a Bit of Newton Physics
---------------------
In order to understand the workings of blackholes we have to
sharpen our skills with Newton physics. There are several features of
the Newton laws of gravity that are often overlooked in other
treatments for the home astronomers, but they are crucial here.
We start with the omnibus statement by Newton that every particle in
the universe attracts every other particle in the universe with a force
of gravity, and so on. But obviously there is some very untidy mess
here. The universe is not made of 'particles' but of huge bulks of
matter. The Earth is made of a giant ball some 12,800Km across. This
is a 'particle'?
Did Newton intend that we work out the gravity force between each and
every bit of dirt of the Earth and each and every bit of dirt of the Moon?
If he ever did, the task is clearly insane and hopeless. No, he realized
that despite the lordly phrasing of his law, it was utterly useless in the
real world. He had to develop some sensible way to apply the law of
gravity to bulks like the Earth, the Moon, and the comet of 1680.
Ideal Spherical Body
------------------
Newton reasoned that the universe (as that world was known in the mid
1600s) consisted of globular worlds. The Sun is round. The planets are
round. Altho the satellites of Jupiter and Saturn appeared only as points
in the crude scopes of the era, he surmised that they were like our own
round Moon.
Could so simplification come from this symmetrical packing of matter
into spheres? Yes. provided that the spheres are of constant density
thruout or had only a radial gradient of density from the center to the
surface. In both cases the sphere can be constructed of thin shells or
layers each of a uniform density so that the mass of any piece of the
shell is proportional to the area of that piece.
Newton could not know for sure but the Earth and the other objects of
the solar system are remarkably close to this ideal sphere. Only Jupiter
displayed in his small telescopes an obvious oblate shape deviating
substantially from a sphere.
Gravity Outside a Spherical Shell
-------------------------------
A sphere can be built up from layers of shell each of constant
density thruout; this caters to worlds that have a radial density
distribution, like the Earth. An axis is drawn from the center of the
shell to the external point. It pierces the shell in its forward and
rearward poles.
The shell has radius r, has mass m, is very thin compared to the
radius of the shell, and stands R distance from the external point.
Note that R > r. We take from this shell a thin hoop or ring of mass
delm placed perpendicular to the axis so it is all around the same
distance x from the external point. So we have the [crude ASCII]
diagram here.
A O is center of shell
+ A, B are top and bottom
r / \ of the ring
/ \x Q is center of the ring
/ Q \ P is the external point
O+------+------------+P x is AP or BP
\ / R is OP
\ /x r is OA or OB
r \ / theta is angle AOP or BOP
+ phi is angle APO or BPO
B
The ring comes out of the page at A, arcs in front of the page
over Q, and recedes into the page at B. It completes back to A behind
the page. The shell is centered on O and has radius r = OA or OB.
The mass in the ring sets up a gravity field at P from every
particle of it. So there is a cone of force vectors from P to the
ring. They point to A and B, for the two points in the diagram.
The radial component of the gravity vectors do not act on P.
They, up & down in the figure, cancel from diametricly opposite parts
of the ring, as A and B.
The axial components do not cancel, but add. Ny symmetry the
summed up axial components act as if located at the center of the
ring, Q. Note that this axial portion is only a part of the whole
gravity force from the ring, the rest bing radial. It is of order
Gring*cos(phi).
The same analysis is applied to a ring of the same radius but on
the far side of O from P. The gravity vectors felt at P are less due
to the grater distance of the ring. But more of these vectors is
directed onto P because the ring is angularly smaller. cos(phi) is
nearer to one. The two factors exactly compensate to make the net
acial vector at P equal that of the nearer ring. The effect is that of
two rings worth of axial gravity acting midway between the rings, or
at o, the center of the shell.
Continuing in the way for all rings over the shell, we end up with
the gravity of the entire shell's mass acting at the center of the
shell. Or
Gshell = gamma * Mshell / R^2
+-------------------------------+
| GRAVITY FORCE OUTSIDE A SHELL |
| |
| Gshell = gamma * Mshell / R^2 |
+------------------------------+
we can replace the bulk shell with a point at its center that
contains the shell's whole mass. The gravity field around that point
is that of the original shell.
