The next in our series of papers describing the extended gravitational lens (extended, that is, in that we are no longer treating the lensing object as a gravitational monopole) is now out, on arXiv.

Here’s one of my favorite images from the paper, which superimposes the boundary of the quadrupole caustic (an astroid curve) onto a 3D plot showing the amplitude of the gravitational lens’s point-spread function.

I was having lots of fun working on this paper. It was, needless to say, a lot of work.

A very nice article about our work on the Solar Gravitational Lens was published a few days ago on Universe Today, on account of our recent preprint, which shows quantitative results assessing the impact of image reconstruction on signal and noise.

Because the SGL is such an imperfect lens, the noise penalty is substantial. However, as it turns out, it is much reduced when the projected image area is large, such as when an exoplanet in a nearby star system is targeted.

While this is good news, the Sun’s gravitational field has other imperfections. We are currently working on modeling these and assessing their impact on noise. Next comes the problem of imaging a moving target: an exoplanet that spins, which is illuminated from varying directions, and which may have varying surface features (clouds, vegetation, etc.) Accounting for all these effects is essential if we wish to translate basic theory into sound scientific and engineering requirements.

So, the fun continues. For now, it was nice to see this piece in Universe Today.

According to the immortal Douglas Adams, God’s final message to His creation is simple: “We apologize for the inconvenience.”

But there’s also another final message of sorts, the answer to the Ultimate Question about Life, Universe, and Everything: 42.

Recently, a researcher by the name of Michael Hippke analyzed the seemingly random bits that are contained in minute fluctuations of the Cosmic Microwave Background (CMB) radiation. His conclusion: there is no discernible pattern, no appearance of constants of nature, no detectable statistical autocorrelation. The message is random.

I beg to respectfully disagree. In the 512-bit segment published by Hippke, the bit sequence 101010 appears no fewer than eight, er, nine times (one occurrence split between two lines).

Now if we only knew the question to which the answer is 42…

The giant Arecibo radio telescope is no more.

Damaged by a broken cable just a few weeks ago, the telescope completely collapsed today.

Completed in 1963, the telescope was 57 years old, just like me. I hope I will last a few more years, though.

Tonight, Slava Turyshev sent me a link to an article that was actually published three months ago on medium.com but until now, escaped our attention.

It is a very nice summary of the work that we have been doing on the Solar Gravitational Lens to date.

It really captures the essence of our work and the challenges that we have been looking at.

And there is so much more to do! Countless more things to tackle: image reconstruction of a moving target, imperfections of the solar gravitational field, precision of navigation… not to mention the simple, basic challenge of attempting a deep space mission to a distance four times greater than anything to date, lasting several decades.

Yes, it can be done. No it’s not easy. But it’s a worthy challenge.

A few weeks ago, Christian Ready published a beautiful video on his YouTube channel, Launch Pad Astronomy. In this episode, he described in detail how the Solar Gravitational Lens (SGL) works, and also our efforts so far.

I like this video very much. Especially the part that begins at 10:28, where Christian describes how the SGL can be used for image acquisition. The entire video is well worth seeing, but this segment in particular does a better job than we were ever able to do with words alone, explaining how the Sun projects an image of a distant planet to a square kilometer sized area, and how this image is scanned, one imaginary pixel at a time, by measuring the brightness of the Einstein-ring around the Sun as seen from each pixel location.

We now understand this process well, but many more challenges remain. These include, in no particular order, deviations of the Sun from spherical symmetry, minor variations in the brightness of the solar corona, the relative motion of the observing probe, Sun, exosolar system and target planet therein, changing illumination of the target, rotation of the target, changing surface features (weather, perhaps vegetation) of the target, and the devil knows what else.

Even so, lately I have become reasonably confident, based on my own simulation work and our signal-to-noise estimates, as well as a deconvolution approach under development that takes some of the aforementioned issues into consideration, that a high-resolution image of a distant planet is, in fact, obtainable using the SGL.

A lot more work remains. The fun only just began. But I am immensely proud to be able to contribute to of this effort.

Our most comprehensive paper yet on the Solar Gravitational Lens is now online.

This was a difficult paper to write, but I think that, in the end, it was well worth the effort.

