Duck and Cover

[I considered "Don't Look Up," but I have a soft spot for Red Scare propaganda.]

This week there was news of an asteroid with a 1-in-82 chance of hitting the Earth in 2032. Most reports seemed to spin this as a reassuringly low chance, but as someone well-versed in low probabilities, I found it to be uncomfortably high, so I wanted to dig into it a bit. NASA's Small Body Database has an entry for this asteroid, which gives its orbital parameters and a nice interactive plot of its path through space. In order to do some calculations with this info though, I found a tool developed by MIT called poliastro, which can connect to the database and read the orbit definitions.

As a first pass, I tried duplicating the type of plot NASA included on its page, showing the orbit of the asteroid compared with a few of the planets around the time of the possible collision:

This shows the orbits of Earth in blue, Venus in cyan, Mars in red, and the asteroid in magenta. The planets' orbits are all fairly circular, while the asteroid's is sharply elliptical, and crosses the paths of the planets. The predicted flyby is around December 2032, and if you watch the animation you can see the asteroid skirts behind the Earth with a fair gap.

All predictions include errors though, so what happens if this orbit isn't quite the right one? Poliastro offers an object called a maneuver that can perturb a given orbit. Starting with the orbit above, I applied a random change in velocity with magnitude up to 10% of the initial value. Then I measured the distance between the Earth and the asteroid in time for each variation in the orbit:

The x-axis is the Modified Julian Date, a measure of time often used in astronomy. The black line is the predicted orbit, which only gets down to around 200 million km separation. For reference, that's about 1,000 times the distance from Earth to the Moon. However, some of those variations get a lot closer. To see how many of them are getting close, we can look at a histogram comparing the closest approach for each of the variants to the original:

Values below zero indicate the variant gets closer than the base prediction, and the red line marks the distance that would result in a collision with the Earth. Several of those variants are getting awfully close, so much like my pressure cooker post, I'm not sure I feel better at the end of this!

A Mass of Incandescent Gas

[Title from They Might Be Giants.]

This week, I got a question from my father Steve: We're able to identify the source of nuclear materials used in reactors and weapons from their isotope ratios. Could we do the same thing to figure out which star material that hits Earth came from?

First, let's talk about isotopes: Atoms are made up of a nucleus or protons and neutrons, surrounded by a cloud of electrons. The number of protons tells you what element the atom is – one for hydrogen, two for helium, and on down the periodic table. The number of electrons tells you the charge of the atom – neutral if it's equal to the number of protons, negative or positive for more or fewer electrons. Finally, the number of neutrons tells you the isotope – These are variations on the same element. For example, most carbon on Earth is called carbon-12, which has 6 protons and 6 neutrons for a total atomic mass of 12. However, some is carbon-14, which has 6 protons (since it's still carbon), but 8 neutrons. This configuration is unstable, and gradually decays to carbon-12. The mixture of carbon-14 and carbon-12 leads to radiocarbon dating, which is used in archeology to measure the age of excavations.

Natural uranium is almost all U-238, with small amounts of U-235 and a few other isotopes. Putting it in a nuclear reactor though will change those ratios. As the U-238 decays, it loses neutrons, raising the amount of U-235 present. The amount of U-235 in a sample can be further increased through enrichment, which uses various methods (often advanced centrifuges, which come up in nuclear policy) to separate the lighter U-235 from the heavier U-238. There can also be other isotopes of other elements mixed in depending on the exact process a reactor was using.

Now to stellar compositions: Stars are mostly made up of hydrogen, but the star's mass causes the hydrogen to fuse into helium, releasing energy that helps keep the star from collapsing. Helium can fuse too, and that can continue a few steps down the periodic table, but it's limited, typically petering out near iron:

Wikipedia (Click to enlarge)

The elements in yellow may be present in an active star, and will be spread around the universe when the star eventually explodes. We can find which are in a given star by looking at their absorption spectra:

Wikipedia

The star emits light in a black-body spectrum due to its heat, but the elements it contains will absorb some of that light, leading to dark bands on the spectrum. The frequencies (colors) of those bands correspond to different elements that let us determine the composition of the star.

