Monday, July 5, 2021

A Goldilocks Universe

Anyone with experience in astronomy has encountered the term 'Goldilocks planet'.  It pertains to a planet that is not too near a star, nor too far, such that it may have liquid water on the surface.  Many scientists believe that this is a necessary pre-cursor for life.  Earth is the only Goldilocks planet in our solar system, but exoplanet searches have identified others across this galaxy.

This morning I was thinking about the Universe, and noting that there could be no Goldilocks planets without, what we might call 'Goldilocks stars'.  I would define a Goldilocks star as one that has a main sequence that endures for billions of years at the least.  There are countless such stars in our galaxy.

Why are billions of years of stable star output important?  It is because such is the timeframe that it takes for the development of life (itself an unlikely event) on a planet (which itself may take hundreds of millions of years to develop into a potential host for life).

A star's main sequence describes its stable state where the gravity that holds it together is in balance with the internal pressure that pushes it outward.  It is achieved during the period of time when the core of the star is largely a mass of protons zooming about (these protons are denoted as H-1, as they are hydrogen isotopes that lack a neutron, known as 'protium' as they are effectively just protons).  Energy is created via nuclear fusion when these protons collide and enter into what is known as the unfortunately named 'p-p cycle'.

A complete p-p cycle is a complex series of nuclear fusion reactions that eventually convert six protons into two protons and one Helium atom.  Each link in the fusion chain spits out other matter including positrons, neutrinos, and gamma particles.  Most importantly, the fusion reaction releases thermal energy because the nuclear by-products have less mass than the nuclear fuel - the fusion process produces energy E in the amount of dm multiplied by the speed of light squared (Einstein's uber famous equation) where dm is the quantity of annihilated mass.

The big picture is far less complex than the details: hydrogen fuel converts to helium and releases energy at a specified rate until it runs out.  The amount of time that this dance will play out for is determined by just one thing: the star's mass.

Red Dwarfs are small stars and are the most common; they can burn for trillions of years.  Yellow Dwarfs (like the Sun) are medium-sized and less common but not uncommon; these burn for billions of years.  Supergiants are far more massive than the Sun and are far less common; these burn for just millions of years before they exhaust their fuel supply.

Given the brief period of time (in cosmological terms) that Supergiants undergo their main sequence, it is unlikely that its planets can ever harbor life.  We can deem these stars too big.  We do not yet know whether Red Dwarfs can sustain life on the planets that orbit them.  These stars might be too small.  We do know for certain that planets orbiting Yellow Dwarfs can harbor life (we know of one clear example of this).  These stars, it seems, are just right: Goldilocks stars.

But it all comes back to that p-p cycle.  The rate at which our Sun burns through its fuel depends upon the probability that a p-p cycle can be completed.  Smashing two protons (H-1) together does not guarantee that a deuteron (H-2) will be synthesized (step one in the p-p cycle)... Far from it!  It is actually extremely unlikely.  The probability that it will occur is on the order of 1 in 10 to the power of 26!  The reason that the Sun produces energy at such a high rate is that despite the low fusion rate, there are some 10 to the power of 57 protons zooming about.

It is the 1 in 10 to the 26 rate that confounds me.  I mean, like, why that rate?  Each proton-proton collision is a quantum event.  The particular fusion rate seems so random, arbitrary even.  But it is ultimately critical to our existence.  If this rate were, say, ten times higher than it is, our Sun would have burned out long before life emerged on this planet.

Physics reveals many instances where the conditions of the Universe, its matter and the laws that govern how it interacts, seem to be just right.  If the strong nuclear force that binds the nucleus of an atom were slightly weaker, the electrostatic repulsion of protons would exceed it and prevent the existence of any atom not called Hydrogen.  No atomic variety means no life, just as no long-burning stars means no life.

One can imagine a universe not so perfectly tuned; a universe where life is impossible instead of improbable.  We may live on a Goldilocks planet that orbits a Goldilocks star, but if we widen our gaze, we see that we reside in a Goldilocks universe.  Not that it matters, but it is a funny coincidence that like Goldilocks herself, I ate porridge for breakfast today.  I mixed it with leftover brownies.  It became just right

Saturday, June 19, 2021

AATIP Reveals Compelling Videos of UFOs

Some weeks ago, as my class was discovering notions of relativity, a student asked what I thought of the bizarre videos that were making its rounds on the internet - they reveal what appears to be some kind of unusual aerial vehicle.  I watched these black and white videos with curiosity.  In the background, you can hear some excited voices expressing genuine confusion about what they are witnessing.  With final exams looming and little free time, I did not pursue this rabbit hole any further.  Then weeks later, a friend we'll call 'Phil', asked what I thought about the UFOs.