We are still investigating the spherical Sun (the gravitational field of the real Sun deviates ever so slightly from spherical symmetry, and that can, or rather it will, have measurable effects) and we are still considering a stationary target (as opposed to a planet with changing illumination and surface features) but in this paper, we now cover the entire image formation process, including models of what a telescope sees in the SGL’s focal region, how such observations can be stitched together to form an image, and how that image compares against the inevitable noise due to the low photon count and the bright solar corona.

Yesterday, we posted our latest paper on arXiv. Again, it is a paper about the solar gravitational lens.

This time around, our focus was on imaging an extended object, which of course can be trivially modeled as a multitude of point sources.

However, it is a multitude of point sources at a finite distance from the Sun.

This adds a twist. Previously, we modeled light from sources located at infinity: Incident light was in the form of plane waves.

But when the point source is at a finite distance, light from it comes in the form of spherical waves.

Now it is true that at a very large distance from the source, considering only a narrow beam of light, we can approximate those spherical waves as plane waves (paraxial approximation). But it still leaves us with the altered geometry.

But this is where a second observation becomes significant: As we can intuit, and as it is made evident through the use of the eikonal approximation, most of the time we can restrict our focus onto a single ray of light. A ray that, when deflected by the Sun, defines a plane. And the investigation can proceed in this plane.

The image above depicts two such planes, corresponding to the red and the green ray of light.

These rays do meet, however, at the axis of symmetry of the problem, which we call the optical axis. However, in the vicinity of this axis the symmetry of the problem is recovered, and the result no longer depends on the azimuthal angle that defines the plane in question.

To make a long story short, this allows us to reuse our previous results, by introducing the additional angle β, which determines, among other things, the additional distance (compared to parallel rays of light coming from infinity) that these light rays travel before meeting at the optical axis.

This is what our latest paper describes, in full detail.

Here is a thought that has been bothering me for some time.

We live in a universe that is subject to accelerating expansion. Galaxies that are not bound gravitationally to our Local Group will ultimately vanish from sight, accelerating away until the combination of distance and increasing redshift will make their light undetectable by any imaginable instrument.

Similarly, accelerating expansion means that there will be a time in the very distant future when the cosmic microwave background radiation itself will become completely undetectable by any conceivable technological means.

In this very distant future, the Local Group of galaxies will have merged already into a giant elliptical galaxy. Much of this future galaxy will be dark, as most stars would have run out of fuel already.

But there will still be light. Stars would still occasionally form. Some dwarf stars will continue to shine for trillions of years, using their available fuel at a very slow rate.

Which means that civilizations might still emerge, even in this unimaginably distant future.

And when they do, what will they see?

They will see themselves as living in an “island universe” in an otherwise empty, static cosmos. In short, precisely the kind of cosmos envisioned by many astronomers in the early 1920s, when it was still popular to think of the Milky Way as just such an island universe, not yet recognizing that many of the “spiral nebulae” seen through telescopes are in fact distant galaxies just as large, if not larger, than the Milky Way.

But these future civilizations will see no such nebulae. There will be no galaxies beyond their “island universe”. No microwave background either. In fact, no sign whatsoever that their universe is evolving, changing with time.

So what would a scientifically advanced future civilization conclude? Surely they would still discover general relativity. But would they believe its predictions of an expanding cosmos, despite the complete lack of evidence? Or would they see that prediction as a failure of the theory, which must be remedied?

In short, how would they ever come into possession of the knowledge that their universe was once young, dense, and full of galaxies, not to mention background radiation?

My guess is that they won’t. They will have no observational evidence, and their theories will reflect what they actually do see (a static, unchanging island universe floating in infinite, empty space).

Which raises the rather unnerving, unpleasant question: To what extent exist already features in our universe that are similarly unknowable, as they can no longer be detected by any conceivable instrumentation? Is it, in fact, possible to fully understand the physics of the universe, or are we already doomed to never being able to develop a full picture?

I find this question surprisingly unnerving and depressing.

A while back, I wrote about the uncanny resemblance between the interstellar asteroid ‘Oumuamua and the fictitious doomsday weapon Iilah in A. E. van Vogt’s 1948 short story Dormant.

