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Some Consequences of Kepler, Galileo, and Newton Equations

§ Determining mass of earth:

earth_mass.png

§ Determining earth's velocity around the sun: 

earth_velocity.png

§ Determining the mass of the sun:

The centripetal force exerted by the sun keeping earth in its near perfect circular orbit is equal but opposite to earth's centrifugal force as follows

centripetal_force.png.

And because of Newton's Law of Universal Gravitation, we equate the following:

sun_mass.png

Notice that if we take the ratio of sun mass to earth mass we get the following:

sun_earth_ratio.png

which means that the sun's mass is at least 300,000 times greater than earth's!!

§ Determining the mass of any other solar system planet:

Simply use the following Newton equation where the sun's mass and centripetal force will determine the planet's orbital velocity as follows

planet_mass.png

which is equivalent to

planet_mass2.png.

Now, in our solar system all of the planets except Mercury, Mars and the outer, dwarf - planet Pluto are almost perfectly circular. And for these non - circular elliptical orbits, radius r is simply replaced by the semi-major axis of the orbital ellipse.

Finally, notice that Newton's equations validate Kepler's 3rd Law ( Harmonic Law ) - i.e., the square of a planet's orbital period is directly proportional to the cube of the planet's mean distance ( semi-major axis of the planet's elliptical orbit ) from the sun because of this simple rearrangement of the above terms:

kepler_3rd_law.png.

§ The best method for determining the mass and surface acceleration of any solar system body:



Deep Impact's July 4, 2005 encounter with comet 9P/Tempel 1. When the impactor separated and flew into Tempel 1, Deep Impact spacecraft was orbiting at about 10,000 km above Tempel 1's surface. Both Deep Impact spacecraft and Tempel 1 at time of impact was approximately 0.89 AU from Earth and 1.5 AU from the Sun, the comet's perihelion elliptical distance.

Given today's Space Age technology, by far the best method for determining the mass of any solar system body such as an asteroid object, is by means of placing an artificial orbiting satellite into a near - perfect circular orbit of known radial distance to the object's center of mass and astronomically observing its period of revolution T  about the object's body. Then by employing Kepler's 3rd Law ( Harmonic Law )

kepler_3rd_law3.png

and knowing the artificial satellite's mass, mass_satellite.png , we get

planet_satellite_masses.png.

Since,  less_than.png this effectively yields

planet_mass_from_satellite.png.

And from instrument measurements onboard the artificial satellite as well as astronomical observations of the planet, it's possible to further determine the planet's radius, planet_radius.png, and at the planet's surface any mass has weight W and acceleration g as follows

planet_acceleration.png

§ Determining the distance of the earth to the moon:

In this calculation we basically only need to know what is the time period for the moon's rotational orbit from astronomical observation together with Euclid's geometry and apply the following Kepler's 3rd Law ( Harmonic Law ) as validated by Newton's Universal Law of Gravitational Attraction:  

earth_moon_distance.png

§ Determining mass of earth's moon:

As was mentioned above, the very best way of determining the mass of any solar system body is by placing an orbiting artificial satellite around that object and measure its period and distance to the center of the body - e.g. the moon. 

However, in times past this was not feasible and so therefore other means such as utilizing Euclid's geometry of proportions gave some approximate answers. See: "Copernicus of Antiquity" - Aristarchus of Samos [ circa 310 B.C. - 230 B.C. ]. 

(i). Nevertheless, here is the "quick and dirty" modern method:

moon_mass1.png

(ii). 2 - Body System Method number_5.png :

moon_mass2.png.

Notice, again, that essentially all we need to know to determine the mass of the moon are the astronomically observed orbital moon period ( sidereal month ) and the earlier derived quantities, moon_orbital_radius2.png and earth mass earth_mass3.png !!

(iii). The accepted derived quantities for earth and moon therefore are

derived_quantities1.png

(iv). Deriving the barycenter  for earth - moon ( common center of mass for earth - moon ):

Since this again involves a 2 - body system analysis, it's recommended that you first visit "The Two - Body Problem" web page number_5.png.

barycentric_equation.png

earth_moon_center_mass.png

Notice: the earth - moon barycenter is approximately 3/4 of earth's radius and resides 1/4 of the way inside earth's crust!