Now to Steve's suggestion: When massive particles hit the Earth, could we use their makeup to associate them with a particular star? To my understanding, the answer is no, the particles that hit us are typically single protons or neutrons, not entire atoms, and certainly not the collection of atoms that would be needed to find a concentration of certain isotopes. There's another problem too: Uranium and other elements typically associated with isotopic signatures aren't present in active stars – If you look at the table above, you see those need neutron star collisions to form.

So it seems this idea won't work for distant stars, but if we screw things up badly enough on Earth, future scientists will be able to figure out where things went wrong, and curse that we ever trusted Mr. Clevver.

The How of the Wow

In 1977, the Big Ear radio telescope at Ohio State University [alma mater-mandated "Boo!"] picked up a burst of energy that was dubbed the "Wow! signal" thanks to researcher Jerry Ehman's note on the printout:

Big Ear

There's been a lot of speculation on possible sources – It was an exceptionally strong signal in a narrow bandwidth. On top of that, the frequency of the signal is associated with hydrogen's emission line, which had been identified by the first SETI paper in 1959 as a likely choice for extraterrestrial species to send messages. If you look closely at the image above, the circled bit reads "6EQUJ5". As this story got around, some people misunderstood this as a message in the signal, but it's actually the measurement of the signal-to-noise ratio in time: Each character is the SNR for a 10-second span, with letters A-Z representing SNRs of 10-35 – Quite high values!

A lot of the mystery surrounding this signal has been its transient nature. Followup searches found no recurrence of the signal, which ruled out a pulsar as the source, but also calls the ET explanation into doubt – Wouldn't a message like that be repeated? However, one of the arguments for the signal being artificial is its narrow bandwidth. Coincidentally, I've been listening to old episodes of the Futility Closet Podcast, and I came across an episode from almost 10 years ago about this signal. They liken the Big Ear telescope to a row of radios in a line, each tuned to a slightly different station – These correspond to the different columns in the printout above. One of the surprising things about the Wow! is that it only appears in one column. Most astrophysical signals would have more variation.

The signal is back in the news thanks to a new paper identifying its source. The authors were using the Arecibo Telescope (RIP) to search for similar signals to the Wow!, and were able to find several with the same spectral qualities, though never with the high intensity of the original signal. The hypothesis they arrived at was that an object like a magnetar or a soft gamma repeater released a burst of photons, which passed through a cloud of hydrogen, causing stimulated emission (the se in "laser") of photons at the detected frequency:

Figure 4

It might seem disappointing to have the possibility of a signal from aliens ruled out, but it's also an opportunity to understand the universe better, which in turn gives us better chances of finding a real signal in the future. Alternatively, we could end up making our own version of the Wow! for another species out there.

Gaussamer Threads

At my lab's group meeting this week, one of my colleagues was showing off the tungsten cube he had purchased to be used in a satellite designed to measure the Earth's mass distribution, similar to the GRACE mission I discussed earlier.

(May contain Infinity Stone)

Since we want the cube to be only affected by gravity, one of the steps in fabrication is degaussing, or removing any residual magnetism. I was curious about this process, since it didn't align with my previous context for degaussing: In high school, I worked for the IT department during the summers, and I was once assigned the task of erasing a collection of video tapes that had been used for a media class. This was done using a degausser, which was essentially an electromagnet that I ran over the surface of the tapes. However, this would put all the magnetic fields pointing in the same direction, not zero them.

One technique I found for driving the field to zero is to apply a large external field, then repeatedly reorient it while decreasing the magnitude. I decided to try this in 2D, similar to the Ising model, but more classical: The magnetic spins can point any direction in the plane, and experience a torque from the surrounding spins and external field. The animation below shows each spin's direction in black, the average direction in red, and the total magnitude of the field as time progresses in the lower plot. The external field is shown on the outer edges.

The way I applied the external fields I think results in the diagonal bias you can see near the end, but overall I'm impressed I was able to reduce the field to 25% of the original value – Not nearly enough for the sensitivity we need though!

Chain Change

Recently I've been rereading the book Seveneves by Neal Stephenson, which opens with the Moon being destroyed by an unknown object, forcing humanity to flee the surface of the Earth before the debris comes through the atmosphere. One of the engineers that goes into space notices something interesting about the chain he wears:

[He] had got it spinning around his neck. It had opened up into a broad, undulating oval that didn't touch his neck or collar anywhere, so it was just orbiting around him in free space. [...] He had learned a few tricks for speeding it up and coaxing it into different shapes by blowing on it with a drinking straw or flicking it with a fingernail. [...He] poked an index finger up into the chain's path. It caught on his knuckle, hiccupped, and suddenly wrapped around his hand in a chaotic tangle.