And now I am reading that Iilah’s, I mean, ‘Oumuamua’s trajectory changed due to non-gravitational forces. The suspect is comet-like outgassing, but observations revealed no gas clouds, so it is a bit of a mystery.

Even if this is purely a natural phenomenon (and I firmly believe that it is, just in case it needs to be said) it is nonetheless mind-blowingly fascinating.

There is an excellent diagram accompanying an answer on StackExchange, and I’ve been meaning to copy it here, because I keep losing the address.

The diagram summarizes many measures of cosmic expansion in a nice, compact, but not necessarily easy-to-understand form:

So let me explain how to read this diagram. First of all, time is going from bottom to top. The thick horizontal black line represents the moment of now. Imagine this line moving upwards as time progresses.

The thick vertical black line is here. So the intersection of the two thick black lines in the middle is the here-and-now.

Distances are measured in terms of the comoving distance, which is basically telling you how far a distant object would be now, if you had a long measuring tape to measure its present-day location.

The area shaded red (marked “past light cone”) is all the events that happened in the universe that we could see, up to the moment of now. The boundary of this area is everything in this universe from which light is reaching us right now.

So just for fun, let us pick an object at a comoving distance of 30 gigalightyears (Gly). Look at the dotted vertical line corresponding to 30 Gly, halfway between the 20 and 40 marks (either side, doesn’t matter.) It intersects the boundary of the past light cone when the universe was roughly 700 million years old. Good, there were already young galaxies back then. If we were observing such a galaxy today, we’d be seeing it as it appeared when the universe was 700 million years old. Its light would have spent 13.1 billion years traveling before reaching our instruments.

Again look at the dotted vertical line at 30 Gly and extend it all the way to the “now” line. What does this tell you about this object? You can read the object’s redshift (z) off the diagram: its light is shifted down in frequency by a factor of about 9.

You can also read the object’s recession velocity, which is just a little over two times the vacuum speed of light. Yes… faster than light. This recession velocity is based on the rate of change of the scale factor, essentially the Hubble parameter times the comoving distance. The Doppler velocity that one would deduce from the object’s redshift yields a value less than the vacuum speed of light. (Curved spacetime is tricky; distances and speeds can be defined in various ways.)

Another thing about this diagram is that in addition to the past, it also sketches the future, taking into account the apparent accelerating expansion of the universe. Notice the light red shaded area marked “event horizon”. This area contains everything that we will be able to see at our present location, throughout the entire history of the universe, all the way to the infinite future. Things (events) outside this area will never be seen by us, will never influence us.

Note how the dotted line at 30 Gly intersects this boundary when the universe is about 5 billion years old. Yes, this means that we will only ever see the first less than 5 billion years of existence of a galaxy at a comoving distance of 30 Gly. Over time, light from this galaxy will be redshifted ever more, until it eventually appears to “freeze” and disappears from sight, never appearing to become older than 5 billion years.

Notice also how the dashed curves marking constant values of redshift bend inward, closer and closer to the “here” location as we approach the infinite future. This is a direct result of accelerating expansion: Things nearer and nearer to us will be caught up in the expansion, accelerating away from our location. Eventually this will stop, of course; cosmic acceleration will not rip apart structures that are gravitationally bound. But we will end up living in a true “island universe” in which nothing is seen at all beyond the largest gravitationally bound structure, the local group of galaxies. Fortunately that won’t happen anytime soon; we have many tens of billions of years until then.

Lastly, the particle horizon (blue lines) essentially marks the size of the visible part of the universe at any given time. Notice how the width of the interval marked by the intersection of the now line and the blue lines is identical to the width of the past light cone at the bottom of this diagram. Notice also how the blue lines correspond to infinite redshift.

As I said, this diagram is not an easy read but it is well worth studying.

There is a very interesting concept in the works at NASA, to which I had a chance to contribute a bit: the Solar Gravitational Telescope.

The idea, explained in this brand new NASA video, is to use the bending of light by the Sun to form an image of distant objects.

The resolving power of such a telescope would be phenomenal. In principle, it is possible to use it to form a megapixel-resolution image of an exoplanet as far as 100 light years from the Earth.