§ Determining the masses of near and distant disk galaxies:

Amazingly, we use Newton's and Kepler's Laws of motion! This is accomplished first by astronomical observations of the outer most "spiral tail" of disk galaxies to determine both the radial distance to the galactic center as well as the period of rotation about the galactic center for these tails. Of course, both in terms of distances and rotational periods these quantitative elements are neither easy to obtain nor are they anything within normal human experience. In fact, everything regarding cosmic expanses is totally behind normal human experience. But the further amazing thing is that these cosmic quantities and their related cosmic galaxies are not beyond human knowledge and understanding!! This in turn leads to applying Newton's Law of Gravitational Attraction and Kepler's 3rd ( Harmonic ) Law as we previously did in the following manner:

The problem with determining the mass of our Milky Way ( The Galaxy; Latin: "Via Lactea" derived from Greek word "Kiklios Galaxios" which translates as "milky circle" ) is that we can't exactly "see" our Milky Way tail since our platform for observation is earth within our own solar system. Our solar system in turn is traversing a nearly circular orbit within the inner rim of the Orion Arm of The Galaxy at about 40 - 50% distance away from the Galactic Center with an approximate velocity of 220 kilometer per second equivalent to one (1) light year in about 1,400 earth - years or one (1) AU every 8 earth days and completing one "galactic year" in about 225 - 250 million earth years. Nevertheless, Newton's and Kepler's Laws determine Milky Way galactic mass and with some extra luminosity observations we can further tweak the mathematics for Milky Way mass.

Notwithstanding what amounts to a "galatic year" for earth's solar system, there is now a seminal study by the astrophysicists at the Cardiff Center for Astrobiology, Cardiff, Scotland, proposing that earth's solar system transits the plane of the Milky Way galaxy approximately every 35 - 40 million years with the consequent result of sending comet collisions into the earth itself on a regular 35 - 40 million year time scale. The meaning of this is that astrobiologists as well as earth scientists can now better understand the periodicity for crater occurrences on earth's surface as well earth's recurring mass extinctions, especially for the dinosaurs some 65 million years ago. source: http://www.world-science.net/othernews/080503_galaxy

Milky Way Galaxy
Milky Way Galaxy
Artist's concept of the Milky Way galaxy, with the "galactic bar" visible in the center. (Image by R. Hurt) source: NASA Press Release by Bob Silberg/PlanetQuest, February 14, 2006

Still, from greater astronomical observations of near and distant galaxies we discern that spiral and disk tails are traveling at doppler - shifted velocities [ note: using hydrogen's 21 cm radiation line ] greater than can be justified by the observed amount of light emanating from these galactic masses. That is, the strength of gravitational attraction between masses of bodies within any galaxy will determine spiral tail or other disk velocities. Therefore since there is such a wide disconnect or disparity in so many galaxies as between observed velocities which otherwise would tear these galaxies apart and observed [ light illuminated ] total mass of bodies providing glactic [ gravitational ] "glue", hence logically there must exist within so many galaxies hidden or "dark" gravitational forces which in turn can only arise from hidden or "dark matter".

Dark Matter
dark matter
Estimated distribution of dark matter and dark energy in the universe. source: NASA Press Release by Bob Silberg/PlanetQuest, February 14, 2006


Solar System Attributes
Name Equatorial
diameter[a]		 Mass[a] Gravity[a] Orbital radius (AU)
(semi-major axis)		 Orbital period[a]
(sidereal)		 Inclination
to Sun's equator (°)		 Orbital
eccentricity		 Rotation period[a]
(sidereal)		 Moons Rings Atmosphere Black-box
temperature[a] (K)
Terrestrials Mercury 0.383 0.0553 0.378 0.38709893 0.241 7.00487 0.2056 58.785 — no vacuum 1.740
Venus 0.949 0.815 0.905 0.72333199 0.615 3.39471 0.0067 -243.686[d] — no CO2, N2 0.911
Earth[b] 1.00 1.00 1.00 1.00 1.00 7.25[c] 0.0167 1.00 1 no N2, O2 1.00
Mars 0.533 0.107 0.379 1.52366231 1.881 1.850 0.0935 1.029 2 no CO2, N2 0.826
Gas giants Jupiter 11.209 317.83 2.530 5.20336301 11.862 1.304 0.0489 0.415 63 yes H2, He 0.433
Saturn 9.449 95.159 1.065 9.53707032 29.457 2.485 0.0565 0.445 56 yes H2, He 0.319
Uranus 4.007 14.536 0.905 19.19126393 84.011 0.772 0.0457 -0.720[d] 27 yes H2, He 0.229
Neptune 3.883 17.147 1.14 30.06896348 164.79 1.769 0.0113 0.673 13 yes H2, He 0.183
Dwarf planets Pluto 0.187 0.0021 0.059 39.48168677 247.68 11.880 0.2488 -6.405[d] 3 no N2, CH4 0.147
Eris 0.19 0.0025 0.0816 67.6681 - 44.187 0.44177 > 8 h? 1 no temporary -
Makemake 0.12 0.0008 0.0510 45.791 unknown 28.960 0.1590 unknown 0 no temporary -
Barycenter Sun 109.2[e] 333,000 28.0 distance to earth[f] - - - 25.449 - - H, He 17.3024
a  measured relative to the Earth.
b  earth reference = 1
c  earth reference = 0
d  retrograde rotational motion contrary to similar bodies within earth's solar system
e  volumetric mean diameter
f  mean: 149.6 x 106 km
minimum: 147.1 x 106 km
maximum: 152.1 x 106 km