We did not go thru the full derivation that newton needed, and the
argument above is a bit loose. It is a lot more mature than the usual
way it is explained in ho,e astronomy.
By building nested shells Newton showed that for a solid sphere
the point mass at its center could replace the full sphere. The shells
could be of different densities, so long as they were uniform within
each self. The gravity vectors of the shells add to make the
centralized point mass for the sphere..
Gravity Inside a Spherical Shell
------------------------------
We next examine the case where the point P is INSIDE the shell, so
R is less than r. All of the analysis is the same as or an outside
point. In this case it's easier to compare the force of rings cut by
a force cone from P that cuts the shell in opposite sides. The rings
are of different sizes, and mass, and also of different distances from
P.
The critical caution is that the gravity on point P is generated
in opposing directions by rings on either side of the point. The maths
demand careful attention to the signa of the gravity force from each
ring. Else the conclusion is nonsense.
The result is sturrange. Yet most works on gravity for home
astronomers don't even mention it.
Within the shell the gravity field at point P is ZERO! We can see
this plausibly by noting that for every part of the shell on one side
of us there is a part opposite it on the other side. Their two
contributions to the gravity field at our location tend to cancel.
+------------------------------+
| GRAVITY FORCE INSIDE A SHELL |
| |
| Gshell = 0 |
+------------------------------+
This does NOT allow us to build a zero-G chamber inside a shell.
Gravity can not be shielded like electric and magnetic fields. An
object inside a shell feels the ambient gravity of the material
outside, like the Earth's mass. It just does not fell extra gravity
from the enclosing sphere.
Gravity Field within a Solid Globe
--------------------------------
When we penetrate into the bulk of a solid globe by, say, drilling
a well toward the center, we stand on the outer surface of a smaller
sphere and on the inside of a shell. The sphere is the globe's body
with radius equal to the distance from the center to our location in
the well at the instant. Its mass is governed by the volume of this
sphere and its density. For simplicity we let the globe have a
uniform density thruout, not made of assorted layers.
The shell is the globe's body above us up to the surface of the
globe. Its inner radius is the distance we are standing out from the
center.
The shell above us contributes NO gravity field at our location.
We are inside the shell. The mass of the sphere contained in the
shell is removed from gravity efficacy.
The smaller globe below us produces a gravity field on us, being
that we are outside of it. Its mass and radius are less than those of
the complete globe.
As we descend farther to the center in the well, we eventually
reach the very geometric center of the globe. Here the shell of mass
above us is as thick as the whole radius of the planet, yet it has no
gravity influence on us. There is no more mass under us in a sphere and
so there is no downward gravity field left.
Hence, at the center of the globe there is NO gravity field left
at all! We would float with no weight at the bottom of the well.
For a solid sphere of uniform density the gravity force inside is
surprisingly easy to work out. The method works for a fluid sphere as
ell, so long as the density is constant over its whole volume. We
start with relating the mass of the sphere to its radius, which is
density times voume,
Mr = (4 * pi * r^3 / 3) ( rhp
Gr = gamma * Mr / r^2
= (gamma * 4 * pi * rho * r^3 / 3) / r^2
= gamma * 4 * pi * rho * r / 3
+--------------------------------------+
| GRAVITY FIELD INSIDE A SOLID SPHERE |
| |
| Gr = gamma * 4 * pi * rho * r / 3 |
+-------------------------------------+
The gravity field inside a solid sphere increases linearly with
radius out from the center to the surface. Mind well that this assumes
a uniform density rho. For a layered sphere the analysis is done
piecewise for each layer as a shell with its own density.
Range of Applicability
--------------------
Newton knew that his simplification of a particle-to-particle
interaction to one of bulk matter interaction was an approximation. It
looked good because the celestial objects known to him were pretty
much globes. He had no idea if they were of constant or layered
density, He hoped they were somewhat like the Earth, which the
embryonic geology showed was made of strata of different rocks.
He already figured out by the mass of the Earth that its density
was about 5-1/2 times water, or 5-1/2 ton/m3. But the rocks on the
surface were only of density 2-1/2 to 3. On this evidence he hazarded
that the interior of the Earth must be made of some extra dense
material to make up the defect from under-dense surface rocks. This
was an incredible deduction for science of the mid 1600s!
