-------- 3812
- MERCURY -
closest planet to the Sun?
Messenger Spacecraft is orbiting Mercury so we can calculate the
planet’s mass. On May 26, 2012, the Messenger spacecraft is in
orbit. The math to put it in orbit was
developed in the 16th century.
-
------------------------- 3812
- MERCURY -
closest planet to the Sun?
- So Bob, Here is how we know the planet Mercury is mostly iron. Other metals evaporated.
- Tycho
Brahe used the newly developed telescope to accurately measure the orbits of
the planets. He died in 1601 and his
assistant Johannes Kepler inherited his data that he had collected. Kepler spent 20 years analyzing the
data. He concluded that to match the
data planets had to be in elliptical orbits about the Sun and not the perfect
circular orbits that the consensus at the time believed.
-
- His
mass analysis concluded that equal time intervals were swept by the radius
vector from the Sun to the planet as the planet orbited the Sun.
-
-
Kepler’s third law in math was that the time it takes for a planet to
make one full revolution around the Sun is proportional to:
-
-------------
Time^2 / Radius^3 = Constant.
R is half the major axis of the Ellipse.
-
- If
factors are proportional they can be turned into an equality with the right
Constant of Proportionality. Isaac
Newton did the math to accomplish this for Kepler’s laws. He first assumed that the orbit were
ellipses. He knew that a stable orbit
was a balance of forces.
-
- The
force of gravity obeyed the inverse square law. The force varied inversely as
the square of the distance between the masses.
-
------------------------- Gravity Force
= Constant / Radius^2
-
-------------------------- F
= Constant / R^2
-
------------------------- Acceleration of object in circular orbit =
velocity^2 / Radius
-
------------------------- a
= v^2 / R
-
------------------------ Newton had is own law of motion = Force
= mass * acceleration
-
------------------------- F =
m*a
-
-------------------------- Substituting: F
= m * v^2 / R
-
-------------------------- Setting forces equal --------
Constant / R^2 = m* v^2 /
R
-
------------------------- Solving for v^2 =
Constant * R / m * R^2
-
---------------------- Calculating the period of an orbit =
T = circumference / velocity
-
------------------------ T
= 2 * pi * R / v
-
------------------------ T^2
= 4 * pi^2 * R^2 / v^2
-
----------------- Substituting v^2 --------
T^2 = ( 4*pi^2 * m
/ Constant) * R^3
-
- This
was Kepler’s 3rd law of orbits because with a constant mass the
factor :
-
---------------------- ( 4*pi^2 * m
/ Constant) , is simply another constant.
-
---------------------- T^2
/ R^3 =
Constant
-
- Newton
defined the Constant of Proportionality with the Gravitational Constant ,
G.
-
----------------------- T^2
/ R^3 = 4 *
pi^2 /
G * m
-
--------------------- G
= 6.67*10^-11 meters^3 /
(kilograms * seconds^2)
-
-------------------- T^2
/ R^3 =
(5.91*10^11) * m
-
- We will
use this equation to solve for the mass of Mercury, but, first let’s try it out
on the International Space Station that we know is orbiting at 6,738 kilometers
radius. And we know the mass of the
Earth is 5.9 *10^24 kilograms.
-
------------------ T^2
= 5.91 ( 6.738*10^6)^3 /
5.9*20*10^24 = 30,640,000 seconds
-
------------------- T
= 92.3 minutes.
-
- The
International Space Station circles the Earth about every 90 minutes. You can see it. The scheduled pass over’s for your area are found
on the web.
-
- Now we
have confidence to use this math on the planet Mercury On April 25, 2011 the Messenger spacecraft
completed one orbit in 12 hours and 2 minutes.
The radius of the orbit was 10,124 kilometers.
-
---------------
T^2 / R^3
= 5.91*10^11 / m
-
---------------- m
= 5.91*10^11 * (
10.124 * 10^6 meters )^3 /
(43.32*10^3 seconds)^2
-
----------------
m = 3.27*10^23 kilograms
-
---------------
Another orbit on September 14, 2011
-
----------------
R - 10,085 kilometers
-
----------------
T = 11 hours , 58 minutes
-
----------------
m = 3.27 * 10^23 kilograms
-
- A third
orbit on May 25, 2012, yesterday:
-
----------------
R - 7,715 kilometers
-
----------------
T = 28,800 seconds
-
----------------
m = 3.27 * 10^23 kilograms
-
- The
elliptical orbits around Mercury are changing but the mass remains
constant. The radius, velocity, and
period all adjust to keep the equation constant working according to the laws
of physics.
-
- My
astronomy textbook had the mass of Mercury at 3.3 * 10^23 kilograms. Mercury is just 5.5% the mass of the
Earth. The old guys had it right.
-
-
Mercury smallest, densest, fastest planet. Here are some more messages from
Mercury. The Messenger Spacecraft has
been in orbit around Mercury since March 18, 2011. Mercury is the closest planet to the Sun and
it is the smallest and densest planet.