I was really curious if I could simulate some of these effects, so I came up with a simple model for the chain: A circle of points that are connected to each other by simple springs. The force applied by the springs is proportional to how far each point is from its neighbors, so the circle will tend to collapse to a point, but if we set it spinning, then just like an orbit we can keep the points from falling inward. We can see what happens if we tap one of the nodes:

This shows some really nifty behavior: Part of the poke moves around the circle in the direction of rotation, but there's also a dent that stays fairly stable until the other part of the poke comes around again. You might also notice when the animation loops that the chain has moved off-center by a fair amount. That made me wonder what would happen if I poked it symmetrically on each side:

Now the chain stays centered, but becomes more ellipsoidal, with the long axis rotating with the nodes. So far, so good – Let's try sticking a finger in it!

Hmm, not quite what we were looking for. To model the "finger," I checked at every step whether each node was inside the finger's radius. If it was, I would move it to the nearest point outside. Unfortunately, this meant that if I moved the finger much further in than I do above, links would start popping to the outside, which sounds painful for our simulacrum! I do like the faux-3D helix the chain forms on the other side of the finger though. 

Vanishing Trick

Back in March, I was at the LIGO/Virgo/KAGRA meeting, and during breakfast one morning, a fellow researcher posed an interesting question: Suppose a cluster of stars are all moving in the same direction. What can we learn from their vanishing point?

The concept of a vanishing point is often used in art: When parallel lines recede into the distance, they appear to converge at a point on the horizon. It seems logical then that our parallel stars will also converge to a point in the sky, which could tell us about how they're arranged in space. First though, we need to be able to generate a set of parallel lines in 3 dimensions. In 2D, this is fairly simple: Any lines with the same slope but different intercepts will be parallel. I found a page giving a simple way to express 3D parallel lines using a "double-equals" form:

The intercepts for each line are different, but the slopes associated with each dimension must be proportional between the lines.

This format is a bit difficult to imagine plotting, but we can fix that by setting the three terms equal to a parameter t, and then solving:

Once we have the paths in x, y, z, we can transform them into the right ascension and declination angles used by astronomers:

We have all the machinery in place now, so let's try it on a trio of lines. First, we can look at them in 3D, to check that we got parallel lines as expected:

Looks good! Now we'll run it through the projection...

Uh oh, the three lines don't share the same vanishing point! For a while I was sure I had made a mistake somewhere, but I think this can be explained by the fact that we're projecting onto a sphere, not a plane, as is usually done. I made a feeble effort at proving this to myself, but we're still running ourselves ragged getting things set up in our new home in Florida (hence the long silence here)!

Pulsar-Teacher Association

On Thursday, the NANOGrav project, along with international partners, made the announcement that they had detected a stochastic gravitational-wave background! This week, I thought I'd talk a bit about the news, and how the discovery was made.

First though, we should talk about what a stochastic gravitational-wave background is. Gravitational waves are produced whenever large amounts of mass move around in an asymmetric way. In the case of (still undetected) continuous waves, a bump on the neutron star, or for CBCs a pair of black holes or neutron stars. In the case of stochastic waves, we're talking about galaxies colliding, which is a much slower process. Since the movement is slower, that means the frequency is lower, on the order of nanohertz, or about 1/(32 years). That range of frequencies is far below what LIGO, or even LISA can detect:

Wikipedia

The orange region on the left is the background signals we're talking about, and the type of detectors used are called Pulsar Timing Arrays (PTAs). Pulsars are rapidly-spinning neutron stars, which produce pulses of radio-frequency signals at extremely regular intervals. They were initially referred to (jokingly) as LGMs, or "little green men", since it seemed like regular radio bursts would be a hallmark of an intelligent species.