He knew that Jupiter was a flattened globe. Small scopes today
show this quite plainly. He was very lucky that the satellites of
Jupiter run around the equatorial bulge. Symmetry made Jupiter act on
the satellites like a spherical globe.
He had his first challenge with the comet of 1680. This thing was
not at all round and it was immense in physical size including the
tail. He was very lucky that the mass of a comet is almost entirely
within its nucleus with essentially nothing in the coma and tail. He
treated the coma and tail like a luminous atmosphere of no substance.
This enabled him to suss out the motion of the comet -- the first
ever analyzed -- to be a parabola with the Sun at one focus. By this
example he proved that Kepler's laws could include parabolae as well
as circles and ellipses for orbits.
The proof that the 1680 comet moved in accordance with Newton's
gravity theory demolished contemporary critics. They claimed that
the planets and Moon were well observed for thousands of years and
that Newton somehow fitted his theory to this long legacy. But
the comet was an entirely new visitor to the Earth with no prior
history to fudge the theory with.
Nonspherical Bodies
-----------------
With the space exploration of the 1960s we ran into the problem of
gravity around nonspherical bodies. These included the Moon, Earth,
asteroids, minor satellites, and comet nuclei. The orbits of
artificial satellites of the Earth mapped out an extraordinary field
of gravity, with weak and strong spots along the orbit. These are due
to concentrations of mass in the Earth's crust and depletions of it
under the oceans. Today, satellites placed in special orbits, like for
navigation, must have maneuvering rockets to keep nudging them back
into proper place after being tugged off-base by the irregular gravity
field of the Earth.
The exploration of asteroids and comets, beginning in the 1980s,
required careful and cunning models of the gravity field around lumpy,
tumbling, conglomerate bodies. The Giotto probe of Halley's comet was
more of a brute force flyby than an effort to close in and orbit
Halley's comet. But the Soviet's Phobos probe tried to land on Phobos,
Mars's larger moon, and get tossed about by the shifting gravity
field. It most likely crashed on the surface much like a boat is dashed
by rough water currents against rocks near shore.
By the mid 1990s enough had been learned about such irregular or
chaotic gravity fields that we are sending orbiters around various
asteroids, close fly-alongs of comets, and hoverings over small moons.
No Defense Against Gravity
------------------------
Gravity is fundamentally different from electricity and magnetism.
The latter two can be ducted, diverted, shielded, manipulated by
various substances. The proper choice of these and their arrangement
in 'circuits' makes electricity and magnetism so mature an industry
and science. There is no known way to apply gravity in so handy a
manner.
There is no known substance that masks off gravity, refracts or
reflects it, amplifies or reduces it, or in nay other way modifies its
action. There are only two ways to alter the effect of gravity at a
point: Vary the mass of the attrahent body or vary the distance to
that body. That's it.
There are no 'gravity circuits' comparable to electric circuits,
no 'gravity optics' comparable to light optics, no 'gravity storage
cells' comparable to batteries. We can only enjoy or suffer gravity in
a completely passive mode with no human intervention with its workings.
Gravity is Cumulative
-------------------
Gravity penetrates every known substance interposed between the
source and the observer without suffering any attenuation. So every
point in the universe is subject to the full force of gravity from
each and every single particle of mass in the universe, regardless of
distance or intervening material.
This singular behavior of gravity makes it possible to allow the
sphere to considered a layering of thin shells; each shell acts
totally independently of the others and adds its gravity force to that
of the others. The farthest shell from the observer has exactly the
same effect with the others shells in between as when it is the only
shell with nothing else.
Thus, we are confident that on the grand scope of the universe
gravity is the one operative force. Any arbitrary volume will enclose
mass, electric charge, and magnets. There is no such a thing as a magnet
monopole, an isolated north or south pole by itself. Hence any volume
of space will enclose a net magnetism of zero for there will be equal
numbers of north and south poles within it.
There may be local separations of charge. Plasmas of ionized nuclei
and clouds of free electrons do abound in space. On the local scale it
is possible to surround a net positive or negative charge from the
localized imbalance of ions and electrons. But on the larger scale the
charges tend to net out because of the sheer impossibility to keep the
separation stable over vast distances. So the larger the volume you
take, the enclosed charge tends to zero.
