of blackholes as a feature of Einstein physics. The body doesn't have to actually be a blackhole. Any object has an event horizon that would surround it if the body some how became a blackhole. The event horizon for a given mass is R| = 2 * gamma * mass / (lightspeed)^2 Putting in values for the Sun, R| = (2) * (6.674e-11m3/kg.s2) * (1.99e30kg) / ((3e8m/s) ^2) = 2.96e3 m -> 3 km Since R| is proportional to mass, it is easiest found for any other body by ratio to the Sun. R| = (3km/Sun) * (Sun mass) The Milky Way blackhole has event horizon of radius R| = (3km/Sun) * (Sun mass) = (3km/Sun) * (4.1e6 Sun) = 12.30e6 km -> 0.0822 AU. If this blackhole replaced the Sun, its event horizon in Earth's sky would be some9-1/2 degrees diameter. The interior would be void of all radiation. There would be extreme gravitational lensng around the event horizon. radial Doppler shift -------- --------- From the discovery of spectral shifts in the 1830s thru the 1940s there was only one astronomy cause for the shift. It was produced by real spatial motion in the line of sight, radially, of the source. During this span all astronomy was carried out in the optical band, where the displacement was toward the red or blue end of the spectrum, whence 'redshift' and 'blueshift'. Einstein physics was well established, specially in atomic labs, but astronomers almost completely ignored it. Only in peculiar instance did any mention of relativity come into traditional astronomy. When in the 1940s radio, ultraviolet,X-ray, and other spectral zones were explored, the terms were applied to them. Redshift/blueshift refers to shifts toward longer or/shorter wavelength, lower/ higher frequency, lower/higher photon energy. The displacement of wavelength is a demonstration of time dilation that should be easily comprehended by home astronomers. it is, but hardly ever it is presented as such in the usual tuition. This is hardly ever revealed in typical tuition for home astronomers. The formula, missing out the derivation, is LAMBDA[ms] / LAMBDA[mm] = 1 / sqrt(1 - (vl / c)^2) * (1 + (vl / c)) This looks odd compared to the usual formula carried in astronomy for over a full hundred years. The statement has two components. The square root term is the Einstein time dilation generated by the radial movement of the source. The (vl/c) term is an anomaly that factors in for the travel time of radiation from the source while the source is moving along the line of sight. This anomaly increases with reproach; decreases, approach. The formula is arranged for a positive, wavelength displacement for a recession of the source. LAMBDA is the wavelength of radiation, such as a spectral line. [me] means the wavelength from the 'moving' source as perceived by the 'standing' observer. [mm] is the wavelength of the moving source as perceived by that same source. It is commonly called the 'rest' wavelength. vl (letter l, not number 1) is the line-of-sight speed and c is lightspeed. LAMBDA[ms] is the wavelength of the moving source radiation as experienced by the standing observer. LAMBDA[mm] is the wavelength from the moving source as experienced by the very moving source itself and is often called the 'rest' wavelength. In traditional astronomy vl is very small against c. Even a extreme speed of 1,000km/s is only about 1/3 of 1 percent of c. Stars and other bodies within the Milky Way have speeds of tens to hundreds of kilometers per second. For small vl, (vl/c) is small and (vl/c)2 is far smaller. The square root dwindles to (1/sqrt(1-(~0)) -> 1. This leaves the familiar formula LAMBDA[ms] / LAMBDA[mm] = 1 + (vl / c) vl / c = (LAMBDA[ms] / LAMBDA[mm]) - 1 Because for astronomy speeds the square root term is always very close to unity, it was not separately recognized in traditional spectrometry. All the wavelength displacement was assigned only to the (1+(vl/c)) term. When vl is large, a couple percent of lightspeed, the formula starts to break down. The square root term is not so nearly unity and the maths yield an excessive wavelength displacement . If the entire displacement is treated as a speed, the source has excess radial speed. of recession. Tangential redshift ----------------- The source can move at any angle of flight, or attack, against our line of sight. For the radial motion the radiation's travel time modulates the time dilation to cause a net red or blue shift. For tangential motion the radiation travel time is the same for all the arriving waves. vl is zero and the (1+(vl/c)) term collapses to unity. (1+(vl/c)) -> (1+(0/c)) -> (1+(0)) -> (1). Only the pure time dilation term is in force. lAMBDA[ms] / LAMBDA[mm] = 1 / sqrt(1 - (vt / c)^2) vt is the speed across the sightline. All other symbols are those for radial Doppler shift. Tangential redshift is also called transverse, orthogonal, redshift. From its association with the radial Doppler shift, 'Doppler' is commonly added into the name, like 'transverse Doppler shift'. Classical astronomy ignores tangential redshift because in all instances it is so small that it's just integral with the radial Doppler shift. When vt is large enough to make the square root term significantly off of unity, the tangential redshift shows up as a wavelength displacement thrown into a spurious radial speed. Tangential movement of the source is handled by