The strength of a gravitational wave depends in part on the size of the masses that are moving. Since this background signal is due to entire galaxies moving, the gravitational waves are a million times stronger than those detected by LIGO! You might wonder then, why they were not detected before the CBCs that LIGO found. While I was thinking about this myself, an analogy occurred to me: Shifts in the Earth's tectonic plates are responsible for both earthquakes and continental drift. Even though the drift is on a significantly higher scale than the earthquakes, it's much harder to detect, due to the long periods (low frequency) involved, while earthquakes are picked up every day.

Since the first detection by Jocelyn Bell in 1968, many more pulsars have been found. The regular signals from these pulsars can be thought of as distant clocks ticking, from which the idea of pulsar timing arrays was conceived. A passing gravitational wave will cause a change in the signal's arrival time on Earth, but that change will depend on the direction of the pulsar, and the direction and polarization of the gravitational wave.

An isotropic signal means it should be the same in all directions. In 1983, Hellings and Downs suggested a method to detect such a signal: If two pulsars are affected by the same gravitational wave background, then the measurement of those pulse deviations on Earth should depend on the strength of that background, the noise in our measurements, and the orientation of the pulsar relative to Earth. By averaging the correlation between two pulsars over a long period, we can reduce the noise (which should be uncorrelated) and increase the background signal. Hellings and Downs derived a specific curve that that correlation should follow, according to the angle between the pairs of pulsars. After 15 years of collecting data from 67 pulsars, the collaboration presented this comparison to the expected curve:

Figure 1c

The points clearly deviate from the straight line that would result from no stochastic background signal, and instead follow the predicted curve, indicating a background signal is present. It's exciting to have another part of the gravitational wave spectrum filled in, and I look forward to more results from PTA groups!

Scanning the Skies

Last week, the LIGO-Virgo-Kagra Collaboration held a town hall with electromagnetic observers to discuss the status of the ongoing 4th observing run. Among the presenters were representatives of the Swift Burst Alert Telescope, or BAT, a satellite designed to detect gamma rays like the ones released by the binary neutron star collision LIGO picked up in 2017. They caught my attention with the name for their analysis tool for BAT: GUANO, and I'm a sucker for Dr. Strangelove references. What I started thinking about though, was what would be the best strategy for observing the whole sky, given that the satellite can only make detections in a small patch at any given time.

The theory is to use the same type of effect I discussed several years ago, where a spinning object tumbles in unexpected ways, thanks to Euler's equation:

According to this, if the angular velocity ω is not aligned with the symmetries of the object, represented by I, the velocity will change, even if the torque τ is zero. While it bears little resemblance to BAT, I decided to see what happens if I take a simple cylinder, and spin it off-axis. The plot below shows the cylinder in 3D, with a line marking a constant point on the outside to show rolling motion (though a plotting quirk makes it hard to see sometimes). Under that is a skymap of the parts of the sky the telescope has passed over.

You can see the color rescales to account for the telescope retracing areas it's seen before. I wondered though whether I could pick a particular rotational velocity that would allow the telescope to scan the whole sky without ever needing to apply a torque, which would require fuel. After failing to get an optimizer to figure out the best choice, I just tried a bunch of values, and settled on this one, which makes a nice latticework:

Of course, you can imagine the nauseating sort of view this pattern will give you! If this were actually the way the satellite operated, it would need a lot of post-processing, but the whole point of LIGO's public alerts is that detectors like this one can rapidly refocus on possible events, so I don't expect Swift to adopt this technique anytime soon.

Maybe Avoid the "Nuke" Idiom

This week I saw the news that Caltech had completed a proof-of-concept mission demonstrating power transfer from space using microwaves. I was instantly transported to my childhood playing SimCity 2000, which offered a microwave solar power plant for more advanced cities. I hadn't realized such a plan was actually feasible, but it's been considered since the 90s. The main obstacle has been cost, since the project requires many solar panels, all delivered to space. With the price of solar panels going down though, it's come back into the realm of possibility. However, I was curious about the possible risks of such a system, since SimCity (a credible source if ever there was one) suggested the possibility of accidental fires set by the system.

Before getting into that though, let's discuss how these systems work. To get the maximum amount of power from a solar cell, we want it to be constantly illuminated, but for a panel on Earth this isn't possible, since it spends roughly half its time in night. To get around this, we could put the panel is space, where it can always face the Sun, but now we have a new problem: How do we get the power it produces back to Earth? The simplest solution is to send back electromagnetic waves, but why choose microwaves? To answer that, we need to look at the absorption spectrum of the Earth's atmosphere (click to enlarge):

Wikipedia

We're interested in the regions with low absorption, since we want our beam to go through the atmosphere to a receiver on the surface. Microwaves have a wavelength around 12 cm, which falls neatly in the dip on the right side of the plot.