But mass can only accumulate with ever larger volume and the
gravity field set up by that mass can only grow. Hence, as we look at
cosmic regions, the magnetic and electric forces are zero or nearly so
while the gravity force swells to immense values.
This is the seminal feature of Einstein's cosmology. For the
universe as a whole, only gravity matters. It governs the
development, behavior, and fate of the cosmos.
Gravity Field Strength
--------------------
The strength of the gravity field at a given point is the
on a unit mass at that point. This comes directly from Newton's law of
gravity
+-------------------------+
| NEWTON'S LAW OF GRAVITY |
| |
| G = gamma * M / r^2 |
+-------------------------+
M here is the mass of the attrahent or central body. The units are
newton/kilogram. gamma is the Newton constant, 6.672E-11 n.m^2/kg^2,
determined by cunning and delicate experiment. It remains today among
the crudest known of the fundamental constants of physics.
Newton also developed more general laws of motion without regard to
gravity. His second law states the force experienced by a body is the
body's mass times its resulting acceleration.
+-----------------+
| NEWTON'S SECOND |
| LAW OF MOTION |
| |
| F = m * a |
+-----------------+
m is the mass of the test body, not of the central body. This law is
not confined to gravity but is a general one valid for any force.
Gravity is only one possible force that can act on a body. From this
we have
F / m = a
whose units are meter/(second^2)
A Useful Equivalence
------------------
Look again at Newton's second law. It relates an acceleration to
mass and force. Force is in newton, mass is in kilogram, acceleration is
in meter/(second^2).
Please take care to use the units of acceleration properly. Other
works on astronomy and physics may say 'meter per second per second'
and 'meter/second/second'. While the author can argue purple that this
is correct, it is immensely misleading. As a general rule you can
manipulate a maths expression to contain one and only one division
operation or to be written with one and only one division symbol.
Stuff like 'm/s/s' is immature maths. This principle avoids the silly
mistakes of undoing the first division with the second or duplicating
the two into a squaring operation, which ever is the wrong choice.
You will, therefore, be sure to cite acceleration as 'meter per
second squared' or, OK, 'meter per square second' and never as 'meter
per second per second'.
Back to Newton's second law. Inserting units for both sides, we have
newton/kilogram = meter / (second^2)
+----------------------------------------+
| VERY USEFUL EQUIVALENCE |
| |
| newton / kilogram = meter / (second^2) |
+----------------------------------------+
This looks unbalanced but it isn't. The two are exactly equivalent. In
fact in any other physics formula you may freely exchange the one for
the other in t he same way you can exchange, say, sin^2(angle) for 1-
cos^2(angle). They sure look different but are the same thing stated
two ways.
Gravity Work and Energy
---------------------
A body in a gravity field left to itself will fall thru that field
toward the attrahent body. As it falls it picks up speed and kinetic
energy. This acquired energy can be exploited to perform useful work.
This energy is a function of the distance the body falls thru the field.
On Earth this gravity energy is widely used in water works. Water
falling down a waterfall is channeled to a paddle wheel. The rotating
paddle wheel is geared or belted to machinery. Such water works were in
use since the Babylonian era and many survive from the Roman and
mediaeval eras.
Where there is no natural fall to exploit, an artificial one formed
by backing up water behind a dam can be used. The water is passed thru the
dam via pipe or flume to the paddle wheel.
Gravitational Potential Energy
----------------------------
The potential energy of a point in the gravity field is measured by
the work required to move a test particle from the center of gravity to
that point. This is a positive amount; work is done on the particle by
some external means. Because the gravity field itself is a function of
distance r from the center of gravity, the work must be integrated on this
function. So we have
Wg = intl(gamma * M / r^2, delr)
= (-gamma * M / r) + const
+-------------------------------+
| GRAVITY POTENTIAL ENERGY |
| |
| Wg = (-gamma * M / r) + const |
+-------------------------------+
We could take the Earth's sea level as the zero point and this is done
in engineering, like for hydroelectric projects. But with the coming
of the space age, this proved quite impractical. For one thing, there
is no 'sea level' on most of the other planets. You could define
uniquely for each planet its own zero point radius. On Mars, for
example, the zero point level is a certain atmospheric isobar. But
this is totally ad hoc and arbitrary. Not good science. Or politics
once the planets get settled with colonies.