measuring the source's proper motion across the sky. This first determined as an angular motion in arcsecond/year and converted to linear motion in kilometer/second by knowing separately the distance to the source. proper motion is best observable for nearby sources, else the angular displacement is too tiny to detect. Gravitational redshift -------------------- The radial and transverse Doppler shifts are features of special relativity affecting sources in motion against the observer. Gravitational shift is part of general relativity. The source is standing in a gravity field different from the observer's. If the source field is weaker, the shift is a blueshift; stronger, redshift. Astronomy applications were weak because until blackholes were known there was no really strong gravity field to show the redshift well. Near a blackhole the gravity increases toward infinite strength at the singularity. Before then, at the event horizon, the field is so strong that the emitted radiation is shifted to infinite length. it takes forever for one wave to arrive at the observer. The gravitational redshift, is LAMBDA[ms] / LAMBDA[mm] = 1 / sqrt(1 - (R| / R)) R| is the radius of the blackhole's event horizon; R, the distance of the radiation source from the singularity. [ms] and [mm] here refer to 'mobile' source and 'safe' observer. For all astronomy functions, the 's' observer is on Earth, a thoroly safe distance away, in gravity essentially zero compared to the blackhole. One hideous misunderstanding is that the source actually radiates at the longer wavelength, which travels unaltered to the safe observer. A person next to the source would see it redder than it should be. Not true.The radiation is produced at its physicly proper wavelength, being that physics works the same every where. It's only when the radiation is received in a different gravity regime that the gravitational shift is produced. Parallax and dimensions --------------------- Parallax in astronomy is the angular swing of our line of sight on a celestial target as we orbit the Sun. Ideally this angle is found by astrometry on the target from opposite sides of the Earth's orbit. This ideal situation is not realized in practice.n Measurements are taken as catch can over many years. The measure are then reduced to the radius of Earth's orbit. The angle between lines of sight is the apex of a long slender triangle: target-Sun, Sun-Earth, Earth-target. The Sun-Earth side is the base of the triangle, of length one orbit radius or one AU. The other two sides are essentially of equal length, the distance to the target. Because the parallax in astronomy is so small, none among the stars being so large as one arcsecond, we can apply the small-angle rules to work with parallaxes. In addition we define a new unit of distance, the parsec, as the distance of a target of one arcsecond parallax. Other distances are inversely proportional to this base definition. That is (parsec distance) = 1 / (arcsecond parallax) In familiar measure, 1 parsec is 206,265AU, commonly rounded to 200,000AU. It is also about 3.08 lightyeaars. An other way to understand parallax is to stand at the target and look at Earth's orbit. The angular radius of the orbit, the very apex angle of the triangle, is the parallax or the inverse of the parsec distance. Any other angular dimension at that distance is its linear dimension in AU. This is exploited in binary star work where the orbit parameters are cited in either angular or linear measure. We have (AU linear dimension) = (arcsec ang dimension) * (parsec dist) In fact, many parameters of the star S2 orbit were determined in this way. The distance to it is substantially that of the galactic center, some 8,000 parsecs. Gravity field at S2 ----------------- One initial question we can answer easily is: How strong is the gravity field at star S2? This is answered by proportion from Sun's field at Earth. If the Sun was replaced by the MWBH, of 4.1 million Suns of mass, the field at Earth would be 4.1 million times greater. Next, remove Earth to the distance of s2 from MWBH, 120AU. The gravity field weakens inversely with the distance squared, so (field at S2) = (4.1 million) / ((120)^2 = 284.72 -> 285 Star S2 suffers the blackhole's gravity field 285 tomes stronger than Earth does from the Sun. And it does so at three times the distance of Pluto from the Sun. Gravitational redshift at pericentron ----------------------------------- The strong gravity field of the MWBH shows up for the safe observer as an increased wavelength of radiation emitted from S2. From the formula for gravitational redshift we have LAMBDA[ms] / LAMBDA[mm] = 1 / sqrt(1 - (R| / R)) = 1 / sqrt(1 - ((0.0822AU) / (120AU))) = 1 / sqrt(1 - (6.850e-4)) 0 1 / = 1 / sqrt(0.99932) = 1 / (0.99965) = (1.000343) At first look this seems to be an awfully small excess of redshift. Was it some good feat to measure it? No. Redshifts of this order were routinely captured thruout the 20th century on spectrograms taken with chemical film and passive optics. The feat wasn't detecting the redshift but in the redshift being large enough to tell apart from the wavelength displacements due to s2's orbital motion. Pretending this redshift is a Doppler redshift, we have (v / c) + 1 = LAMBDA[ms] / LAMBDA[mm] (v / c) = (LAMBDA[ms] / LAMBDA[mm]) - 1 = (1.000343) - 1 = 0.000343 v = (0.000343) * c v = (0.000343) * (300,000km/s) = 102 km/s A miserable mistake is to think that star S2 is moving ~100KPS faster in its orbit than it should. The conversion of a redshift having nothing to do with motion into one caused by motion is not a sound and fair astronomy practice. This 102KPS is about 1/2 the generally cited 200-or-so km/s excess redshift. There are TWO components to the excess redshift, but in many astronomy media only the 'gravitational' part is described. 