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Monday, December 11, 2017

Kepler's Laws from Newton


Friday, December 1, 2017

Brief History Of Gravity

This is something I've been working on for a while and I hope you will find something new in it and you will almost certainly find some place where I've made an error (contact me on twitter @ColdDimSum to report errors). I don't think I've made any grievous errors but possibly have some things out of order, misattributed, or wrong in the fine details or due to my clumsy wording (I'm not a professional scientist nor writer nor science writer, I'm a professional software developer and I have been writing programs for over 35 years). For any errors I apologize -- but please consider these my notes on the subject and always find a good Primary source to substantiate anything specific. Thankfully, scientific theories are valid based solely on the authority of the evidence and not on who thought of them or when.

But I do think I have some insights I can share based on my studies and I hope my errors do not detract from the overall story.


In all likelihood, nothing that we see, feel, taste, or otherwise experience actually exists exactly as it seems to our human senses. There is no color 'pink' in Nature, it is a mixture of a very tiny slice of the electromagnetic spectrum that depends on the peculiarities of human visual senses (pigments in our eyes that stimulate the rods and cones in our retina) and processing in our brain in order to be experienced as 'pink'. Even amongst humans, the experience of 'pink' is not universal, some people lack the pigments necessary to sense the light frequencies necessary. Some humans lack taste buds, some humans lack auditory senses, some lack all visual senses, some lack nociception (a sense of pain) but still have normal somatosensation (touch) -- I cannot say with certainty that there is any sense that is truly universal to all humans, even the way we think and experience 'Self' might well be different in the extremes.

So it likely is with Gravity. In fact, in the Einsteinian view, Gravity is a consequence of the shape of Spacetime itself rather than a proper force, and this view is supported by the fact that, when seemingly being accelerated by Gravity, a body does not measure a proper acceleration, unlike all other sources of acceleration. As shown below, the acceleration drops to zero during the free fall (indicating a near weightless free fall).

Accelerometer recording during iPhone Free Fall
By the end of this article we will be able to estimate the height from which this iPhone was released, noting that it falls for about 12/20ths of a second.
However, this doesn't mean that Gravity doesn't exist. There is some underlying phenomena which we observe, measure, and number and we relate to this experience indirectly. It's unlikely that will or can ever truly know what exists because we are stuck inside of reality. Imagine an AI in a simulated world -- how could it ever know that underlying its experience is trillions of logic operations? It would instead observe that objects do not instantly appear from one place to another, but seem to move at some limiting speed -- only that you also don't seem to be able to make assumptions about 'where' the object is except when you are measuring it. But it could never, through experiment alone, deduce the logic underlying its own experience.

(Read more about Solipsism)

Whatever else the case may be, we can observe and measure the effects of Gravity and assign these observations to Laws and form Theories that attempt to explain 'why' at the level of experience to which we have access. The AI would not necessarily be wrong to conclude there is a maximum speed within the realm of physics to which it has access, this apparent 'maximum speed' would be emergent from the deeper physics to which it necessarily remains ignorant.

So this question of the reality of Gravity (and is it a fundamental force or an emergent force) is really irrelevant to the question at hand -- this doesn't mean that Gravity doesn't exist -- when you drop an object it clearly accelerates towards the center of the Earth (and we can measure how the motion of atoms are affected by nearby large masses, using a process called Cold Atom Interferometry). So we have direct empirical data for Gravity - it behaves like no other Force having no proper acceleration, cannot be shielded, doesn't have a polarity, does not depend upon the alignment of spin, is proportional only to the mass (not the material or its composition).

It doesn't really matter "what" Gravity is, what matters is what we do know about how it works and what are the unavoidable consequences of that knowledge, in our experience of the world -- our Physics.


So I would like to take a look at the history of this concept, explore how we know what we know, what the consequences are, and maybe get a sense some of what we don't know.

Gravity


Allow me to jump to the end of the story briefly so that we might see where we are headed...

Law of Universal Gravitation

every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them
This is summed up mathematically as \[F = G\frac{mM}{r^2}\] The consequences of this discovery are remarkable and not immediately obvious from such a lowly description. The impact of this simple Law is far reaching, touching on almost everything we experience. Without Gravity we wouldn't have stars, and without stars we wouldn't have planets, and without planets there would be no us to wonder.

And here we have to be careful to understand that this formula applies to a point mass, and while Newton will show that a large rigid body can be effectively treated as a point mass at the Center of Gravity - in the real world rocks and planets are not perfect rigid masses; so the deeper truth is that things get even more complex in the fine details but even that complexity is the product of our humble formula acting over a multitude of little masses.

Now let's glimpse the tiniest bit of the story behind how we got here...

Long before Copernicus, an Indian philosopher, Yajnavalkya (circa 9th Century BCE) might be credited as the first person to put down into words that the planets follow the Sun when he wrote:

The sun strings these worlds – the earth, the planets, the atmosphere – to himself on a thread. —Shatapatha Brahmana, 8.7.3.10

Lending weight to this being a heliocentric expression, and not just a poetic one, he also measured the distances between the Sun-Moon and Earth-Moon and recognized the Earth was spherical.

[See Shatapatha Brahmana, Full Text]

During the same period it was also written:
The Sun never sets nor rises. When people think the sun is setting, it is not so; they are mistaken. It only changes about after reaching the end of the day and makes night below and day to what is on the other side. —Aitareya Brahmana
Hundreds of years later, Greek astronomers, such as Aristarchus of Samos in the 3rd century BCE also proposed heliocentric models but, at that time, the models of Aristotle (384-322 BCE) and Ptolemy (~2nd century BCE) won out because we didn't have the data of sufficient accuracy to distinguish the models. Ptolemy's model is purely empirical based on cycles upon cycles and is accurate enough for basic astronomical observations -- the problem, as we will see, is that it isn't right. You have to keep adding and adding unexplained corrections to the model.

Unfortunately we don't have Aristarchus' full work - but we have mentions of it by Archimedes (287-212 BCE) and perhaps fragments of copies.