The technical difficulties are, however, challenging. For starters, a probe would need to be placed at least 550 astronomical units (about four times the distance to Voyager 1) from the Sun, precisely located to be on the opposite side of the Sun relative to the exoplanet. The probe would then have to mimic the combined motion of our Sun (dragged about by the gravitational pull of planets in the solar system) and the exoplanet (orbiting its own sun). Light from the Sun will need to be carefully blocked to ensure that we capture light from the exoplanet with as little noise as possible. And each time the probe takes a picture of the ring of light (the Einstein ring) around the Sun, it will be the combined light of many adjacent pixels on the exoplanet. The probe will have traverse a region that is roughly a kilometer across, taking pictures one pixel at a time, which will need to be deconvoluted. The fact that the exoplanet itself is not constant in appearance (it will go through phases of illumination, it may have changing cloud cover, perhaps even changes in vegetation) further complicates matters. Still… it can be done, and it can be accomplished using technology we already have.

By its very nature, it would be a very long duration mission. If such a probe was launched today, it would take 25-30 years for it to reach the place where light rays passing on both sides of the Sun first meet and thus the focal line begins. It will probably take another few years to collect enough data for successful deconvolution and image reconstruction. Where will I be 30-35 years from now? An old man (or a dead man). And of course no probe will be launched today; even under optimal circumstances, I’d say we’re at least a decade away from launch. In other words, I have no chance of seeing that high-resolution exoplanet image unless I live to see (at least) my 100th birthday.

Still, it is fun to dream, and fun to participate in such things. Though now I better pay attention to other things as well, including things that, well, help my bank account, because this sure as heck doesn’t.

The Internet (or at least, certain corners of the Internet where conspiracy theories thrive) is abuzz with speculation that the extrasolar asteroid ‘Oumuamua, best known, apart from its hyperbolic trajectory, for its oddly elongated shape, may be of artificial, extraterrestrial origin.

Some mention the similarity between ‘Oumuamua and Arthur C. Clarke’s extraterrestrial generational ship Rama, forgetting that Rama was a ship 50 kilometers in length, an obviously engineered cylinder, not a rock.

But then… I suddenly remembered that there was another artificial object of extrasolar origin in the science-fiction literature. It is Iilah, from A. E. van Vogt’s 1948 short story Dormant. Iilah is not discovered in orbit; rather, it lays dormant on the ocean floor for millions of years until it is awakened by the feeble radioactivity of isotopes that appear in the ocean as a result of the use and testing of nuclear weapons.

Iilah climbs out of the sea and is thus discovered. It becomes an object of study by a paranoid military, which ultimately decides to destroy it using a nuclear weapon.

Unfortunately, the energy of the explosion achieves the exact opposite: instead of destroying Iilah, it fully awakens it, making it finally remember its original purpose. Iilah then sets itself up for a tremendous explosion that knocks the Earth out of orbit, ultimately causing it to fall into the Sun, turning the Sun into a nova. Why? Because Iilah was programmed to do this. Because “robot atom bombs do not make up their own minds.”

Artist’s impression of ‘Oumuamua

So here is the thing… the Iilah of van Vogt’s story had almost the exact same dimensions (it was about 400 feet in length) and appearance (a rock, like rough granite, with streaks of pink) as ‘Oumuamua.

Go figure.

Today, a “multi-messenger” observation of a gravitational wave event was announced.

This is a big freaking deal. This is a Really Big Freaking Deal. For the very first time, ever, we observed an event, the merger of two neutron stars, simultaneously using both gravitational waves and electromagnetic waves, the latter including light, radio waves, UV, X-rays, gamma rays.

From http://iopscience.iop.org/article/10.3847/2041-8213/aa91c9

The significance of this observation must not be underestimated. For the first time, we have direct validation of a LIGO gravitational wave observation. It demonstrates that our interpretation of LIGO data is actually correct, as is our understanding of neutron star mergers; one of the most important astrophysical processes, as it is one of the sources of isotopes heavier than iron in the universe.

Think about it… every time you hold, say, a piece of gold in your hands, you are holding something that was forged in an astrophysical event like this one billions of years ago.