-
- You
would think it was the hottest (806F), but, Venus gets that honor due to its
greenhouse gases (854F).
-
- Mercury
is 3,000 miles in diameter and slightly larger than our Moon which is 2,159
miles diameter. There are two other
moons in the Solar System that are larger, Ganymede and Titan.
-
- Mercury
is mostly rock and metal. The purpose of
this calculation is to try to learn the proportions of each that make up the
core and the crust of the planet.
-
-
Messenger spacecraft’s orbits have allowed astronomers to calculate the
mass of the planet. The mass of Mercury
is 3.31 * 10^23 kilograms. Earth is 60
*10^23 kilograms (18 times more massive).
-
- This is
counter-intuitive, however , even though Mercury’s temperature rises to 806F it
is the fastest cooling planet. This is
because of the ratio of its surface area to its volume. By “cool” we mean the internal heat escapes
into space from the surface until both are at the same temperature.
-
- Mercury
cooled much faster than Earth because it is a much smaller planet. The total amount of heat contained in a
planet depends on its volume. However,
the amount that escapes into space only happens at the surface. The total time it takes a planet to lose its
internal heat is directly proportional to the surface area to volume ratio.
-
--------------------------- Ratio
= surface / volume
-
--------------------------- Surface area
= 4 * pi* r^2
-
------------------------- Volume
= 4/3 * pi * r^3
-
------------------------- Ratio
= (4 * pi* r^2 ) / (4/3 * pi * r^3) = 3 /
r
-
- Because
the radius is in the denominator larger objects have a smaller surface area to
volume ratio. Smaller objects cool
faster. Crushed ice will cool your drink
faster than larger ice cubes.
-
---------------------------- Radius of Mercury =
1,506 miles
-
---------------------------- Mercury Ratio
=
3 / 1506
-
---------------------------- Radius of Earth =
3,963 miles
-
---------------------------- Earth Ratio
= 3 / 3963
-
-
Comparing the two ratios:
-
--------------------- Mercury Ratio
/ Earth Ratio = (3 /
1506) /
(3 / 3963) = 2.6
-
- This
comparison tells us that mathematically Mercury has cooled 2 ½ times faster
than the Earth.
-
- Gravity
separates material by density, the heavier material sinks to the bottom. The result is layers of different material
spread through the planet like onion skins.
The denser material like iron sank to the center of the planet when
everything was molten.
-
- Because
Mercury cooled rapidly the tectonic and volcanic activity was able to cease
after only 1 billion years. Earth is 4.5
billion years and it still has some of this activity. In addition to their cooling ratio, bigger
planets simply contain more heat.
-
- But,
in the case of Mercury ancient craters remain intact. There is little out gassing and low gravity
means gas escapes into space easily leaving almost no atmosphere to cause
erosion. ( A 100 pound high school
student on Mercury would weigh only 38 pounds).
-
- Extreme
heat and no atmosphere also means there is no rain or snow to cause
erosion. As a consequence the geological
activity on Mercury’s surface looks very much like the Moon’s.
-
- The
mass of Mercury is 3.31*10^23 kilograms.
To calculate the density we take the ratio of mass to volume. The radius is 2,425 kilometers average. Volume = 4*pi*r^3. Therefore the average density is 5,542
kilograms / meter^3.
-
- If we
assume an iron rich core has a density of 7,800 kg/m^3. And, we assume the crust being mostly rock
has a density of 3,000 kg/m^3, how big would the core be to have the average
density we have measured?
-
------------------ Outer radius of Mercury =
2,425 kilometers
-
-------------------- Inner radius of the core = Rc
-
------------------- Core mass
+ Crust mass =
Planet mass
-
---------
Core Density * Volume + Crust Density * Volume = 3.31
* 10^23 kilograms
-
------------------- 7800 kg/m^3 * (4/3 *pi*Rc^3) + 3000
kg/m^3 ( 4/3 * pi * (2.425*10^6)^3 - (4/3*pi*Rc^3) =
3.31*10^23 kg
-
------------------------ Rc^3
= 7.55*10^18 m^3
-
------------------------ Rc
= 1,960 kilometers
-
----------------- The total radius is 2,425 kilometers, so the
core is 81% of the planet.
-
----------------- The crust is 289 miles thick which is 19% of
the planet.
-
- Mercury
should have a solid iron core, yet, it has a small magnetic field ( 10% that of Earth’s) which means there is a liquid iron core to
cause electric currents to flow and a magnetic field to be generated.
-
- To
explain this astronomers conclude that Mercury’s extreme elliptical orbit and
the extreme tidal forces being so close to the Sun heat up the planet through
stretching and friction. There have been
many surprises as we learn more about Mercury.
-
- Jamuary 7, 2023 MERCURY -
closest planet to the Sun?
1478 1479
2695 3812
----------------------------------------------------------------------------------------
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--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for
copies or Index of all reviews
--- to:
------
jamesdetrick@comcast.net
------ “Jim Detrick” -----------
---------------------
--- Saturday, January 7,
2023 ---------------------------
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