But these were all speculative or philosophical models, however genius in their time, lacking sufficient empirical data to judge them on their strengths and weaknesses. We would have to wait about 1500 years before the next leap forward.

Bede (c.672-735) - deduced that the Moon’s gravitational pull on the Earth was responsible for causing our tides and worked out a formula for predicting the tidal times.

Nicolaus Copernicus (1473-1543) we flash forward to around 1514 with Copernicus quietly writing a manuscript (De revolutionibus orbium coelestium / On the Revolutions of the Celestial Spheres) that would change the world with his new ideas about planetary motion placing the Sun at the center, at least of his Universe (he didn't have a telescope, so he didn't yet know about other galaxies or moon around other planets).

The Copernican Revolution (or shift away from the Ptolemaic system) was built on the following assumptions (some of which proved to be incorrect, others deeply insightful):

  1. There is no one center of the celestial spheres.
  2. The center of the Earth is not the center of the universe, but is the center of the lunar sphere.
  3. All other spheres revolve about the sun as their midpoint, therefore the sun is the center of the universe.
  4. The ratio of the Earth's distance from the sun to the height of the firmament (outermost celestial sphere containing the stars) is so much smaller than the ratio of the Earth's radius to its distance from the sun that the distance from the Earth to the Sun is imperceptible in comparison with the height of the firmament.
  5. Whatever motion appears in the firmament arises not from any motion of the firmament, but from the Earth's motion. The Earth, together with its circumjacent elements, performs a complete rotation on its fixed poles in a daily motion, while the firmament and highest heaven abide unchanged.
  6. What appear to us as motions of the Sun arise not from its motion but from the motion of the Earth and our sphere, with which we revolve about the Sun like any other planet. The Earth has, then, more than one motion.
  7. The apparent retrograde and direct motion of the planets arises not from their motion but from the Earth's. The motion of the Earth alone, therefore, suffices to explain so many apparent inequalities in the heavens.

This would irrevocably remove the Earth from the center of the Universe, despite it's errors and failings. The idea was too powerful and technology was about to catch up.

Ships were sailing to previously unfathomably distant lands powered by TIME -- that is, accurate clocks (and knowledge of the Trade Winds). Sailors had long known how to take the position of stars and figure out their Latitude fairly accurately -- the angle of Polaris over the horizon is good to within a degree (Polaris is 2/3 of a degree off center so it's altitude, or angle of the horizon, varies over each day slightly. But once you got close to home you could narrow it down to a dozen or so miles and easily find your way... if only you could find your Longitude. Longitude is much more difficult to find with the stars alone. You need to know exactly what time it is -- a few seconds off and your position drifts further and further away from true. And sailors needed accurate star maps and computations of the tables that enabled them to use a Sextant to find their position sufficiently. It's a complex bit of spherical trigonometry that most sailors couldn't directly perform (nor most modern people either!) But that's another story (see Marine Chronometer).


Tycho Brahe (1546-1601) was wealthy and owned an island near Copenhagen where he built an observatory and designed new versions of the sextant and quadrant allowing him to make naked-eye observations accurate to within about 1 arcminute - unparalleled in his day and working without the aid of telescopes he very carefully recorded the position of the stars and planets over time. His work, especially observations of Mars, would prove critical to Kepler in developing his laws of planetary motion (despite Brahe personally being opposed to the heliocentric model). He died from a burst bladder (some suspected poison) before his data could bear fruit, he was, above all, an empiricist and likely would have become convinced by the data.

Johannes Kepler (1571-1630) worked briefly with and analyzed Brahe's data (and there are stories within stories behind that, included the Catholic church trying to force Kepler to convert). He first assumed planets went in circles but that would have put Brahe's observations of Mars about 8 arcminutes off, which was too great an error give how methodological and careful Brahe had been, so he started over and formulated his three Laws of planetary motion:


  1. Law of Ellipses: Planets went in ellipses with the Sun near one foci.
  2. Law of Equal Areas: Planets sweep out equal areas over equal times.
  3. Law of Harmonies: The ratio of the squares of the sidereal periods of revolution of the planets (\(T^2\)) to the cubes of their mean distances from the Sun (\(R^3\)) is a constant (\(T^2/R^3\)).

Galileo Galilei (1564-1642) investigating the laws of motion discovered the principle of inertia - which says that an object that is moving in a straight-line will continue to move in a straight-line unless some Force acts upon it. The Aristotelian view of motion up to that time was that some force must be continuously applied to keep an object in motion. Galileo observed that it was forces such as air resistance and friction tend to apply forces to slow objects.

He also discovered Galilean invariance which says that Laws of Physics are the same in all inertial frames of reference -- which has far reaching consequences in itself.

The example Galileo considered was that on a ship travelling perfectly smoothly an observer enclosed below deck wouldn't be able to tell if the ship was moving or stationary.

And finally, Aristotle has proposed that objects fall at a rate proportional to their mass and Galileo showed that this was not correct and that, ignoring air resistance, all objects accelerate at the same rate.



Sir Isaac Newton (1642-1726?) Much of what transpired previously would come to fruition in the mind of Newton and consequently this is our longest entry.

LAW I: Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

~ Newton's Laws of Motion from Principia

We can see that when Newton pens "If I have seen further it is by standing on ye sholders of Giants" in 1675, he is referring to a vast body of work that preceded him. Years of toil, in the heat or frozen air, carefully measuring out the positions of stars and planets ever more carefully and precisely, all the mechanical work to improve the instruments, and all the intellectual thought he inherited. Even the concept expressed is attributed to Bernard of Chartres (Latin: nanos gigantum humeris insidentes) in the 12th century.

Newton studied what happens when a Force acts upon objects and discovered the principles behind accelerated motion -- that a Force is needed to change the velocity. If you apply the Force in the direction of motion then it will speed up, if it changes direction then the Force must have been at some angle to the direction of motion.

Newton also discovered that the Force can be measured as the product of two effects -- how much the velocity changes over some interval of time (the acceleration) and the inertial coefficient, or mass of the object.

