Monday, March 22, 2021

3100 - GALAXIES - Don’t Follow the Laws of Gravity?

  -  3100  - GALAXIES  -  Don’t  Follow the Laws of Gravity?    Astronomers do not have a good idea of what Dark Matter really is.  It is something that has mass and is an attractive force for gravity.  But, it does not interact with the electromagnetic forces.  It does not absorb or emit light or any other part of the electromagnetic spectrum.

-------------------   3100 -   GALAXIES  -  Don’t  Follow the Laws of Gravity?

- Johannes Kepler ( 1571 - 1630 ) and Isaac Newton ( 1642 - 1727 ) invented the laws of gravity.  They developed the math that defines the force of gravity which is believed to be the same everywhere in the Universe.  

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-  Kepler developed his laws to explain the orbits of the planets.  Newton extended these laws to include any orbiting body.  The force of gravity’s attraction between any two bodies was proportional to the product of their masses and inversely proportional to the square of the distance between them.

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-    These laws work perfectly for the planets orbiting the Sun.  Why don’t they work for “suns” orbiting the Blackhole in the center of a galaxy?

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-  Kepler’s laws state that the planets orbit in ellipsis and not in perfect circles.  These elliptical orbits sweep out equal areas in equal times.  For example, a planet travels faster when it is nearer the Sun and slower when it is farther from the Sun.  Planets that are more distant orbit the Sun at a slower average speed.  Their orbits obey this precise mathematical relationship:

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-----------------  (period)^2  =  ( radius )^3

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-----------------  p^2  =  r^3

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-  The radius is measured in “Astronomical Units” and the “Period” is measured in years.  The radius is the average distance from the center, called the semi-major axis in the case of ellipsis.  Here is the data for the 8 planets in our Solar System:

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-----------  Mercury  -------------  0.387 AU  ------------------  0.2409 years.

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-----------  Venus  ----------------  0.723 AU  ------------------  0.6152 years

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-----------  Earth  ----------------  1.000 AU  -------------------  1.000 years

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-----------  Mars  ----------------  1.524 AU  -------------------  1.881 years

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-----------  Jupiter  --------------  5.203 AU  ------------------  11.86 years

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-----------  Saturn  --------------  9.539 AU  ------------------  29.42 years

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-----------  Uranus  -------------  19.19 AU  ------------------  84.01 years

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-----------  Neptune  -----------  30.06 AU  ------------------  164.8 years

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-  ********************   cubing the radius, AU^3,  and squaring the period , years2:

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-----------  Mercury  -------------  0.058 AU^3  ------------------  0.058 years^2.

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-----------  Venus  ----------------  0.378 AU^3    ----------------  0.378 years^2

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-----------  Earth  ----------------  1.000 AU^3    -----------------  1.000 years^2

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-----------  Mars  ----------------  3.54 AU^3    -------------------  3.67 years^2

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-----------  Jupiter  --------------  140.7 AU^3    --------------  140.9 years^2

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-----------  Saturn  --------------  866 AU^3    -----------------  868 years^2

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-----------  Uranus  -------------  7058 AU^3    --------------  7067 years^2

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-----------  Neptune  -----------  27,159 AU^3    -----------  27,162 years^2

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-  It is harder to measure the periods for galaxies since suns take millions of years to orbit their galaxy.  Our Sun takes 220,000,000 years for one revolution.  So, astronomers use the redshift in the Doppler Effect to measure their orbital velocities.  From the velocity we can calculate the period using the radius as the average distance from the center.

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---------------  Circumference  =  2*pi*r

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----------------  Velocity  = distance / time

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---------------  Velocity  =  2*pi*r / period

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----------------  Period^2  = radius ^3

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---------------  Period =  r^3/2

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---------------  Velocity  =  2*pi*r /  r^3/2

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--------------  Velocity  =  2*pi / r^½

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-----------  Mercury  -------------  0.387 AU  ------------------  107,376 miles per hour

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-----------  Venus  ----------------  0.723 AU  ------------------  78,295

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-----------  Earth  ----------------  1.000 AU  -------------------  67,110

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-----------  Mars  ----------------  1.524 AU  -------------------  53,688

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-----------  Jupiter  --------------  5.203 AU  ------------------  29,081

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-----------  Saturn  --------------  9.539 AU  ------------------  22,370

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-----------  Uranus  -------------  19.19 AU  ------------------  15,229


-----------  Neptune  -----------  30.06 AU  ------------------    3,798

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- Notice that the planets farther from the Sun have a slower orbital velocity.  That is how orbiting planets sweep equal areas in equal time during their orbits. 