So here it is: another gravitational wave event detection by the LIGO observatories. But this time, there is a twist: a third detector, the less sensitive European VIRGO observatory, also saw this event.

This is amazing. Among other things, having three observatories see the same event is sufficient to triangulate the sky position of the event with much greater precision than before. With additional detectors coming online in the future, the era of gravitational wave astronomy has truly arrived.

Here is a belated picture of yesterday’s solar eclipse, taken by my friend David in New York City:

His equipment is (semi-)professional but the solar filter that he used wasn’t. Still, it is a heck of a lot better than anything I was able to see (or project with a makeshift pinhole camera). I suggested to him to obtain a quality solar filter by 2024. Who knows, we may meet in Watertown to watch totality.

There is a brand new video on YouTube today, explaining the concept of the Solar Gravitational Telescope concept:

It really is very well done. Based in part on our paper with Slava Turyshev, it coherently explains how this concept would work and what the challenges are. Thank you, Jimiticus.

But the biggest challenge… this would be truly a generational effort. I am 54 this year. Assuming the project is greenlighted today and the spacecraft is ready for launch in ten years’ time… the earliest for useful data to be collected would be more than 40 years from now, when, unless I am exceptionally lucky with my health, I am either long dead already, or senile in my mid-90s.

Here is a spectacular photograph of the Moon made last night by my good friend David Ada-Winter in light-polluted New Jersey:

David explains: “I took this picture of the Moon using the so-called Sunny 16 rule, the essence of which is the following: On a clear day, with an aperture of 16, the exposition time must be the reciprocal of the ISO value. In the case of this picture, the ISO was 200, so the exposition time was 1/200 with an aperture of 16. In front of my telescopic lens, I also used a doubler that extended the focal length to 800 mm. The picture itself was made with the Canon Rebel t2i camera, which has a crop factor of 1.6, allowing the Moon to appear even larger in the image.”

Apparently, David’s wife disapproves of his pricey hobby. I’m tempted to remind her that other men of David’s age often acquire even pricier hobbies, which usually involve brightly colored sports cars and lightly clad ladies…

The reason for my trip to China was to participate in the 3rd workshop on the TianQin mission.

TianQin is a proposed space-borne gravitational wave detector. It is described in our paper, which was recently accepted for publication in Classical and Quantum Gravity. The name, as typical for China, is poetic: it means a zither or harp in space or perhaps (sounds much nicer in English) a celestial harp. A harp that resonates in response to continuous gravitational waves that come from binary pulsars.

Gravitational waves are notoriously hard to detect because they are extremely weak. To date, we only have indirect confirmation of gravitational waves: closely orbiting binary pulsars are known to exhibit orbital decay that is consistent with the predictions of Einstein’s gravity.

Gravitational radiation is quadrupole radiation. It means basically that it simultaneously squeezes spacetime in one direction and stretches it in a perpendicular direction. This leads to the preferred method of detection: two perpendicular laser beams set to interfere with each other. As a gravitational wave passes through, a phase shift occurs as one beam travels a slightly longer, the other a slightly shorter distance. This phase shift manifests itself as an interference pattern, which can be detected.

But detection is much harder in practice than it sounds. Gravitational waves are not only very weak, they are also typically very low in frequency. Strong gravitational waves (relatively speaking) are produced by binaries such as HM Cancri (aka. RX J0806.3+1527) but even such an extreme binary system has an orbital period of several minutes. The corresponding gravitational wave frequency is measured in millihertz, and the wavelength, in tens or hundreds of millions of kilometers.

There is one exception: inspiraling neutron star or black hole binary systems at the very end of their lives. These could produce detectable gravitational waves with frequencies up to even a kilohertz or so, but these are random, transient events. Nonetheless, there are terrestrial detectors such as LIGO (Laser Interferometer Gravitational-wave Observatory) that are designed to detect such events, and the rumor I heard is that it may have already happened. Or not… let’s wait for the announcement.

But the continuous waves from close binaries require a detector comparable in size to the wavelength of their gravitational radiation. In short, an interferometer in which the laser beams can travel at least a few hundred thousand kilometers, preferably more. Which means that the interferometer must be in space.