Giving us the famous equation for Newton's Second Law of Motion (note that Newton himself didn't state it this directly) \[\text{Force} = \text{mass} \times \text{acceleration}\] This is a direct consequence of the more general law stated earlier: given G (the gravitational constant), M=Earth's mass, and r=our distance from the center of Earth we find:
\[\begin{align*}F &= m \times \left[\frac{G \times M}{r^2}\right] \\ F &= m \times \left[\frac{(6.67408 \times 10^{-11} \; \mathrm{m^3 \; kg^{-1} \; s^{-2}}) \times (5.972 \times 10^{24} \mathrm{kg})}{(6371393\;\mathrm{m})^2}\right] \\ F &= m \times \left[9.818\;\mathrm{m/s^2}\right] \end{align*}\] Note: Now the cool thing is that when your Theory gives you a different result (\(9.818\;\mathrm{m/s^2}\)) than you actually measure you have to account for that difference or your Theory is wrong. In this case, we already know that the actual value for 'g' varies by latitude, altitude, Earth's exact density variations, your relative speed, etc -- so we've already accounted for all of those differences in reality. I'm just giving the average value of 'g' in this case, to get the actual value you would have to sum up force vectors between every particle in the Universe -- but that's never required in practice. In practice, \(9.8\;\mathrm{m/s^2}\) gives us about as much accuracy as we're likely to be capable of measuring.

And because this works we can also weigh an object by knowing how much force it takes to accelerate that object by some amount. This is, in fact, how scales work. They measure the force resulting from the acceleration of gravity and divide it by \(9.8\;\mathrm{m/s^2}\), converted to whatever units your scale works in (pounds, kg, grams, micrograms, etc). There are also balance scales which work on the principle of directly comparing the force generated by your test mass against some reference mass until it's balanced.

What we find is that, if you take the same mass to different latitudes on the Earth and use the same scale calibration as your starting latitude, you can actually see the change in weight, as shown here:



It also turned out that these generic and simple empirical laws of Forces and Motions, combined with a simple Law of Gravity (the mutual attraction proportional to mass and inversely proportional to the square of the distances) explained the observed motions of the Planets deduced by Kepler.

Edmond Halley travelled from London to Cambridge in August of 1684 and asked Newton what kind of orbit a planet would follow if it were subject to a attractive force towards the Sun that were inversely proportional to the square of the distance between to the two bodies. Newton had already shown it would be an ellipse but couldn't find his notes and ended up redoing his analysis. He sent them in a short treatise entitled, On the motion of bodies in an orbit.

Halley would then go on to push Newton to publish his Principia.

But first Newton would need to show that you could treat large, mostly spherical masses as one unit with a point mass at the center. Without this it becomes computationally infeasible to consider anything more than a trivial point mass.

And this is indeed one of the most astonishing products of Principia and begins in Section XII where he devises a clever proof by considering a series of 'evanescent orbs' and showing how the forces would sum; for example, Prop. LXXI. Theor. XXXI:
Iisdem positis, dico quod corpusculum extra Sphæricam superficiem constitutum attrahitur ad centrum Sphæræ, vi reciproce proportionali quadrato distantiæ suæ ab eodem centro.

The same things supposed as above, I say that a corpuscle placed without the sphærical superficies is attracted towards the centre of the sphere with a force reciprocally proportional to the square of its distance from that centre
Newton avoids directly using his Calculus in Principia feeling that it has not yet found a sound footing, but his arguments (such as dividing a spherical mass into a series of evanescent shells) clearly hint that his insights spring from it.

Newton also showed that the Gravitational explanation worked for bodies here on Earth, the planets of the Solar system, and the moons of Earth, Jupiter and Saturn.

Newton discovered the equivalence of inertial mass and gravitational mass.

Newton estimated the ratio of the masses relative to the Sun for several planets, getting Jupiter correct to within about 2%.

Newton showed that the center of the solar system is not the Earth or Sun, but as a center of gravity near the Sun that constantly changes position as the planets orbit. The center between two masses is called a barycenter.

This, combined with the rotation of the Earth, explained the Tides whereas Bede had only calculated the cycles.

Later, it would be noted, through careful observation of the orbits of the moons of Jupiter, that they were about 8 minutes ahead of the Newtonian calculated schedule when Jupiter was closest to the Earth and about 8 minutes late when Jupiter was furthest from the Earth. This would turn out to be a consequence of the speed of light rather than an error in the calculations.

Calculus


Kepler's earlier work left many questions of analysis open, new approaches to mathematics would be required to solve the unanswered questions -- that is, to know what the consequences of the observations are. These questions and more drove Newton to invent a new calculus - the "ultimate ratio of evanescent quantities".

For the layperson, suffice it to say that the area under the curve of well-behaved, continuous functions can usually be well-approximated by breaking it up into rectangles and adding up the area. And that if you do that very thoughtfully you'll see that patterns form that allow you find the function to calculate that area directly instead of approximating it.

Here is an animation that explains this concept of breaking things up into smaller rectangles to approximate the area, as you carry this out to infinitely many rectangles you approach the correct answer:


Image Credit: MathWarehouse

For example, the integral (notated as \(\int\)) of \(x^2\) is found to be \(\frac{x^3}{3}\), this means that the area under the parabola \(x^2\), let's say taking x from 0 to 3 is \(\frac{x^3}{3}\) which is 9. This becomes an incredibly powerful tool of analysis and you can verify it for trivial cases like \(x = x\) and see that it simplifies to the same as the area of a right triangle with equal length sides.
\[\int x\; \mathrm{d}x = \frac{x^2}{2}\] so the area of x=1 is 1/2, area of x=2 is 4/2, etc. It is easy to see by inspection that this is true.

Likewise, we can ask about how much a function changes instantaneously at some point (known as the tangent) by finding the derivative of that function. This is approximated similarly but by taking the "RISE OVER RUN" over increasingly tiny intervals.