Note that the potential energy is the greatest at the farthest
distance from the center of gravity and decreases with lesser distance. To
accommodate the needs of space travel the zero point is set at 'infinity'
radius away, so all planets have the same datum to measure from. The
numerical value of the energy is in this scheme always negative.
This 'negative energy' is nothing special. It's the same sort of
negative number as that of temperature or stellar magnitude. It's merely a
mathematical artifice to extend the number system lower than zero.
The energy between two points in the gravity field is merely the
difference in the two points's energies. Or, what amounts to the same
thing, the work needed to move a particle from a one point to an other in
a gravity field is the difference in the energies at these two points.
Gravity Well
----------
The notion that you must acquire energy or have work done on you to
rise up from a planet toward infinity promotes the concept of a 'gravity
well'. This was popularized in the 1950s by the early books on space
travel. The wells for each planet were dips or pits in an otherwise smooth
horizontal plane. The depth of the pit is scaled to the energy of the
gravity field at the planet's surface. (In those days, the surface of a
planet was its visible boundary, regardless of any clouds.) In practicality
once you got far enough from a planet to be in the influence of others you
were effectively free of that planet's gravity, so the horizontal plane was
that of interplanetary space rather than truly at infinite distance away.
The vertical scale of this diagram was loosely defined. Properly
speaking the slope of the plane is 1/r function of energy. So the 'well'
looks like a lily flower or flute, whose bottom is the energy level
of the planet's surface. Hence to pass from the Earth to Mars you 'climb
out' of Earth's well by acquiring enough energy, by firing your rockets,
to reach the zero level of interplanetary space. You coast freely over
to Mars. Then you 'climb down' Mars's well by riding your retro rockets
to the Martian surface.
Escape Velocity
-------------
It doesn't matter how we obtain the energy to leave the Earth's
gravity. If it were feasible we could simply build a ladder to the Moon
and walk to there. But it so happens that our civilization has no means to
continuously and steadily supply energy to move about in interplanetary
travel. There are thinktank musings about sending energy produced on
Earth to a remote spaceship by a super powerful laser beam. On the
ship the energy is used to generate ions for thrust.
So far our only methods is tp give all the required energy to the
test particle in one impulse. This is done nowadays by rockets, which burn
furiously for a few minutes to lift the craft off of the Earth and give it
the energy to leave the gravity well. For the greater part of its journey
the spaceship coasts with no ongoing power.
For the Earth the energy to remove the rocket from the ground to
infinity is
Wg = (-gamma * M / infy) - (-gamma * M / re)
= 0 - (-6.673E-11n.m^2/kg^2) * (5.976E24Kg) /( 6.378E6m)
= 0 - (-6.252E7n.m/kg)
= 6.252E7n.m/kg
= 6.252E7j/kg
-> 62.57 million j/kg
This is a very substantial amount of energy to give to a body in one
impulse. To do this by present-day means, we give this energy to the
body in the form of kinetic energy. This is, we impart to the body a
velocity large enough to equal this 6.252R7 joules for each kilogram
of the body's mass.
The kinetic energy per unit mass is
Ek = v^2 / 2
v^2 = 2 * Ek
v = (2 * Ek)^(1/2)
= (2 * (6.252E7j/kg))^(1/2)
= (12.504E7j/kg)^(1/2)
= 11.182E3(j/kg)^(1/2)
= 11.182E3m/s
-> 11.1km/s
The shift from j/kg to m/s may not be obvious. Noting that one joule
is one newton-meter and that one meter/(second)^2 is one newton/kilogram,
we have
j / kg = n * m / kg
= (n / kg) * m
= (m / s^2) * m
= m^2 / s^2
= (m / s)^2
(j / kg)^(1/2) = m / s
It is necessary to get the spaceship to this speed, the escape
velocity, in order to free it from the gravity of the Earth. Similarly for
the other planets with their mass and radii put into the formula above.
Note well that this is a requirement only because we have jut the one way
to move spaceships off of the Earth by a single impulse of speed. If we
had a means to supply the energy slowly and steadily, analogous to
walking up stairs in the stead of jumping up, we would not need this
escape velocity