This is the idea behind LISA, the Laser Interferometer Space Antenna project. Its current incarnation is eLISA (the “e” stands for “evolved”), a proposed European Space Agency mission, a precursor of which, LISA Pathfinder, was launched just a few days ago. Nonetheless, eLISA’s future remains uncertain.

Enter the Chinese, with TianQin. Whereas eLISA’s configuration of three spacecraft is designed to be in deep space orbiting one of the Earth-Sun Lagrange points with inteferometer arm lengths as long as 1.5 million kilometers, TianQin’s more modest proposal calls for a geocentric configuration, with arm lengths of 150,000 km or so. This means reduced sensitivity, of course, and the geocentric orbit introduces unique challenges. Nonetheless, our colleagues believe that it is fundamentally feasible for TianQin to detect gravitational waves from a known source with sufficient certainty. In other words, the primary mission objective of TianQin is to serve as a gravitational wave detector, confirming the existence of continuous waves emitted by a known binary system, as opposed to being an observatory, usable to find previously unknown sources of gravitational radiation. Detection is always easier: in radio technology, for instance, a lock-in amplifier can be used to detect the presence of a carrier wave even when it is far too weak to carry any useful information.

 Theoretical sensitivity curve of the proposed TianQin mission.

The challenges of TianQin are numerous, but here are a few main ones:

• First, precisely controlling the orbits of shielded, drag-free test masses such that their acceleration due to nongravitational forces is less than $$10^{-15}~{\rm m}/{\rm s}^2$$.
• Second, precisely controlling the optical path such that no unmodeled effects (e.g., thermal expansion due to solar heating) contribute unmodeled changes more than a picometer in length.
• Third, implementing time-delay interferometry (TDI), which is necessary in order to be able to compare the phases of laser signals that traveled different lengths, and do so with sufficient timing accuracy to minimize the contributions due to fluctuations in laser frequency.

Indeed, some of the accuracy requirements of TianQin exceed those of eLISA. This is a tall order for any space organization, and China is no exception. Still, as they say, where there is a will…

 Unequal-arm Michelson interferometer.

One thing that complicates matters is that there are legal barriers when it comes to cooperation with China. In the United States there are strong legal restrictions preventing NASA and researchers at NASA from cooperating with Chinese citizens and Chinese enterprises. (Thankfully, Canada is a little more open-minded in this regard.) Then there is the export control regime: Technologies that can be utilized to navigate ballistic missiles, to offer satellite-based navigation on the ground, and to perform remote sensing may be categorized as munitions and fall under export control restrictions in North America, with China specifically listed as a proscribed country.

The know-how (and software) that would be used to navigate the TianQin constellation is arguably subject to such restrictions at least on the first two counts, but possibly even the third: a precision interferometer in orbit can be used for gravitiational remote sensing, as it has been amply demonstrated by GRACE (Gravity Recovery And Climate Experiment), which was orbiting the Earth, and GRAIL (Gravity Recovery And Interior Laboratory) in lunar orbit. Then there is the Chinese side of things: precision navigation requires detailed information about the capabilities of tracking stations in China, which may be, for all I know, state secrets.

While these issues make things a little tricky for Western researchers, TianQin nonetheless has a chance of becoming a milestone experiment. I sincerely hope that they succeed. And I certainly feel honored, having been invited to take part in this workshop.

In Douglas Adams’s immortal Hitchiker’s Guide to the Galaxy, someone builds a device called the Total Perspective Vortex. This device invariably drives people insane by simply showing them exactly how insignificant they are with respect to this humongous universe.

The Total Perspective Vortex may not exist in reality, but here is the next best thing: A model of the solar system, drawn to scale.

The scale of this page is set so that the Moon occupies one screen pixel. As a result, we have an image that is almost a thousand times wider than my HD computer monitor. It takes a while to scroll through it.

Thankfully, there is an animation option that not only scrolls through the image automatically, but does so at the fastest speed possible, the speed of light.

Oh, did I mention that it still takes well over five hours to scroll all the way to Pluto?

By the way, the nearest star, our closest stellar neighbor is roughly 2,000 times as far from us as Pluto.

Or, once again in the words of Douglas Adams, “Space is big. Really big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.”