Tom is a staunch believer in the scientific method and a skeptic when it comes to conspiracies and the like.  But he found these videos to be very compelling.  He informed me about Luis Elizondo and the Advanced Aerospace Threat Identification Program (AATIP) and suggested I watch his recent interviews.  I did.  I also came across a clip of Barack Obama giving credence to the notion that the highest levels of American intelligence have come across aerial vehicles whose origins confound them.  It appears that AATIP is indeed a genuine Pentagon program and they will issue an official response to the aforementioned videos.

I impressed upon Phil that I am typically not drawn into stories of this nature due to the extreme unlikelihood of alien visitation.  However, if these videos were real artifacts, free of manipulations, they reveal technology that is far beyond current human capability.  The aerial vehicle in the videos:

1. Has no visible means of propulsion and whatever does propel it shows no sign of interacting with the environment.

2. Transfers from air to water without disturbing the water.

3. Banks extremely sharp turns at impossibly high speeds.

Let us, for instance, analyze point 3.  The vehicle is tracked at speeds in excess of Mach 5 (five times the speed of sound in air, so about 1,650 m/s).  In order to not experience violent accelerations in excess of 5g (about 50 m/s/s), the minimum radius that its circular path would require is 36 km!  Points 1 and 2 are even more bewildering.

If these videos are authentic, how did the vehicles get here undetected by our radio astronomers?  Elizondo theorizes they emerged from the deep ocean.  Phil asked me where we should purchase our aluminum hats.

Passing on the hats for the moment, I went to the library later that day, and returned home with They Are Already Here: UFO Culture and Why we See Saucers, by Sarah Scoles.  The book is a historical account of the human obsession with UFOs and the possibility of alien intelligence, from Roswell and Area 51 to AATIP.  The title to the book is misleading: the author confides on the last pages that she remains unconvinced that any interplanetary intelligence has ever visited Earth, and that the plethora of reported human encounters with aliens are either honest mistakes or fabrications.

I am interested in honest mistakes, as they force us to apply the scientific method within this thought-provoking context.  These range from explainable celestial events, to high-tech military operations, and a wide range of optical illusions.  I also understand and do not fault claims of UFOs that are entirely psychological, whether they be drug-induced or convincing dreams.   

On the other hand, fabrications offend me.  They are an affront to my senses.  They degrade the entire process of discovery.  Muddying the evidence, manipulating the data, unfalsifiable claims masked as truths... These acts of dishonesty, whatever their motivation, highlight the fly in the ointment, which is human corruption.  Such acts of deceit serve only to spoil the earnest endeavor of identifying UFOs.  One key take-away from Scoles' book is that distinguishing genuine science from hoaxes is half the battle in the search for alien intelligence.

When it comes to the matter of extra-terrestrials, we must be extra skeptical of information emanating from sources who have a vested interest in making the first human contact with them.  One such player is Robert Bigelow, a wealthy American who has initiated numerous 'scientific teams' whose primary outputs have been UFO fabrications.  When I discovered that Bigelow has a connection to AATIP, I began to doubt the authenticity of the internet videos.

It is improbable that we have been or will ever be visited by interplanetary beings during our species' tenure on this pale blue dot.  The chances that intelligent life exists in our neighborhood of this galaxy during the small window of time comprising human existence are very low.  But not zero.  And that is what distinguishes the topic of aliens from other human obsessions, like paranormal activity.  The former is entirely conceivable according to our current understanding of nature.  

Evidence that confirms the existence of aliens would cause a dramatic shift in our understanding of the universe and our place in it.  That is why this conversation is so alluring.

I await the Pentagon's response to the videos that have captured the attention of so many.  If their  assessment does not support the alien intelligence theory (and I highly doubt that it will), conspiracy-theorists will be unmoved.  Government history does include cover-ups, which merely confirms the general prevalence of human weakness.  This history of dishonesty injects doubt into the UFO conversation.  

I will not be buying an Aluminum hat just yet.

Tuesday, June 8, 2021

Enforced Rotation of Tarzan Rope (Solution)

The semester has ended, and alas, nobody posted a solution to the difficult problem I posed months ago (see problem here).  In short, we have a rope that is suspended from the top and is being moved along a circular path in the horizontal plane with constant angular velocity.  Aerodynamic effects shall be neglected.  We are seeking a lateral deflection function.  Here is my solution...