102im/s is just about 1/2 of the 200km/s stated in the litterature. There must be a factor of 2 missing? A wrong formula? A maths mistake? This gravitational redshift was only one of the two components in the excess redshift exhibited by star s2. he other is a redshift due to s2's high speed near pericentron. By chance this happens to be also quite 100KPS! The sum is spot on with the full 200KPS. Speed at pericentron --------- ------- One of the observational feats in the study of the MWBH was to confidently measure the displacement of S2 in space over only a day or two around pericentron. That's from a distance of some 8,000 parsecs! S2 was flying! The orbital mean speed, averaged over the whole orbit, of s2 comes from ratio on Earth's mean speed v = (30 km/s) * (1,030 AU) / (16.05 yr) = 1,925 km/s We didn't need the gamma or mass of the blackhole because we took the distance and time for a single orbit, regardless of how these parameters were generated by the blackhole. The star run a course 1,030 times in length and 16.05 times in duration than Earth's orbit. This is the 'circular' speed, that for a circular orbit of radius equal to semimajor axis. We could proportion off of Earth because it has an almost circular orbit with low exvcentricity. From orbital mechanics the speed in an orbit at radius r is vr / v = sqrt((2 / r) - (1 / a)) vr is the speed at radius r; v, the mean or circular speed; a, semimajor axis. For star S2 a is 1030AU and r is the pericentron distance of 120AU. We have vr / v = sqrt((2 / r) - (1 / a)) = sqrt((2 / 120) - (1 / 1031)) = sqrt((2 / ((0.116)) - (1 / 1)) = sqrt((17.167) - (1)) v = sqrt((16.167) = (4.029) vr = v * (4.029) = (1,925 km/s) * (4.029) = 7,740 km/s This is almost the cited value for the pericentron speed of star S2 -- and it is FAST! it's 2.58% of lightspeed! I'm surprised that this is agreeable with the litterature because i did not try to factor in the strike and dip of the orbit relative to our line of sight. Star S2 is as at mid 2018 the fastest known star in a ballistic trajectory. The previous record-holder is star US708, an 18th magnitude white dwarf about 2,000LY away near iota & kappa Ursae Majoris. Altho discovered in 1982, its radial velocity was first reliably determined in 2015 as 1,200km/s in recession. The star may be flung from a binary system when its companion supernovated. It seems that only the radial component of total speed was ever measured, so the star could be moving rather much faster. Assuming the tangential speed is also 1,200km/s, the flight angle on our sightline being 45 deg, the total speed would be about 1,700km/s. transverse redshift at pericentron -------------------------------- I did not work out the strike and dip of the star's orbit to separate the radial and tangential components of the pericentron speed. For now I let the entire speed tbe tangential. The redshift is the straight time dilation of motion LAMBDA[ms] / LAMBDA[mm] = 1 / sqrt(1 - (vr / c)^2) = 1 / sqrt(1 - ((7740km/s) / (3e5km/s))^2) = 1 / sqrt(1 - (0.0258)^2) = 1 / sqrt(1 - (6.656e-4) = 1 / sqrt(0.9933) = 1 / (0.9967) = 1.000330 Treated as a 'speed' we plug this into the radial Doppler formula v / c = (LAMBDA[ms] / LAMBDA[mm]) - 1 = (1.000330) - 1 = (0.000330) v = (0.000330) * (3e5 km/s) = (99.3896km/s) -> 100 km/s Total excess redshift ------------------- The total excess redshift, beyond that from a flat Kepler orbit is the sum of the gravitational and tangential redshifts (total excess redshift) = (gravitational) + (tangential) = (102km/s) + (100kn/s) = 202 km/s which is consistent with reported values. I remind that it is very misleading to convert redshifts from causes other than radial Doppler effect into 'speeds'. It is too easy to think tat star s2 some how moves 200 km/s faster than it should. Conclusion -------- News about blackholes always excites home astronomers. For the most part, they are wowed by the reports and pictures but feel they could not actually understand how blackholes work. Blackholes are part of relativity, a subject home astronomers still shy from. The behavior of star s2 near the Milky Way's central blackhole holds out a capital episode where home astronomers can exercise some of the calculations of the campus astronomers. And get results pleasantly close to those calculations. These calculations are a mix of familiar orbital mechanics and the surprisingly simple forms of Einstein physics. Only high school algebra is needed. It was only by chance that the gravitational redshift, which some news accounts stated as the full redshift, is quite 1/2 of the correct total redshift. This caused many home astronomers to go crazy looking for the error in their formulae or maths. When the two-part nature of the excess redshift is recognized, every thing works out well