Since we want to keep the beam fixed on a single receiving station, we want the satellite to be in geostationary orbit, which requires a distance of 36,000 km. This page gives the size of the receiving antenna as about 3 km in diameter, which corresponds to an angle of about 5 millidegrees from the spacecraft. That page also gives the total power transfer as about 1 GW. Given how tiny that angle is, it's easy to imagine the beam being knocked off center, so how much damage could it do?

With the numbers above, the spot would have a power density of 141 W/m^2. For thermal radiation, this is well below the level that can burn you. Of course, these are microwaves, commonly used for cooking, so how does it compare to what you have in your home? A typical microwave oven has an area of around 20" x 24", and puts out around 1000 W, which comes to 3200 W/m^2, almost 23 times what our beam is sending!

So you'd be able to take a nap on the receiving dish without getting cooked, but your WiFi uses the same 2.4 GHz frequency that this system does, so would you be able to read this blog? I found a paper discussing the power density from WiFi base stations as a function of distance:

Figure 9

Note the scale is in milliwatts per square meter, meaning this is several orders of magnitude below the beam's power. Even this weak microwave signal can knock out your wifi!

It seems my childhood fear (or fascination) of fiery beams from the heavens was unfounded, but if they do build one of these, it has the potential to knock out your internet nearby...

Da Breeze of Debris

I recently saw a blog post suggesting publicly available datasets good for testing analysis techniques. Paging through them, I found the US Government's data server included NASA resources, and a connection to my own research occurred to me: One of my colleagues at the University of Florida has been working on simulating the effect of micrometeorite impacts on the LISA spacecraft. At a recent meeting, he was discussing the direction the meteorites might hit the spacecraft – They're generally falling inward toward the Sun, while the satellites (and the Earth) are orbiting around the Sun:

According to this model, very few meteorites should hit from the side facing the Sun. Less obvious though is the other 3 sides: Do more hit the side opposite the Sun, or is there a greater effect from the orbit taking us into the meteorite's path?

NASA's datasets include a record of meteorite landings on Earth, spanning the last 2 centuries, but unfortunately only provides the year, which means we can't find the Earth's position in the orbit. I almost gave up, but then I found a list of Fireball and Bollide Reports, which gives the precise date. Unlike the previous table, these are objects that completely burned in the atmosphere. We can look at the locations where these events were reported, using one of the map projections I discussed a while ago:

These appear fairly evenly distributed, but this plot doesn't consider the location of the Sun. Using the Astropy package, we can find the location of the Sun for a given date, then find the angle from the Sun to Earth, to the direction of the report:

This would seem to suggest that the most common angle is 90°, which corresponds to the orbit taking us into the meteorite. However, there are some significant caveats to this conclusion: It may be that there's a bias in this data, since it's easier to see a streak across the sky, while a meteor coming head-on would just appear as a point. Then there are the limitations of my analysis: The table only give the date of the events, not a time, so I may be introducing bias by choosing midnight. I'll be curious to see what results my colleague turns up, and maybe I'll find more datasets in the list to play with in the future.

Ring Around the 'Rora

Recently I started reading a page called Michigan Aurora Chasers, which shares pictures of the aurora taken in our current home state. The pictures are incredible, but I was really interested by a post that came up discussing Newton's Rings, an effect that can sometimes appear when viewing light from a monochromatic source through a series of lenses, like a camera.