With a problem such as this, we must begin with a physical model.  My hand drawing is seen below (I apologize for the crude sketch, but the summer me exerts less effort):


The solid blue line represents the rope whose profile we aim to determine.  At some location (x, y), we will apply Newton's second law to a single mass element dm.  My free body diagram is on the right side.  There are two external forces acting on the element; one is real and the other, a pseudo-force.  The real force, dFg, is gravitational, while the centrifugal load, dFc, is a pseudo-force as it is effectively an inertial term.  Finally, tension acts internally, pulling this element in both directions tangent to the rope's profile at (xy).  The upward pointing tension is (correctly) assumed slightly higher than the downward one, by some amount dT.  One useful, though limiting facet of the assumed model, is that, at a given vertical location x, each element simply displaces horizontally - in reality, it also shifts up vertically, ever so slightly.  This simplification allows an elegant solution, but whose accuracy is limited as we shall see.

Applying Newton's second law to that element on both axes, we get:

dFc = dTsinθ                                                                                                    (1)

dFg = dTcosθ                                                                                                   (2)

We can express the elemental forces as:

dFc = dm(ω2y)                                                                                                 (3)

dFg = dm(g)                                                                                                     (4)

The angular velocity of the enforced circular motion is denoted by ω.  If we divide equation 1 by equation 2 and then divide equation 3 by equation 4, we get the relationship

tanθ = ω2y/g                                                                                                     (5)

The key realization to move forward is that the derivative dy/dx = tanθ.  This yields the governing equation:

dy/dx = ω2y/g                                                                                                    (6)

The particular solution to equation 6, after having applied the boundary condition y(0) = R0, the radius of the enforced circular path, is given by:

y(x) = R0exp(xω2/g)                                                                                          (7)

This solution is quite interesting.  We first notice that the density and area of cross-section of the rope have no effect on the shape it takes.  This is not surprising because both external forces were proportional to the elemental mass.  The more important takeaway here is that the lateral deflection becomes exponential.  The faster we spin the top of the rope, the more dramatic the curve.  This makes sense, but there is a serious flaw: the rope has a finite length.  As this function is exponential, there is no limit to the lateral deflection it describes.  As the imposed angular velocity increases, the lateral deflection can quickly become greater than the total length of the rope, which is physically impossible.  

I suspect that I ran into this problem because, in my original model, I neglected the gain in altitude that a particle driven laterally inevitably experiences.  For fun, I included this effect in a subsequent attempt.  After a page of work, I saw that numerical tools would be required to solve.  Again, it's summer, and I am content to move on and not pursue this problem further, especially when a closed solution appears impossible. 

Equation 7 may be a good approximation of the rope's profile for fairly slow rotation rates.  An experiment is difficult to conduct for multiple reasons. While air effects lead to a three dimensional profile, so to would inertial effects when it comes to establishing planar motion.  In principle, it may be possible to enforce the theoretical equilibrium configuration as well as a uniform angular velocity for all string elements, but it is not practical.  Failure to do this would inevitably lead to a helical 3D profile.

You may be thinking I did all that work for nothing.  It is important to realize that simplified approaches teach us a lot about complex problems.  They give confidence to the more strenuous, complex solutions that follow them.

And now, out of my cave.  Summer beckons.

Sunday, January 24, 2021

Enforced Rotation of Tarzan Rope (Problem)

I had so much fun with the rope problem I posted to start off 2021 (which was subsequently solved by Anthony Attia - see his elegant solution here), that I want to continue to explore this theme.  While that problem seemed tough (seeking the steady state profile of a uniform rope pinned at its top end and suspended vertically in a uniform horizontal wind), it turned out to be fairly simple.  It was almost disappointing.  To remedy the situation, consider an even more intriguing problem...

Imagine a Tarzan rope (bulk density 'p') that you suspend vertically in uniform surface gravity 'g'.  You then take the top end of the rope with length 'L' and move it with uniform circular motion in the horizontal plane (radius 'R' and angular frequency 'w').  Ignoring aerodynamic effects (because that would cause a 3D problem and have no clean analytical solution), what profile will the rope assume?  That is, if we froze the video at any given instant, what lateral deflection function, y(x), describes the rope's shape?  Treat the rope like a string (cannot support shear loads).

I spent some time on the problem, and it turns out to be even more interesting that I expected.  I will not give any hints this time.  I am curious to see if anyone will post a solution.  If you do, please provide a description of how you did it.

I am excited to share my solution, but I will be patient, and see what, if anything, gets submitted here.

Saturday, January 23, 2021

A Tarzan Rope in the Wind (Solution)

This is very exciting: a former student of mine, Anthony Attia, has submitted a solution to the Tarzan rope problem I posted some weeks ago.  Anthony was in my Mechanics class at Vanier College in 2016.  He is now pursuing undergraduate studies in mechanical engineering and simultaneously doing a stage at my former employer, MDA Space.