The derivative, notated by Newton as \(\dot{f}\) (fluxion) or Leibniz as (\(\dfrac{\mathrm{d}}{\mathrm{d}x}\)) or sometimes as \(x'\), of \(\frac{x^2}{2} = x\) -- the opposite of the Integral. So there is a deep relationship between the tangent and the area of functions.

And with that tiny bit of calculus we can already begin to understand accelerated motion much more clearly than we could without it.

If you have a function \(f(t)\) that gives the position of an object at each point in time (t) then the derivative of that function \(f'(t)\) will be the velocity (or speed, which is the magnitude of velocity), and the derivative of speed is the acceleration.

In the "simple" case of a rigid body accelerated by gravity, the acceleration is a constant \(g = 9.8\;\mathrm{m/s^2}\) (technically, as we get further from Earth gravity becomes weaker but for human-scale events the difference is miniscule and we can ignore it for now), so we have:

\(f(t) = gt²/2\) -- this gives us the expected displacement of our object after time (t), in seconds
\(f'(t) = gt\) -- this is the velocity, or the rate of change in our position (the magnitude of which is speed)
\(f''(t) = g\) -- this is our acceleration, \(9.8\;\mathrm{m/s^2}\)

These give us the Equations for a falling body. The derivative of position is velocity, and the derivative of velocity is acceleration. We can do the same in 3-dimensional space using vectors, but the result is, in essence, the same (just more complex to understand).

See more on the Fundamental Theorem of Calculus.

Newton also didn't try to explain why gravity works the way it does, but rather just explained what was directly observed about it:

I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis, and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.

With all this as the background, Newton observed that if the same Gravity we observe here on Earth was supplying a centripetal force to the Moon and planets, then Kepler's Laws fall out of this Law of Universal Gravity.

Newton didn't know the gravitational constant \((G)\), nor the mass of the Earth or the Sun \((M)\), so he worked in terms of ratios \[ \frac{C}{M}=\frac{c}{m}=\frac{k}{4 \pi^2} \\
\begin{align*} & \text{where,} \\ & k = \text{universal factor of proportionality} \\ & C = \text{constant for the Sun} \\ & c = \text{constant for planet} \\ & M = \text{mass of Sun} \\ & m = \text{mass of planet} \end{align*} \] The \({4 \pi^2}\) arises because when we consider the centripetal acceleration required to keep something in orbit we are considering circular motion around a circumference, where acceleration is \(v^2/r\) and \(v = 2 \pi r/T\)

Let's consider the following orbit using Kepler's laws

\( \color{OrangeRed}{\vec{v} \; \text{planet velocity vector}} \\ \color{Blue}{\vec{r} \; \text{radius vector}} \\ \color{YellowOrange}{\vec{\omega} \; \text{angular velocity vector}} \\ \color{Mulberry}{\vec{a} \; \text{centripetal acceleration, directed inwards towards the sun}} \)
Orbital Diagram

\[ \begin{align} \text{From Kepler's 3rd Law of Harmonies}& \\ \\ \frac{r^3}{T^2} = C \;\;\text{or}\;\; \frac{r}{T} &= \frac{C}{r^2} \;\;\text{or}\;\; T = C r^{3/2} \\ \\ \text{From Kepler's 2nd Law of Equal Areas}& \\ \\ a = \frac{4 \pi^2 r}{T^2} &= \frac{4 \pi^2 C}{r^2} \;\;\; \text{centripetal acceleration} \\ \\ \text{From Newton's 2nd Law of Force}& \\ \\ f &= \text{mass} \cdot \text{acceleration} \\ \\ f &= m \cdot \frac{4 \pi^2 C}{r^2} \;\;\; \text{sun's force on planet} \\ \\ f' &= M \cdot \frac{4 \pi^2 c}{r^2} \;\;\; \text{planet's force on sun} \\ \\ \text{From Newton's 3rd Law of Equal and Opposite Reaction}& \\ \\ f &= f' \\ \\ m\frac{4\pi^2C}{r^2} &=M\frac{4\pi^2c}{r^2} \\ \\ mC &= Mc \\ \\ \frac{C}{M} &= \frac{c}{m} \;\;\; \text{giving us our ratios} \\ \\ \text{Also, we can see that}& \\ \\ 4 \pi^2 C &= kM \\ 4 \pi^2 c &= km \\ \\ \text{Therefore,}& \\ f \cdot f' &= m \frac{4 \pi^2 C}{r^2} \cdot M \frac{4 \pi^2 c}{r^2} \\ \\ &= m \frac{km}{r^2} \cdot M \frac{kM}{r^2} \\ \\ &= k^2 \frac{m^2 M^2}{r^4} \\ \\ f &= k \frac{m M}{r^2} \;\;\; \text{Newton's Law of Gravitation} \end{align} \]

The planets no longer needed to be "moved" around by Angles but rather they moved inertially forward with a gravitational acceleration towards other masses. Empirical observations of motions here on Earth finally fit the motions observed in the Heavens without appealing to unseen forces.

Astronomy exploded once the telescope (credited to Hans Lippershey) was adapted to this purpose in the early 1600's, allowing ever more accurate observations to be made, with Kepler, Galileo, and Newton all playing roles in improving it. Galileo was merely the first to adapt Lippershey's "Dutch Perspective Glass" to the purpose of astronomy and consequently discovered the four largest moons of Jupiter (Io, Ganymede, Callisto and Europa).

Henry Cavendish (1731-1810) about 100 years after Newton, Henry Cavendish carried out an experiment to measure the density of the Earth (originally conceived by John Michell) by directly measuring the attraction between masses suspended very carefully by a wire.

This gave him an average density for the Earth of about \(5.448 \; \mathrm{g \cdot cm^3}\) which works out to \(G \approx 6.74 \times 10^{-11} \; \mathrm{m^{3} \; kg^{-1} \; s^{-2}}\) -- very close to the value we get today.

Now days, we measure gravity by direct experiment using cold atomic fountains.