Wikipedia has an example of the effect in a microscope, viewing a sodium lamp:

Wikipedia

For aurora viewers, this happens due to using a flat filter over their curved camera lens. When the light passes through the filter, some will bounce between the lens and filter one or more times, changing the phase. This light can then interfere with the light that passed straight through, producing the dark fringes seen above. The extra distance traveled by the light changes depending on how far from the center of the lens it hits:

The wavelength of light also changes how these rings will appear, since the total phase change from bouncing once from each surface is φ = 4πd/λ, where d is the distance between the filter and lens, and λ is the wavelength. We can scan through the visible wavelengths to see how the pattern of fringes changes (thanks to John D. Cook for the wavelength/RGB conversion):

Due to the spherical shape of the lens, as we get farther from the center, the distance changes more rapidly. This means that if we add up several wavelengths (since true monochromatic light is rare in nature), we see that the rings are only visible near the center of the image, as in the aurora photos from the link at the top:

Our area of Michigan is a bit too far south to get to see the aurora in our own sky, so it's been great to get to see the amazing pictures the group members post. On top of that, they introduced me to this really neat optical effect – Thanks Michigan Aurora Chasers!

Dreamt of in Your Philosophy

This week, I got several questions from Papou about the Big Bang and thermodynamics:

Is the Universe a closed system?

In order to answer this, we need to define what is meant by a "closed system": This is where no matter or energy enters or leaves the system under consideration. That means that the Earth, which gets energy from the Sun and radiates it back into space, is not a closed system. The Universe, however, which contains all matter and energy observed, is a closed system.

A different question, which uses confusingly similar terms, is whether the Universe is open or closed in the geometric sense. This refers to what happens if you travel in a straight line: Do you eventually come back to where you started, like you would on the surface of the Earth? Due to the influence of dark energy, for our universe the answer is no, it is geometrically open, whatever Modest Mouse may tell you.

How do the Laws of Thermodynamics apply to the Big Bang Theory?

Before we dive into this, let's talk about what the laws of thermodynamics are:

1. The energy in a closed system remains constant.
2. In any process, the entropy of the system must increase or stay the same.
3. As temperature approaches absolute zero, entropy approaches a constant

These are sometimes summarized as

1. You can't win.
2. You must lose.
3. You have to play the game.

I've mentioned entropy a couple of times on this blog, each time in a slightly different context. In this case, it can be thought of as energy that can no longer be used for work. As an example, if you have a bottle of compressed air, you could use it to propel a cart or turn a pinwheel, but if everything's at the same pressure, the air won't flow. This is actually the plot of a short story I read recently called Exhalation, about robots powered by compressed gas.

The Big Bang theory states that all matter and energy in the Universe started at a single point, which expanded outward. We can check off the first law, since the theory isn't saying anything about where that point came from – The Universe started with some fixed energy, and that energy is still here, just more spread out. That last part applies to laws 2 and 3 – Spreading out means more possible states for the matter, and lower temperature. It's these laws that lead to the heat death of the Universe, which I described before.

What is the impact of the Laws of Thermodynamics on Evolution?

The laws I outlined above are sometimes used as an argument against evolution: Evolution makes things more ordered, but that violates the increasing entropy requirement. The key is that the second law doesn't say entropy everywhere increases, just that it increases in the system as a whole. It's true that a human body has less entropy than a pile of microbes of the same mass, but that skips all the entropy generated in producing a human body. For a (slightly) more detailed discussion, I found this page, written by Robert Oerter at George Mason University.

Can there ever be enough Hawking Radiation to eliminate a Black Hole?

Yes! Hawking radiation is a process by which black holes can emit particles, but according to the first law up above, that means the black hole must lose energy/mass. If this happens over enough time, or the black hole is small enough, it can eventually evaporate. When the Large Hadron Collider first started up, some people (not scientists) were afraid it would create a black hole that would swallow the Earth. Experts were confident that if it did create one of the hypothesized microscopic black holes, they would quickly evaporate under Hawking radiation.

Thanks for more great questions, Papou!

A Breath of Fresh Vacuum

This past week was LISACon 8, a meeting of the LISA Consortium, similar to the LIGO meetings I've attended in the past. One of the presenters showed this daVinci-esque drawing of the LISA constellation:

ESA

A couple weeks ago, I talked about how I was working on a model of the rotation of the LISA satellites. Nominally, the satellites form an equilateral triangle, with 60° angles, but over the course of a year's orbit, those angles "breathe", getting wider and narrower as the spacecraft move along their orbits. That means that we need to change the angles the lasers point, so they can hit the distant sensors.