The Local Group of Galaxies∗ ( LG ) - comprises over 50 galaxies
Satellites distance to solar system[a] diameter[b] mass[c] number of stars shape[d] other characteristics
Milky Way
(The Galaxy)		 approx. 13 2.5 x 104 ly
(galactic center to earth)		 (1.0 - 1.2) x 105 ly
10,000 ly thick		 5.7 x 1011 solar masses
including dark matter: 1.9 x 1012 solar masses		 200 - 400 billion barred spiral Second largest galaxy in the group-most massive of Local Group due to dark matter
Canis Major Dwarf Galaxy 2.5 x 104 ly winding stream 1.0 x 109 solar masses 1.0 x 109 forms a ring around The Galaxy orbits while merging into The Galaxy
Virgo Stellar Stream 3.0 x 104 ly 3.0 x 104 ly (length) -- few 100-thousands stream merging into The Galaxy
Sagittarius Dwarf Elliptical Galaxy (SagDEG) 8.0 x 104 ly 1.0 x 104 ly 5.8 x 107 solar masses 1.0 x 109 elliptical will collide with The Galaxy in next 100 million years
The Large Magellanic Cloud Galaxy ( LMC ) 1.8 x 105 ly 25,000 ly 10.0 x 109 solar masses 10.0 x 109 very prominent center bar probably an early barred spiral
The Small Magellanic Clouds ( SMC ) 2.1 x 105 ly 15,000 ly 2.0 x 109 solar masses 2.0 x 109 contains center bar probably an early barred spiral
Andromeda
(M31)		 approx. 14 or 15 dwarf galaxies 2.54 ± 0.06 M ly 2.2 x 104 ly 7.1 x 1011 solar masses
including dark matter: 1.23 x 1012 solar masses 		1.0 x 1012 spiral largest member of group - nearest spiral galaxy to Milky Way approaching the sun with velocity approx. 300 km/sec
Andromeda I 2.43 M ly 2,200 ly ? 100 variable stars dSph estimated age is approx 10 billion yrs
Andromeda II 2.13 M ly 2,000 ly 2 billion solar masses ( dark matter ) 73 variable stars dSph
Andromeda III 2.44 M ly 3,000 ly ? 56 variable stars dSph
Andromeda V 2.52 M ly ? ? ?
Andromeda VI ( Pagasus Dwarf ) 2.55 M ly 3,000 ly ? 118 variable stars
Andromeda VII ( Cassiopeia Dwarf ) 2.49 M ly 1,900 x 1,500 ly ? dSph dominated by very old stars, with ages up to 10 billion years
Andromeda VIII 2.7 M ly length 30,000 ly ? 400,000 elongated stream widespread and transparent
Andromeda IX 2.5 M ly 3,000 ly probably mostly free-floating dark matter mini-galaxy dSph most diffuse known galaxy - the least luminous galaxy
Andromeda X 2.9 M ly ? dim dwarf most faint satellite galaxy - lies about 280,000 - 450,000 ly from M -31
Triangulum Galaxy ( M33 ) 2.59 M ly 50,000 ly 10 to 40 billion solar masses 10 - 40 billion spiral
SagDIG[e] --- 3.5 M ly 1.5 x 103 ly high gas/mass ratio 665 million ± 5% dwarf irregular at the remotest edge of the Local Group
Barycenter located somewhere between the Milky Way and the Andromeda Galaxy - -
∗ caveat lector! reader, please beware of these unsettled observations and numbers.
a,b  ly = light year; M ly = million light years
c  solar mass = sun's mass
d  shapes are disk, elliptical or non - uniform irregular; dSph = dwarf spheroidal
e  Sagittarius Dwarf Irregular Galaxy; ancient galaxy formed in early universe; highest gas/star ratio ( i.e., low - mass ) making it ideal for study of star formation;

relatively few elements heavier than helium unlike our sun, hence metal - poor galaxy.



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