Now, according to Newtonian physics, the orbits of the planets should not be perfect ellipses because all the planets also pull on each other just a little bit (but not zero). But when astronomers took Saturn, Jupiter, and Uranus into account (Alexis Bouvard published predictive tables in 1821) and calculated out the resulting orbits, substantial deviations began to be noticed.  John Couch Adams and Urbain Le Verrier both performed calculations that predicted the discovery and location of a as yet unseen planet on the basis of this anomaly in the orbit of Uranus -- that planet would be Neptune - located to within about 1 degree of its actual position, solely on the perturbation in the orbit of Uranus.

Around 1859 these ever improving observations would finally call Newton into question when it was noticed that the Perihelion of Mercury (where Mercury is closest to the Sun) was precessing (moving its position around the Sun) in a way that Newtonian gravity did not account for.  It would have to wait until 1905 for a young patent clerk to unravel that mystery.  But that's a story for another time...

With the improvement in telescopes astronomers began to be able to resolve stars that are very close to each other, such as Kruger 60 in Cepheus with an orbital period of 44.5 years, confirming that these same laws of motion applied.


These laws continued to be successful in explaining other phenomena such as great Globular Clusters and Galaxies and even giant clusters of Galaxies.

The story of Gravity continues on, hundreds of books could be written about it.  One of the threads you should follow on your own is to look at how Gravity, along with the Standard Model of physics explains the abundance of the elements, a phenomena known as Nucleosynthesis.  Without the energy potential from Gravity stars could not fuse the simpler elements such as Hydrogen into the heavier elements.  It also requires nova, Super nova, and neutron star collisions to produce the abundance of elements we measure in the Universe.

Experiments such as LIGO (Laser Interferometer Gravitational-Wave Observatory) have detected minute gravity waves from numerous cataclysmic events (black hole mergers and Neutron star collisions).

Other experiments have used Gravity, detected as minute surface variations on the surface of the Ocean, to measure the topography of the Ocean floor.



I've addressed previously why Gravity can hold the Ocean to the Earth but butterflies can fly.

I've shown why the Flat Earth claims that 'Density' or 'Buoyancy' explain our observations do not hold water.

And we've looked at the rotation of the Earth and observed the Eotvos effect with Wolfie6020.  And we've talked about how flights actually work over a curved Earth.

Summary


So we've learned that the same laws of motion observed here on Earth also explain the observed motions of the planets and we've directly measured gravity, many times, using many different methods.

One unavoidable consequence of Gravity is that any large body will pull itself into something roughly spherical until it reaches hydrostatic equilibrium -- and the larger the body, the greater the forces acting on the whole to achieve this. No body with the acceleration we experience on Earth could remain a large flat disc.  So it is with our Moon, the other planets in our Solar System, our own Sun and to the limit of our ability to measure them, the distant stars and their Exoplanets.

Once we account for Relativity, there is simply no experimental, empirical observations that show these Laws are wrong.  Period.  There is room to doubt that the shape of space 'causes' Gravity (the Theory of Gravity) -- but there is simply zero room to doubt that there is a phenomena of mutual attraction of matter that is proportional to the mass and inversely proportional to the square of the distance -- the empirical Law of Gravity.

So, dear Flat Earther, when you say "Gravity is a lie" -- we simply don't believe you in the slightest, and with good cause.  Your claim is vapid and puerile and made without the slightest bit of supporting evidence.  Pardon me if I'm utterly unimpressed.

iPhone Free Fall

Now we easily can solve our iPhone free fall question -- the answer is given above as the first equation of motion for a falling body noting that the displacement over some time (t) is the one-half the acceleration times the time squared , where our time is 12/20ths of a second. \[f(t) = gt^2/2\]\[\left( 9.8\;\mathrm{m/s^2} \right) \left(12\;\mathrm{s}/20\right)^2/2 \approx 1.76 \mathrm{m}\] [wolfram|alpha]

The phone was 2.38 meters off the floor and the top of the pillow was at approximately 0.60 meters high (including the bed), giving us a fall distance of approximately 1.78 meters (2.38-0.60).

Do your own experiment and repeat the experiment several times because your accuracy will vary slightly but you should get to within about 2% for a fall that is a couple of meters high (but don't break your phone).

======
footnote: why do I link to Wikipedia? Because each source has dozens or hundreds of further sources - the entry on Newton has 162 citations. The reader is expected to be able to identify Primary sources and use them appropriately.

Wednesday, November 29, 2017

Celestia Recreation: Apollo 17 'Blue Marble', 7th Dec 1972 at 10:49 UTC, 29,000 km over 30°S 31°E

I love doing these because I found an error in the 'Blue Marble' Wikipedia entry, someone was careless with their timezone conversions and flagged this image as taken at 05:39 UTC which is 6 minutes after launch, clearly wrong.  Also why I always check facts against primary sources.

Blue Marble Wikipedia error


Here is our quarry:

Image Credit: AS17-148-22727 [and Flickr]
The published version that was cleaned up from the scan:



Here is what we know:

Apollo 17 launched on 7th Dec 1972 at 12:33am EST (0533 UTC) [Apollo By The Numbers][Apollo 17]

The 'Blue Marble' frame (aka AS17-148-22727) was taken 5 hours 6 minutes later (probably by Jack Schmitt).

So that puts us at 5:39am EST (1039 UTC).  See the mix up?

There is also an amazing site called Apollo In Real-Time where you can follow along the whole long, view the photos from around that time in the mission, listen to the mission control recordings, and so forth.

So here we are 7th Dec 1972, 10:49 UTC, about 29,000 km (18,000 miles) over about 30°S 31°E [found here].  That's just an incredible level accuracy.


.Celestia CEL

Tuesday, November 28, 2017

Pic Gaspard (3880m)/Grand Ferrand (2758m) from Pic de Finestrelles (2826m) in the Pyrénées

I have some new toys to share in this installment of 'Yes, the Earth really is Curved even though you can see one distant mountain peak from another', so this should be fun!