The laser beams are sent and received by Movable Optical Sub-Assemblies (MOSAs), the tubes in the picture above. We need to rotate those MOSAs to track the other satellites, but there's a problem: Angular momentum is conserved. Usually, we can count on the Earth to absorb extra angular momentum, but that's not possible in orbit. When we turn one of the MOSA, the spacecraft will turn under it. We can figure out how much using Newton's laws:

This says that the sum of the torques on each MOSA and the body of the spacecraft have to cancel out – "Every action has an equal and opposite reaction." We can use another of Newton's laws to express those torques in terms of the angular acceleration:

Here, the Is are the rotational inertia of the MOSAs and the spacecraft. The accelerations are measured in the inertial frame where all three bodies are rotating, but the MOSAs move within a range on the spacecraft, so we really want relative accelerations. We can get this by regrouping things:

We can make a bare-bones model by imagining two rods rotating in a solid disk to get I_M and I_S, then integrate to get the angles:

where m_S and m_M are the masses of the spacecraft and each MOSA, and φ1 and 2 are the angles of the MOSA relative to the spacecraft.

Eule Slick

A couple months ago, I talked about the LIGO work I'm doing here in Florida, but I'm also working on the LISA project, a gravitational wave detector in space. The detector consists of 3 spacecraft that orbit the Sun trailing slightly behind the Earth:

Wikipedia

Recently I've been developing a simulation of the LISA spacecraft, specifically their orientation to face each other. This gets into a part of Physics that I don't have a lot of experience with: rigid-body dynamics.

Most of the time, physicist can get away with considering things to be points or spheres, but things get more complicated with asymmetrical objects. Suppose we have a box that we want to rotate around two axes, x and y. Depending on which we do first, we get different results:

To get around this ambiguity, the mathematician Leonhard Euler (pronounced "oiler") realized you can specify the orientation of a 3D object by defining 3 rotations from a principle set of axes:

Wikipedia

I won't get into the details of how the angles are defined, but the result explains some really interesting effects. Here's a video recorded by a NASA astronaut aboard the space station of a "T-handle" spinning:

That strange tumbling behavior is explained by Euler's rotation equations:

where ω is the angular velocity, and τ is the torque applied. The I is the moment of inertia tensor, which requires some explanation: Simply put, it's the rotational equivalent of mass, which quantifies how a torque relates to an angular acceleration. What complicates things is that I depends on your choice of axes. It's a simple diagonal matrix in the frame where the T-handle is fixed, but that frame rotates. The fact we're working in a rotating frame results in the second term in the equation above.

Here's where the Euler angles come in: The differential equation above tells us how the angular velocity is changing, but we're interested in how the orientation of the handle changes. We need to relate the angular velocity to the change in the Euler angles, and include those in our integration. Putting everything together, we can simulate how the handle reacts to an initial rotation.

First with the spin perfectly aligned, nothing too interesting happens:

If we introduce just a 2% offset in the angle though...

we see precisely the sort of flip from the video! I've said it before here, but this is why I love Physics: With just a few relatively simple equations, you can explain even the weirder parts of reality.

Learn All That is Learnable

[Title from Star Trek: The Motion Picture, V'GER's mission.]

Earlier this week, the Voyager 2 spacecraft broke an 8-month silence to check in with NASA researchers. This is one of the most distant space probes from Earth, currently about 11.6 billion miles away. It was originally launched in 1977, with the primary goal of observing Neptune and Uranus. To get to those planets in 1989, it was given enough velocity to leave the solar system entirely, which it did 2 years ago. The antennas that communicate with Voyager, called the NASA Deep Space Network, were under maintenance for the past 8 months, preventing contact.

After seeing the news article about the renewed contact, I started reading more about the probes, and how they achieved their Sun-escaping speed. They key lies in the idea of a gravity assist, where by flying by a planet in the right way, an object can get a boost of speed:

Wikipedia
   Voyager 2 ·   Earth ·   Jupiter ·   Saturn ·   Uranus ·   Neptune ·   Sun

The principle at work in a gravity assist is actually something usually taught in intro Physics: an elastic collision. This is when two (or more) objects interact in a way that conserves both energy and momentum. An everyday example is a rubber ball that bounces to its original height when dropped. The energy and momentum of an object are given by

where m is the mass and v is the velocity. Conserving these quantities means that before and after the collision, the sums of each one for all the objects involved stays the same. To simplify things, we can consider the case in one dimension: Initially the probe and planet are moving toward each other, the probe slingshots around the planet, and leaves in the opposite direction it came. Then we can use the elastic collision equations to get

where capital variables are for the planet, lowercase for the probe, and i and f for initial and final. Since planets are generally a lot bigger than space probes (though maybe not in the case of the source of this post's title), we can Taylor expand around m/M ≪ 1 to get

Jupiter's mean orbital velocity is about 13.1 km/s, or about 30,000 mph, and we could get double that! Unfortunately, the angles typically don't work out that way, as you can see in the animation above.