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-   Now, let’s compare this result with the orbiting speeds of stars around galaxies, including the Sun around the Milky Way.  The table below has the radius in lightyears distance and the orbital velocity in miles per hour.

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---------------  3,260 lightyears  ----------------  98,965       miles per hour


---------------13,040  ----------------------------  282,757

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---------------26080  ----------------------------   386,777

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---------------39,120  ----------------------------  425,030

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---------------52,160  ----------------------------  439,123

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---------------65,200  ----------------------------  444,380

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---------------78,240  ----------------------------  446,281

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-  Note that the galaxy has a “flat rotational curve”.  As you get further form the center the orbiting velocity becomes a constant 450,000 miles per hour.  This is out to the edge of the galaxy which is radius 78,000 lightyears or 156,000 lightyears across. 

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-   The Milky Way is 120,000 lightyears across.  It too has this constant rotation speed.  All the galaxies measured have the same characteristic, a flat rotational curve.  But, this violates Kepler’s and Newton’s laws for gravity.  How can this be?  

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-  The explanation that astronomers have come up with is that galaxies have a lot more mass than we can see.  Maybe ten times more mass.  It is Dark Matter that surrounds the galaxy in a giant halo. 

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-   The Milky Way halo is 600,000 lightyears across.  Newton rewrote Kepler’s laws of orbit to apply to all orbiting masses.  He introduce the Gravitational Constant , “G”, a constant of proportionality for the units of measurement so we are not confined to Astronomical Units and Years as required by p^2 = r^2.

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----------------  p^2  =  4*pi*r^2  / G * (M1 + M2) 

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--------------  Velocity  =  2*pi / r^½

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--------------    r  =  4*pi^2 / v^2

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---------------  (M1 + M2)  =  r*v^2 / G

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--------------  The outer boundary of this galaxy is 78,240 lightyears.  

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--------------  The orbital velocity is 200 km/sec. 

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---------------  G = 6.67*10^-11 m^3/(kg*sec^2).  9.46 *10^15 meters / lightyear.  

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----------------  The Solar Mass of the Milky Way is 10^12 Solar Mass. 

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---------------- Substituting these factors in the above equation for total mass:


----------------  (M1 + M2)  =  1.65 * 10^13 Solar Mass.

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----------------  The mass of the galaxy is 16.5 times greater inside the boundary of the galaxy we can see and measure the redshifts of starlight.  But, because the velocities are constant regardless of the distance from the gravity towards the center there must be 10 times more mass outside this boundary.

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-  The bottom line is astronomy does not know why the laws of galaxy do not hold up for galaxies.  The theory that has legs is that Dark Matter does exist and that it surrounds every galaxy in the Universe. 

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-   Dark Matter is responsible for the structure of galaxies and galaxy clusters.  Dark Matter makes up 23% of all the matter/energy in the Universe.  Dark Energy being the other 72% and everything we can see only 5%. 

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-   Astronomers do not have a good idea of what Dark Matter really is.  It is something that has mass and is an attractive force for gravity.  But, it does not interact with the electromagnetic forces.  It does not absorb or emit light or any other part of the electromagnetic spectrum.

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-    The astronomers measured the orbital velocity of several more stars in the galaxy than I have in the table above.  They plotted velocity versus radius and then did a least squares regression analysis to get a mathematical equation for the flat rotation curve.  Here is the equation they found:


------------------  velocity  =  200 ( 1 - e^-r/4)

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------------------  where velocity is in kilograms per second.  “e” is the natural logarithm, 

 e = 2.71828.  Where 4 kilo parsec is the radius per 4 kilo parsec. 

   A kilo parsec is  3,260 lightyears.

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----------------  velocity  =  200 - 200*e^-r/4 


---------------  velocity grows logarithmical very rapidly from the center and quickly becomes a constant 200 kilometers / second, or 447,000 miles per hour orbital velocity.

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--------------------------------  What is the rotational period for the perimeter of the galaxy?

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------------- Period  =  2*pi* r / velocity.

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------------  Period  =  6.28* 74*10^19 meters / 2*10^5 meters / second.

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-----------  one year = 3.16 * 10^7 seconds.

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----------  period  =  730 million years.

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-  March 21, 2021      GALAXIES  -  the Laws of Gravity?       1132      3100                                                                                                                                                        

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--------------------- ---  Monday, March 22, 2021  ---------------------------






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