As is the case with some students, Anthony and I have stayed in touch since he graduated from college.  This post, however, is the first one in more than ten years of this blog's existence that someone other than me has written; it is about time.  Watch as Anthony analyzes a uniform rope, pinned at the top and vertically suspended, subjected to a horizontal uniform wind.

The following text appears here with Anthony Attia's consent:

___________________________________________________________________________________

When faced with a complex physical phenomenon, it is quite common to simplify the problem to a point where an analytical solution can be formulated. The simplification is done by stating assumptions throughout the approach. The more assumptions we take, the more likely our approximated answer will diverge from the true value. As students of science, it is our duty to ensure that we are equipped with enough knowledge to apply the proper assumptions.

Tarzan’s rope problem can be as complex as we want it to be. We can treat the rope as either flexible or rigid, we can treat the wind force as a function of time or a constant, we can consider the effects of cold temperature on the characteristic properties of air or we can neglect them. For the sake of maintaining my sanity and that of the reader’s, we shall treat the rope as a pinned rigid body who is subjected to a constant drag force that is acting in the horizontal direction. An important fact about assumptions is that there cannot be an incorrect one per say, however, every single one of them must be justified.

In my preliminary analysis, I will assume the rope to be rigid, effectively assuming that the profile of the rope will be linear when displaced.  Generally, this assumption would not be valid with a rope, but I will make it anyway and check the extent to which it was good later.

With that in mind, we can begin trying to find the velocity of the wind, by relating the drag force FD and the weight W.  Consider the model below, which depicts the scenario:

Given that the net drag force is acting on the center of gravity in the horizontal direction and the weight is acting in the vertical direction, the ratio of these forces, FD/W, ends up being equal to tan(θ).  We can take the sum of all torques about the pin and put them equal to zero.  Then, using the following definitions, we may express the wind speed as a function of the other parameters.

Surface gravity: g
Air density: p
Wind speed: V
Rope angle: θ
Rope mass: m
Rope diameter: d
Rope length: L
Rope shape (cylinder) coefficient of drag: Cd

The wind speed is then given by: 


Knowing this, we may begin computations to determine the wind speed that causes a specific rope deflection.  Assuming some reasonable 'Tarzan rope' values, it takes a 15 m/s wind to rotate the rope by 30°.  This seems reasonable.  But, we can only feel so much confidence in this result, as it is based on an assumption that may not be justifiable.

Say, however, that we now want to treat the rope as a flexible body; how would we proceed? Before answering that question, we must properly understand the behavior of weight and drag. In the previous figure, the drag force was lumped into a single vector whose line of action passes through the center of mass of the rope.  Let us do a quick thought experiment, if we were walking headwind, would our entire body feel pushed by the drag force or just a single point? The answer is the former, so why did we draw a single vector? That vector is actually the resultant or net drag force acting on the rope. If we were to properly illustrate the aerodynamic force that the body is subjected to, we would have to draw many smaller vectors that are acting on the entire exposed surface. These types of forces are called distributed load: though they act on every point of the body, we may sometimes use a single vector to represent the resulting effect (note that gravity is similarly distributed and then a resultant is used). Every segment of the rope has a mass equal to dm and the sum of all segment masses will yield the total mass m. Now, to solve the flexible body problem, we must assess a differential segment dm that is exposed to a differential drag of dFd by drawing its free body diagram.

Newton’s second law in x and in y yields:

These equations simplify to:

Equalizing the two equation we get:


It is evident that the equation obtained for the flexible body problem is the same as the rigid body problem, however, it is in a differential form. To remove the differentials, we must apply an integrating operator to the equation.  If we do so, the same expression linking the angle to the wind speed is obtained.

We conclude that both approaches lead to the same answer, but one requires an understanding of calculus, whilst the other requires only an understanding of mechanics. As one of my professors used to say, the simplest solution is often the best solution!

___________________________________________________________________________________

It gives me much pride to see a former student of mine express himself as he does here.  I get the same result on my end as that which he found.  The reason that the rigid body assumption works is because, due to the symmetry of the scenario, the uniformity of the fields and rope, the rope's profile must be linear.

In this problem, we have the weight force and the drag force.  They act vertically, and horizontally, respectively, onto each element.  While weight acts on dm elements, and drag acts on dA elements, both are uniform: we may think of each as a uniform field.  Effectively, they combine to form a uniform net field, and the rope simply aligns itself with it.  Though I initially thought the rope would have some curvature, it does not.  I am almost disappointed that the result is so simple.  I will try to pose a problem that has a stranger result in my next post.

Still, what I really want to emphasize here, is something greater than the problem itself.  I am thrilled that The Engineer's Pulse just had its first guest writer; he happens to be a fine engineer in the making.