I feel that any discussion of the Voyager Spacecraft has to include the hope that went into creating them, and to that end, I want to end this post with a picture of one of the Golden Disks. Each of the spacecraft was loaded with a message to the stars about Earth and its people. We looked for a way to represent ourselves to the cosmos, and found what was best in us.

Itty Bitty Bang

 Another question this week from Papou: Since a Black Hole can continuously acquire mass (except those cases wherein it loses matter per S. Hawking), does it follow that those Black Hole’s Event Horizon is also continuously getting larger. If that were not the case and the Event Horizon continuously reduced its boundary, does it not follow that Black Hole would become a point mass followed by a Big Bang. If that were the case, then it would be irrational that there was only one Big Bang and we are the product of that singular Big Bang. It is more likely, then, that there may have been other Big Bangs and there are other Universes out in Space. Is there anywhere in space where the Red Shift is not consistent with our Big Bang; which would then imply that there may have been multiple Big Bangs.

I think you get my drift ..... basically I am saying:   “Can a Black Hole become a Big Bang? What is the latest Red Shift evidence?

There are a couple different issues at play here, so let's address them one by one. First off, the event horizon of a black hole: A black hole is a region of space where matter has become so dense, light cannot escape its gravitational pull. The size of that space, called the Schwartzschild radius, is proportional to the amount of mass inside it:

where G is the gravitational constant, M is the mass, and c is the speed of light. You can actually find this yourself by looking for when the escape velocity is equal to c. This radius is sometimes called the event horizon, since in Special Relativity, events are described as points in space and time that are observed through light. If light cannot escape the black hole, we cannot observe events within it.

That brings us to the next part of the question: What happens to a black hole over time? As the equation above states, the event horizon radius is directly proportional to the mass within it, so if it loses mass due to Hawking radiation, or gains mass due to objects falling it, the radius can shrink or grow, but for fixed mass, the event horizon should stay fixed. For small black holes, Hawking radiation can eventually reduce the mass to zero, which is believed to result in the black hole evaporating. As the black hole shrinks, it will cross between the theories of General Relativity, and Quantum Mechanics. In their current forms, these theories are incompatible, but it's believed the evaporating black hole would release a burst of gamma rays as it vanished.

Still, there is a connection between event horizons and big bangs: In 2013, a group of scientists proposed that our universe could exist as the event horizon of a black hole in 4 spacial dimensions. In our 3 spatial dimensions, an event horizon is the surface of a sphere, which is 2D. A 4 dimensional black hole though would result in a 3D event horizon. Of course, that implies the possibility of a 2D universe on the event horizons of our universe.

Finally, the connection to red shift: The universe is expanding at every point, which means every point is moving away from every other point. I often find it helpful to imagine a big rubber sheet being stretched outward; any two points drawn on the sheet will get farther apart. As light moves through the universe, its wavelength gets stretched too, making it "redder", i.e. lower frequency. If you point a radio telescope at an empty part of the sky, as Arno Penzias and Robert Wilson did in 1965, you'll find a constant signal in the microwave band of light, called the Cosmic Microwave Background (CMB). This light is distributed in the blackbody spectrum, the range of photons emitted by objects of a given temperature. That temperature is from 380,000 years after the Big Bang, when things had cooled enough for protons and electrons to combine into hydrogen, about 3000 Kelvin. Over the billions of years that light has travelled, it's been red shifted down to around 2.725 Kelvin, in the microwave range.

If you look at a picture of the CMB, you may notice that it's not entirely uniform:

NASA

These anisotropies are mainly due to gravity pulling particles into clumps, which cool differently. Some have suggested the CMB also contains evidence of "bruises" from collisions between our universe and others existing in a larger multiverse. However, no such collisions have been detected so far.

Thanks for another great question, Papou!