This time we're going to look at a view of two distant mountains from Pic de Finestrelles (2826m) in the Pyrénées, taken by Marc Bret of Beyond Horizons (see also the Flickr album).

Pic Gaspard (3880m) in the Massif des Écrins range at a distance of 443 km.

Grand Ferrand (2758m) at a distance of 392.48 km.

Our view is right around 42.414466°N, 2.132839°E at about 2826 meters elevation, looking right along the coast.

Pic Gaspard/Grand Ferrand from Pic de Finestrelles in the Pyrénées, image by Marc Bret
The EXIF metadata shows this image was taken by a Panasonic DMC-FZ72 with a focal length of 215mm -- given the 5.62 crop factor of the 1/2.3" sensor in this camera, this gives you a 35mm equivalent focal length of 1200mm giving a full frame view of 1.644° wide and 1.215° high.
Since 1521 x 1014 isn't the exact same aspect ratio as the size of image this camera shoots it's likely that some small amount of cropping has taken place.

To double check this we can draw lines from our viewpoint to each peak and we find that, at the distance of Ferrand (392.48 km), our lines are about 1.7 km apart. This gives us an angle of approximately 0.24817° -- if we then apply that over the full 1521 pixels of the image we get an estimate of 1.63° wide -- very close to the 1.644° we expect so let's just use 1.644° / 1.215° as our field of view.
Now we're in a very good position to size the mountain and compare it to our photo.
First, we need the Law of Perspective, which says that the angular size (α) of an object whose height perpendicular to our line of sight (g) at distance (r) is:
α = 2*arctan(g/2/r)
Excellent - we just need to know the height and the distance.
Technically, we should find the straight-line distance but at 440 km it doesn't change the answer much.  If you want to find the straight-line distance from a curved distance (o) along the surface of the Earth you could use this formula (where R is Earth's radius of curvature, for 42°N latitude R ~ 6369km):

g = R*2*sin((o/R)/2) = 6369*2*sin((443/6369)/2) = 442.9 km

So let's just keep it simple since this would be about 1/1000th of a degree.
Pic Gaspard (3880m) @ 443 km = 2*arctan(3.880/2/443) = 0.5018° = 461 pixels
Grand Ferrand (2758m) @ 392.48 km = 2*arctan(2.758/2/392.48) = 0.4026° = 375 pixels

Just for scale, one-half a degree is the apparent size of the Moon, so we're talking about a pinkie-at-arms-length size -- but that that is the size those mountains should appear at the distances they are from this observer.
So we can mark those out on the image now as the full height of the mountain at that distance.

As expected, the "sea-level" bottom for the more distant mountain (Gaspard) is lower than the closer mountain (Ferrand) and since it's a larger mountain the total apparent size is larger even though it's slightly further away.


So please do not try to tell me we're seeing the "whole mountain" here.

Now, Walter Bislin has an amazing calculator that will let you place objects in the world and includes a very advanced refraction module (which I left at defaults except to set the 'surveying standard' 7/6th refraction, about 14%.  This is very modest amount of refraction and since the photographer tells you that this is very rare to be able to see this, the actual refraction is likely greater than shown here.

Here is Ferrand and Gaspard on a Globe model vs a Flat Earth model.  Notice how the Globe model correctly shows just the distant peaks with Ferrand peak above Gaspard even though Ferrand is shorter.  When you put these two mountains on a flat plane then Gaspard clearly stands taller.

There is no contest here, the Globe best fits the observations and really ONLY fits the observations.  If we could see the entire mountain, even at this great distance with perspective, they would stand vastly larger than seen here.

Walter Bislin Horizon Rendering Tool (link)
Another fun tool that comes from the Beyond Horizons page is a 'panorama' rendering tool that will label the mountain peaks.  I've used PeakFinder in the past but this one seems to be more full featured.  Here is our approximate view:

View of Grand Ferrand from Finestrelles via udeuschle panorama generator
So this view comports with the Globe model extremely well and utterly destroys the Flat Earth model.  The problem is that Flat Earthers do not how to do basic estimations like how big something would be in an image (if you could see the whole thing).

Here is also a video made by Sly Sparkane:



Monday, November 27, 2017

Does this iPhone 6s image make my horizon look flat?

A beautiful study in shades of blue of the ocean and open skies just floated across my twitter stream posted by @mcnees, taken on an iPhone 6s.

He was gracious enough to send me a link to the original for this blog (thanks!):


It's so quiet and serene, it really is a gorgeous image... But my inner-scientist just had to check...

A quick check on my phone using the smaller twitter image confirmed a few pixels of drop off on the right side, so I took the full resolution image and rotated it 0.2° counterclockwise to correct for the slight camera rotation and compressed the width to 202 pixels (1/20 of the original size)...

Here is the result:


That is about 5-7 pixels high in the full resolution, 4032 x 3024 pixel image.

However, from this vantage point, let's say about 20 feet over the water, 4032 pixel wide image, ~57.724° Field of View I would expect only about one pixel of 'apparent horizon Sagitta'.  We would have to be about 1400 feet over the water to see 6 pixels of 'apparent horizon Sagitta' with this field of view and resolution.

Sure enough, the iPhone 6- and 7-series cameras have a little bit of pincushion distortion (I tested it myself just now), which is creating a very slight amount of positive curvature here when the horizon is outside of lens center.  This is the exact opposite of the typical GoPro lens, which has barrel distortion (aka FishEye).



But it just goes to show that "looks flat" doesn't mean it actually is flat, you have to very carefully measure it with properly calibrated tools.  And, in this case, "looks curved" doesn't mean it is curved.

That's because when we're this close to the ground our horizon isn't curving sideways, like a sad-face (⌒), but rather is curving 360° around the observer, as we have shown previously.


The fact that subtle lens distortion can throw off our observations is why I came up with the method of taking frames where the horizon pass through dead center of the lens from multiple altitudes and we look at which direction (if any) the horizon bows as we gain altitude.  By only looking at what is changing we can eliminate a lot of variables.

