Some Consequences of Kepler, Galileo, and Newton Equations∗

"Science is a differential equation; Religion is a boundary condition" - Alan Turing ( 1912 - 1954 )

"Too Soon from the Cave, Too Far from the " - Ray Bradbury ( 1920 - 2012 )

[ ∗ note: most of the following examples are used in the future upcoming 'Relativity Science Calculator' Mac software application ]

§ Determining the mass of the earth:

§ Determining earth's velocity around the sun:

Relativity Science Calculator § Determining the mass of the sun:

NASA: Giant solar eruption - April 16, 2012

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

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

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

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

Relativity Science Calculator which is equivalent to

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 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:

§ 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 )

and knowing the artificial satellite's mass, , we get

Relativity Science Calculator Since, this effectively yields

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, , and at the planet's surface any mass has weight W and acceleration g as follows

§ 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:

Relativity Science Calculator § 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 Chapter: "Early Models of the : Aristarchus of Samos [ circa 310 B.C. - 230 B.C. ].

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

(ii). 2 - Body System Method:

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. .

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, and earth mass, !!

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

(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" chapter.

Relativity Science Calculator 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 still 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 ʼseeʻ 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 ( about 30,000 light - years ) away from the Galactic Center, at about 15 light - years above the plane of the Milky Way disk, with an approximate velocity of 220 - 234 kilometer per second, which is equivalent to one (1) light year in about 1,400 earth - years or one (1) AU every 8 earth days. The Sun, and hence the Earth, completes one "galactic year" in about 225 - 250 million earth - years and has made 27 round trips since its earliest formation. 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

Relativity Science Calculator 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 "galactic [ 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

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 Orbital Orbital Inclination Rotation Black-box Equatorial radius (AU) Orbital Name [a] [a] [a] to Sun's [a] Moons Rings Atmosphere [a] [a] Mass Gravity (semi-major period eccentricity period temperature diameter equator (°) axis) (sidereal) (sidereal) (K) 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 [d] — no CO2, N2 0.911 Terrestrials 243.686 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

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 Gas giants 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 14 yes H2, He 0.183

Pluto 0.187 0.0021 0.059 39.48168677 247.68 11.880 0.2488 -6.405[d] 5 no N2, CH4 0.147 Dwarf planets 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 - distance to Barycenter Sun 109.2[e] 333,000 28.0 - - - 25.449 - - H, He 17.3024 earth[f] 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

Relativity Science Calculator The surrounding the Milky Way

source: author Richard Powell, http://www.atlasoftheuniverse.com/localgr.html

Relativity Science Calculator 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 Second largest galaxy in the group - most 2.5 x 104 ly (1.0 - 1.2) x 105 ly 5.7 x 1011 solar masses barred approx. 13 200 - 400 billion massive of Local (galactic center to earth) 10,000 ly thick 12 spiral including dark matter: 1.9 x 10 solar masses Group due to dark matter forms a ring orbits while merging Canis Major Dwarf Galaxy 2.5 x 104 ly winding stream 1.0 x 109 solar masses 1.0 x 109 around into The Galaxy The Galaxy Milky Way 3.0 x 104 ly few 100- merging into The Virgo Stellar Stream 3.0 x 104 ly -- stream (The Galaxy) (length) thousands Galaxy will collide with The 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 Galaxy in next 100 million years very prominent probably an early The Large Magellanic Cloud Galaxy ( LMC ) 1.8 x 105 ly 25,000 ly 10.0 x 109 solar masses 10.0 x 109 center barred spiral bar contains probably an early The Small Magellanic Clouds ( SMC ) 2.1 x 105 ly 15,000 ly 2.0 x 109 solar masses 2.0 x 109 center barred spiral bar Largest member of group - nearest spiral 7.1 x 1011 solar masses galaxy to Milky Way approx. 14 or 15 dwarf galaxies 2.54 ± 0.06 M ly 2.2 x 104 ly 1.0 x 1012 spiral including dark matter: 1.23 x 1012 solar masses approaching the sun with velocity approx. 300 km/sec 100 variable estimated age is 2.43 M ly 2,200 ly ? dSph stars 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 2.52 M ly ? ? ? 118 variable Andromeda Andromeda VI ( Pagasus Dwarf ) 2.55 M ly 3,000 ly ? stars (M31) 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 elongated widespread and Andromeda VIII 2.7 M ly length 30,000 ly ? 400,000 stream transparent most diffuse known Andromeda IX 2.5 M ly 3,000 ly probably mostly free-floating dark matter mini-galaxy dSph galaxy - the least luminous galaxy most faint - lies about 2.9 M ly ? dim dwarf 280,000 - 450,000 ly from M -31 Triangulum 3rd largest member of Galaxy ( 2.59 M ly 50,000 ly 10 to 40 billion solar masses 10 - 40 billion spiral group M33 ) dwarf at the remotest edge [e] --- 3.5 M ly 1.5 x 103 ly high gas/mass ratio 665 million ± 5% SagDIG irregular of the Local Group located somewhere between the Milky Way Barycenter - - and the

∗ 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/ ratio ( i.e., low - mass ) making it ideal for study of ; relatively few elements heavier than helium unlike our sun, hence metal - poor galaxy.

Relativity Science Calculator The Alma Telescope

Relativity Science Calculator § Cassini Virtual Tour - A Fun JPL-NASA Saturn and its Moons Excursion

§ Live Real Time Satellite Tracking and Predictions

§ References 1. "Dwarf Galaxy Planes: The Discovery of Symmetric Structures in the Local Group", authors Marcel S. Pawlowski, Pavel Kroupa and Helmut Jerjen, Monthly Notices of the Royal Astronomical Society, Vol. 435, No. 3, pages 1928 - 1957; November 1, 2013

2. "The Distribution of Satellite Galaxies: The Great Pancake", authors Noam I. Libeskind et al., Monthly Notices of the Royal Astronomical Society, Vol. 363, No. 1, pages 146 - 152; October 11, 2005

3. Instead of WIMPS, weakly interacting massive particles, or axions, rather dark matter may be massive, according to physics professors Glenn Starkman of Case Western Reserve University and David Jacobs of University of Cape Town, in their joint paper "Macro Dark Matter".

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