John Pazmino
 NYSkies Astronomy Inc
 2018 August 3 initial 

    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 

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 
   The other stars in the galactic center seem to be supergiants. This 
could be an instrumental bias forcing less luminous stars to miss 
    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 
    Sadly for classicists, bastard terms are in circulation from the 
Space Age. A spacecraft around Jupiter passes thru a perijov or 
    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 
    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 
    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 
    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 
    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 
    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 
    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' 
    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 
    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 
    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 
    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 
    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. 

    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