If the horizon is truly flat then it should stay the same shape regardless of altitude.  But if the horizon is getting more and more bent then we can see this in the series of frames.

These are three frames from the Rotaflight balloon launch that show the 'apparent horizon Sagitta' increase sharply with altitude:




And, as I have shown previously, this 32 pixels of 'apparent horizon Sagitta' is a near perfect match with our expected curvature and indeed, even if you correct this footage for the GoPro distortion, the horizon curvature remains.


I also did some further analysis on this footage to find out exactly where we're looking and where the sun was during the time of this footage.

Sunday, November 26, 2017

Guest Post - Observational Evidence We Live on an Oblate Spheroid - by C.A.M. Gerlach

This post comes via C.A.M. Gerlach.  The guest author is a degreed research meteorologist specializing in severe convective weather and societal impacts, currently affiliated with the University of Alabama in Huntsville.



Flat Earth Claims: An Observational Perspective
C.A.M. Gerlach, CC-BY-SA 4.0

Many a Flat Earth proponent has invited skeptics to neglect the copious volume of indirect evidence that our planet is a rotating oblate spheroid, and just 'think for yourself.' Very well; coming from someone who has, I too invite everyone to experience things and think for themselves.

Let's consider the International Space Station, and forget the fact that I've personally spoken to the astronauts on it (must have been actors somehow simulating microgravity, who knows?), and just rely on measurements you and I can directly make. Look up when it is due to pass over your location, noting the elevation in the sky at which it appears and disappears and the time the passes between them (any large, low earth orbit satellite, or even an Iridium flare would work, but the ISS is the most visible), and carefully watch it for yourself, as I have, seeing it behave exactly as predicted for such an orbiting body, even weeks in advance. "But," you skeptically ask, "what if it is just a high flying, high speed aircraft with a powerful wide-beam light"?

Okay, fair enough. Now ask a friend a few hundred km/mi away to do the same thing, at the same time as you, keeping note of when and where they first and last observed the object, and whether that all matches the official predictions. If you find, as I did, your observations do match, then you know a few things—first, to cross the distance between you and your friend in the time it did, you know the object must be traveling at a minimum speed close to 28,000 km/h (17,000 mi/hr), Mach 25 or around 8 times faster than the fastest airplane ever flown, the SR-71 Blackbird (even the X-15, essentially a rocket, only traveled a fourth that speed, and only for a very short time). More telling, you can use basic triangulation (really, just the law of sines) to show the object must be around 400 km above your head, well into supposedly inaccessible space, as you and your friend can easily observe the angle the object makes with the vertical from your respective positions, and the distance between you.

Finally, if you are still skeptical, recruit a few Internet friends from different countries and have them look for it too. Does it match the published ground track, consistent with orbital mechanics of a spherical earth? If it does, then assuming a flat earth with the North Pole at its center, the object must be observed to travel over three times faster relative to observers at northern vs. southern latitudes in order to match that ground track; the fact that it doesn't shows the flat earth conjecture to be impossible (and it is important to note, in this scenario the object doesn't even have to be a satellite, it could be a NWO spyplane—all that is required is that two parties observe the angle the object makes with the vertical at a certain time, and note the time it passes overhead; that and the distance between them is all that is needed to calculate its velocity and altitude).

As another example, consider the weather. The Coriolis force is one of the two fundamental forces in our atmosphere, the balance of which with the pressure gradient explains the wind and weather patterns we see in the midlatitudes, from westerly winds and the jet stream to tornadoes and hurricanes is the Coriolis force, which is a fundamental result of the rotation of a spherical Earth. If this were not the case, these phenomena would not exist (we could theoretically still get very weak tornadoes, but the fact that the they can be observed to spin in the opposite direction in the northern and southern hemispheres is a direct consequence of the Coriolis force and its effects on the winds).

Overall; weather around the globe would be vastly different if the Earth was flat and non-rotating; with only the pressure gradient driving the winds, we wouldn't have fronts, blizzards, or any sort of organized weather systems; instead weather would be much like we see in the tropics, dominated by local-scale effects and the heating of the sun, with similar conditions every day and only regular afternoon storms to relieve the boredom. Further, dispelling the notion that these fundamental equations of forces themselves must be in error, they are what is actively used in every single modern weather model; many of which (e.g. WRF, widely used in the US both operationally and for research) you can download, examine the code, and even run for yourself (as again, I have). If the earth was not a rotating sphere, the fundamental equations on which these models are based would be fundamentally wrong, and they would produce widely inaccurate forecasts. The fact that they do, at least on the large scale and for a few days time, produce even remotely reasonable conditions much less relatively accurate predictions (most of the time...) is pretty clear evidence for Earth being a rotating sphere given how important that term is for their calculations.

Beyond that, there's satellite TV and Internet, plate tectonics/geology, GPS, airplane flight paths, space launches, cartography, astronomy, south pole stations, atmospheric refraction, the seasons, the speed of light, ocean sunsets, shipping routes...none of this as we know it is compatible with a spherical earth, and thus a vast proportion of the scientific community (including every single meteorologist, astro-anything, geo-anything, etc, due to how huge a difference it would make on their disciplines; it would be immediately obvious to even a partially trained decently competent student of such), as well as a significant fraction of working professionals in transportation, computers, radio communications, aero/astronautical engineering, etc) would have to not only be aware of it, but actively sworn to secrecy and in many cases actively working to perpetuate the hoax—i.e. millions of people, if not tens of millions, in the US alone.

The fact that not a single credible representative of those disciplines, particularly the first three I mentioned have come forward even on their deathbeds implies that either it cannot plausibly be true, or there is an even vaster group of conspirators keeping an eye on every single one of them, controlling every media outlet, every important political office, and every sufficiently powerful military and law enforcement service to prevent that form happening. But who's keeping _them_ quiet? And so it goes...millions to hundreds of millions of people, billions if not trillions of dollars, all for what? Protecting a hoax that benefits...whom exactly, and how? How do those behind all this benefit from people not knowing the correct shape of the earth? Funneling a mere few billion a year to NASA, when (say) the US military's  annual budget is 30x that, and it would take nearly 100 years of the former to pay for the total cost of, say, the latter's F-35 program? If such folks have for hundreds if not thousands of years been spending vast resources on keeping such a secret from the public at surely enormous risk of discovery, if any untrained online denizen can detect the trickery and put out a video about it to the whole world, what possible payoff would be worth it? Why not just use the money to build up an unstoppable military, or fund a global megacorperation, or develop ultra-advanced technology, or...

But don't take it from me—as we both agree, think and observe for yourself.

Followup to some potential rebuttals:

> But its a plane!

Again, based on your and my observations, it would have to possess the technology to fly 10x faster and 10x higher than any known aircraft, with effectively infinite endurance, and constantly projecting a light of 100x the power used by the average US home. Either that, or it actually is a hunk of metal orbiting the Earth with a few people inside. Besides, it can be imaged and its features fairly clearly made out with a good amateur telescope, tracker, and camera.

> Your math on the triangulation is off as its based on Earth being a sphere.

To the contrary, my math all (up to the last one) assumes the Earth is, in fact, flat, which not only is consistent with FE predictions but also simplifies things. Over a few hundred kilometers of distance, the difference isn't yuge; it only adds up over the final case described, which directly shows the trajectory taken by the object in question is flatly impossible under such assumptions (heh).

> Have you seen how much they're spending on military? They have technology beyond anything we can imagine!

It can be whatever advanced craft you like (or, heck, even is a complete hologram), but it still can't ignore simple geometry. No matter what it is, folks in the Southern Hemisphere would have to see it moving up to 3x the ground-relative speed as those in the North, or else its observed ground track would not be possible if the Earth were a flat disk. This really goes back to my final point—why spend the vast amounts of money, resources and risk that a unparalleled coverup would require when you could just spend it on a bigger military or better technology to take over the (flat) world directly? Why even bother with it in the first place? I suppose to direct billions of dollars to NASA and space contractors, but that's a drop in the bucket compared to military spending already, and the money either has to mostly be spend on hiring real people and building real equipment that will go to waste, or buying their silence and faking everything.

> Those meteorologists can't even predict the weather tomorrow; how are we supposed to trust that they know anything?

I'd suggest that you actually look at the forecast/model and observed weather large-scale weather maps for themselves, and see how well they compare after a few days. Do look even roughly comparable? If the Coriolis force, vorticity, curvature, etc. terms of the fundamental equations the models use did not exist in the real world, all direct products of a spherical Earth and its rotation, errors would accumulate and cascade so such an extent weather maps would look completely unrecognizable after such a time.

> The Coriolis force? That's just made up; no wonder it is called a "fictitious" force!

If Coriolis were not only fictitious force (as it is termed, incidentally, due to it being not precisely a force but the effect of the non-inertial frame of reference we are living on—a rotating sphere), but a nonexistent one, the entire fields of dynamic and synoptic meteorology would be nearly unrecognizable different, and we simply wouldn't observe a whole host the phenomena we do. Considering the forces acting in the situation, basic physics (the pressure gradient force, which can be readily observed with a household vacuum) would dictate that any pressure differences would rapidly equalize and the weather would be stagnant and boring everywhere, driven by only day heating and local scale weather systems. It is easy to observe that this is not the case, and no other conceivable explanation for this exists that fits with what we observe, other than Coriolis. Furthermore, numerous effects would be inexplicable, such as hurricanes and tornadoes spinning in different directions in different hemispheres, and not forming within a certain distance of the equator (which would be as unremarkable a distinction as the Prime Meridian in a flat-Earth model).

> Well, we don't know hurricanes are really spinning, since all our pictures of them are from fake satellites!

Why not just use our magic plane that somehow behaves just like one, while its up there? That aside, besides the fact that there is simply no plausible formation mechanism that matches with their motion and intensity without involving Coriolis (that's why they don't form near the equator, a fact that wouldn't make any sense in the flat earth model), you could readily determine they rotate consistent with imagery, predictions and model plots just by directly observing how the winds shift at different times in the storm; not to mention in intense storms the eye can be directly visible, or at least observable (sudden calm winds) and would be otherwise explainable without rotation around that center, which the winds can be measured to be consistent with. As a matter of fact, you can do the same thing with the everyday midlatitude cyclones that bring most of our weather.

> What do tornadoes have anything to do with Coriolis? They're way too small to be affected by it!

Ah, an educated question—sounds almost like something I would ask... As for tornadoes, their relationship with Coriolis is actually a lot more complicated and requires explaining a lot more than basic force balance (as they aren't driven by Coriolis directly being too small scale, but it has to do with the typical wind field expected from wind balance considerations and the effect that has on mesocylogensis, and in turn tornadogenesis, something that wasn't even understood well until recently. However, there remains no other explanation as to why tornadoes spin different directions in different hemispheres (while dust devils, which aren't driven by larger scale processes, spin about 50/50 in both since Coriolis is too small to have a direct effect at that scale).

> You people keep bringing up how the sun can't light the bottom side of clouds if it never really dips below the horizon—haven't you heard of refraction?

In terms of atmospheric effects limiting resolution and distorting light at long range, refraction is indeed a dominant effect; light "bending upward" due to high refractivity indicies is what gets you mirages, and the opposite can have noticeable effects on Nexrad weather radar, bending the beam into the ground.In general, refraction actually bends EM waves (light, radar, etc) in the same direction as the curve of the earth, though usually not enough to counter it; if the earth were flat the beam would eventually reverse in altitude and eventually encounter the ground (which, of course, doesn't happen outside of unusual situations). In fact, it is refraction coupled coupled with the curve of the earth that produces some of the interesting effects we can readily observe at sunset, effects that would be impossible if the earth were not round (light on the bottom of clouds, etc).

Why the Moon's Shadow moves West to East (Sly Sparkane's Animation)

Made this GIF from Sly Sparkane's video in case anyone finds it helpful:



Here is the full video: