Pulsars, like the one illustrated on the right, may seem an unlikely component of a method to look inside the Sun.
[Left:NASA/Goddard/SDO; Right: Adapted fromNASA/JPL-Caltech]
[Left:NASA/Goddard/SDO; Right: Adapted fromNASA/JPL-Caltech]
Title: Probing the Solar Interior with Lensed Gravitational Waves from Known Pulsars
Authors: Ryuichi Takahashi
First Author’s Institution: Hirosaki University
Status: Published in ApJ
Forget X-ray vision — how about gravitational wave vision? In today’s article, a team of researchers examine the possibility of using gravitational waves
from distant pulsars to learn about our own Sun. Much like the
terrestrial seismologists and geologists who peer into Earth by
listening carefully to the ways waves are distorted as they travel
through its many layers, these imagined future gravitational heliophysicists would use the subtle distortions of gravitational waves that have passed through the Sun to learn about its inner contents.
Figure 1: Depiction of the arrangement of an Earth-based gravitational wave detector and an object that emits continuous gravitational waves that is of interest to the authors of today’s article. When gravitational waves from the source have to pass directly through the Sun to reach Earth, they can be deflected from their original path and distorted in ways similar to how light is distorted when passing through a traditional lens. Credit: Takahashi et al. 2023
Multi-messenger Seismology?
The idea of using the interaction between waves and normal matter to
peer under the surface of otherwise opaque objects has a longstanding
history in physics. Think for example of the idea of seismic tomography, one of the main tools used by seismologists and geologists to understand the makeup of Earth’s interior. As powerful shock waves move through Earth’s interior
after seismic events like earthquakes, they come into contact with
regions of material that may differ in their composition, density,
temperature, and so on. Depending on the wave frequencies and the
properties of the materials with which the waves come into contact,
these waves will reflect, refract, diffract,
or be otherwise altered from their initial waveform. Measuring the
properties of these waves when they make contact with the surface again
at different points around the world can thus tell us a great deal about
what they may have encountered on their journey through the depths.
Similarly, when electromagnetic waves (light) encounter solid objects, they can undergo a number of interactions: certain wavelengths will be scattered or absorbed, and others may pass directly through an object. This is, roughly speaking, how X-ray imaging works: certain high-frequency electromagnetic rays can easily pass directly through your soft skin, but they will be reflected upon encountering denser material like bones, allowing us to reconstruct images of the interior of our own bodies without the need for invasive surgeries (thanks, science!).
Today’s article examines the feasibility of doing similar reconstructions but with waves of a different kind: gravitational waves. Gravitational waves, which have been discussed at length in previous Astrobites, are a hot topic right now within the astrophysics community given that the technology to directly detect their subtle presence has only come to maturity within the past decade or so. This is because gravitational waves, which are produced by the asymmetric motion of massive objects like black holes in binary orbits, produce extremely subtle effects here on Earth due in part to the weakness of the gravitational force and in part to the large distances the waves have typically traversed to reach us. Now that we are measuring these faint signals with regularity in ground-based gravitational wave detectors like LIGO, VIRGO, and KAGRA, interest has been growing in finding more and more exotic ways to use this brand new window into the universe to uncover its many secrets.
To understand the particularly out-there idea behind today’s article, we need to introduce at least one further concept: gravitational lensing. Gravitational lensing occurs when particles (or waves) travel close to a massive source and as a result have their trajectories altered. When a massive source (like a galaxy cluster) sits more or less directly between Earth and some distant object (like an individual galaxy), this deflection can act like a lens, focusing the light from the background galaxy towards Earth to make the galaxy appear bigger, brighter, or even more emoji-like. Additionally, as waves of any kind travel past such a lens, they will appear to take longer to reach the other side than they would have if there were no massive source. This effect is known as gravitational time delay, and it can sometimes manifest in ways not too dissimilar from the way in which light appears to “slow down” when passing through dense mediums like water. Gravitational lensing can cause all sorts of waves including gravitational waves to undergo many distorting effects similar to light traveling through media of varying densities or seismic waves traveling through Earth’s interior.
Taking all of these effects into account, we can begin to see why the idea of using gravitational waves to probe the interior of the Sun isn’t so far-fetched. While electromagnetic waves cannot typically pass into and out of the Sun due to their interactions with dense solar material, gravitational waves from a source behind the Sun would pass through easily, only experiencing distortion due to the aforementioned lensing effects caused by the varying density of the Sun along the line of sight between detectors on Earth and the source of the gravitational waves. While this idea has been explored before, today’s authors attempted a comprehensive analysis of what these distortions might look like for a set of real sources and examined how feasible it would actually be to detect them in present or future gravitational wave observatories. So can it be done? As it turns out, with some upgraded detectors, a dash of new millisecond (very fast-spinning) pulsar discoveries, and a bit of luck — it can!
Similarly, when electromagnetic waves (light) encounter solid objects, they can undergo a number of interactions: certain wavelengths will be scattered or absorbed, and others may pass directly through an object. This is, roughly speaking, how X-ray imaging works: certain high-frequency electromagnetic rays can easily pass directly through your soft skin, but they will be reflected upon encountering denser material like bones, allowing us to reconstruct images of the interior of our own bodies without the need for invasive surgeries (thanks, science!).
Today’s article examines the feasibility of doing similar reconstructions but with waves of a different kind: gravitational waves. Gravitational waves, which have been discussed at length in previous Astrobites, are a hot topic right now within the astrophysics community given that the technology to directly detect their subtle presence has only come to maturity within the past decade or so. This is because gravitational waves, which are produced by the asymmetric motion of massive objects like black holes in binary orbits, produce extremely subtle effects here on Earth due in part to the weakness of the gravitational force and in part to the large distances the waves have typically traversed to reach us. Now that we are measuring these faint signals with regularity in ground-based gravitational wave detectors like LIGO, VIRGO, and KAGRA, interest has been growing in finding more and more exotic ways to use this brand new window into the universe to uncover its many secrets.
To understand the particularly out-there idea behind today’s article, we need to introduce at least one further concept: gravitational lensing. Gravitational lensing occurs when particles (or waves) travel close to a massive source and as a result have their trajectories altered. When a massive source (like a galaxy cluster) sits more or less directly between Earth and some distant object (like an individual galaxy), this deflection can act like a lens, focusing the light from the background galaxy towards Earth to make the galaxy appear bigger, brighter, or even more emoji-like. Additionally, as waves of any kind travel past such a lens, they will appear to take longer to reach the other side than they would have if there were no massive source. This effect is known as gravitational time delay, and it can sometimes manifest in ways not too dissimilar from the way in which light appears to “slow down” when passing through dense mediums like water. Gravitational lensing can cause all sorts of waves including gravitational waves to undergo many distorting effects similar to light traveling through media of varying densities or seismic waves traveling through Earth’s interior.
Taking all of these effects into account, we can begin to see why the idea of using gravitational waves to probe the interior of the Sun isn’t so far-fetched. While electromagnetic waves cannot typically pass into and out of the Sun due to their interactions with dense solar material, gravitational waves from a source behind the Sun would pass through easily, only experiencing distortion due to the aforementioned lensing effects caused by the varying density of the Sun along the line of sight between detectors on Earth and the source of the gravitational waves. While this idea has been explored before, today’s authors attempted a comprehensive analysis of what these distortions might look like for a set of real sources and examined how feasible it would actually be to detect them in present or future gravitational wave observatories. So can it be done? As it turns out, with some upgraded detectors, a dash of new millisecond (very fast-spinning) pulsar discoveries, and a bit of luck — it can!
Figure 2: Models for the amplification (left) and phase offset (right) of a potential continuous wave signal as their line of sight appears to pass behind the Sun from four pulsars that are known to be eclipsed in this way once each year. Credit: Takahashi et al. 2023
Looking for Magic Millimeter Mountains
Unlike with light waves or even some seismic waves, humans do not have
the capacity to generate gravitational waves of sufficient power to be
measured and manipulated for the purposes of doing experiments. Instead,
if we want to use these new waves to our benefit, we have to get clever
with what nature has provided for us. Firstly, what we will need is a
source of continuous gravitational waves. Unlike most of what LIGO sees right now, which are the signature “chirps”
of compact objects undergoing their last seconds of merging, the
continuous waves (meaning gravitational wave signals that are
continuously emitted from a source and detectable for some appreciable
amount of time) that are expected to be seen by ground-based detectors
are most likely to come from tiny (sub-millimeter scale) deformations (sometimes called “mountains”) on the surface of rapidly rotating pulsars.
Once gravitational wave detectors increase in sensitivity enough to
finally detect these waves, researchers will want to find sources that
occasionally pass behind the Sun from our perspective here on Earth
(Figure 1). Over the course of several hours as Earth moves along its
orbit, detectors on Earth may be able to observe how this continuous
signal changes as it appears to pass behind different parts of the Sun,
like watching a straw appear to bend when lowered into a glass of water.
The authors of today’s article employ mathematical formulas (that are not too dissimilar from what one would see in an introductory optics course!) to calculate the amount of deflection, convergence, and time delay experienced by gravitational waves passing behind the face of the Sun at various angles relative to its center. Further wave-optics calculations are employed to find the corresponding amplification factors and phase offsets that waves of different frequencies would experience at these various angles. Their results are clearly shown in Figure 2: as each candidate pulsar (labeled by the different colored lines) appears to pass behind the face of the Sun, its corresponding gravitational wave signal will be amplified and offset in phase in complicated ways determined in part by the frequency of each gravitational wave and how closely the signal gets to passing directly behind the center of the Sun. By understanding how strong these effects are for continuous waves with different frequencies and amplitudes, the authors can begin to assess what it will take to detect them.
As it turns out, the ideal continuous-wave-emitting pulsars are those with high rotational frequencies (>10 Hz) that pass as close behind the Sun’s center as possible. When applying this cutoff to catalogs of known pulsars, only four currently fit the bill. While this doesn’t sound ideal, the authors go on to acknowledge that there are expected to be thousands more fast-spinning millisecond pulsars within our own galaxy that we have yet to discover, many of which could also turn out to pass behind the Sun on occasion.
With all this background knowledge in hand, the primary question left to tackle is this: can these slight deformations in continuous waves actually be detected with enough confidence to infer the density of different layers of the Sun? As to whether the lensing signal could be detectable at all, the authors of today’s article find that such a detection could be made with a high degree of confidence using known pulsars with about one year’s worth of observation time given a signal-to-noise ratio of around 100 or greater. The signal-to-noise ratio can depend on many factors including the loudness of a given source, the sensitivity of the detector, and the timespan of data collection, but to put this number in perspective, current signal-to-noise ratio upper limits for continuous wave detections from LIGO searches are estimated to be around 10.
Unfortunately, accurately measuring the solar density at several different solar depths is even trickier, as the accuracy of each measurement depends on a variety of factors including how many total layers of the Sun one attempts to measure and how one sets the distance between each layer (Figure 3). For one year of continuous wave observation of the three best pulsar candidates, the signal-to-noise ratio needed to accurately measure the solar density at two different layers is found to be ~104, which is an order of magnitude higher than is even expected from the next-generation gravitational wave detectors Cosmic Explorer and the Einstein Telescope. To measure densities across 6 or 10 different layers of the Sun’s interior, the signal-to-noise ratio requirements grow to ~106 and ~107, respectively (Figure 4), well beyond the capabilities of planned detectors unless more rapidly spinning pulsars passing behind the Sun can be found and loudly heard in these future detectors.
Figure 3: Estimates of the solar density profile will have to be made at different chosen slices called “annuli” (depicted here as the numbered concentric circles). Choices for how to pick these annuli affect how accurately their densities can be recovered for a given continuous wave pass. Credit: Takahashi et al. 2023
Figure 4: This plot depicts the uncertainty in solar density measurements at various solar depths using the expected lensing signatures of three known pulsars as they pass behind the Sun. The solid black line depicts the expected solar density as a function of solar radius (measured here as the angular position on the face of the Sun relative to its center). In the case where one attempts to measure six distinct densities with a very high signal-to-noise ratio of 106 over one year of observation (left plot), uncertainties can become fairly low close to or far away from the Sun’s center. To achieve similar uncertainties across 10 annuli, a comparative signal-to-noise ratio of 107 is necessary.Credit: Takahashi et al. 2023
Prospects for Gravitational Wave Vision
In recent years, combining gravitational waves with gravitational lensing has been proposed as a way to learn all sorts of new things about our universe, from constraints on the Hubble constant to independent measurements of the masses of distant stars. While it may take many more years for this sort of analysis to mature to a point where it can give us useful information about the Sun’s density profile, the fact that it may be possible at all is remarkable. For many decades the detection of gravitational waves was thought to be impossible. Now, not only are we detecting them with regularity, but we are finding all sorts of new ways to learn about our universe with each passing day — and that’s something to be excited about, even if you won’t be seeing gravitational wave vision goggles in stores anytime soon!
Original astrobite edited by Jessie Thwaites.
The authors of today’s article employ mathematical formulas (that are not too dissimilar from what one would see in an introductory optics course!) to calculate the amount of deflection, convergence, and time delay experienced by gravitational waves passing behind the face of the Sun at various angles relative to its center. Further wave-optics calculations are employed to find the corresponding amplification factors and phase offsets that waves of different frequencies would experience at these various angles. Their results are clearly shown in Figure 2: as each candidate pulsar (labeled by the different colored lines) appears to pass behind the face of the Sun, its corresponding gravitational wave signal will be amplified and offset in phase in complicated ways determined in part by the frequency of each gravitational wave and how closely the signal gets to passing directly behind the center of the Sun. By understanding how strong these effects are for continuous waves with different frequencies and amplitudes, the authors can begin to assess what it will take to detect them.
As it turns out, the ideal continuous-wave-emitting pulsars are those with high rotational frequencies (>10 Hz) that pass as close behind the Sun’s center as possible. When applying this cutoff to catalogs of known pulsars, only four currently fit the bill. While this doesn’t sound ideal, the authors go on to acknowledge that there are expected to be thousands more fast-spinning millisecond pulsars within our own galaxy that we have yet to discover, many of which could also turn out to pass behind the Sun on occasion.
With all this background knowledge in hand, the primary question left to tackle is this: can these slight deformations in continuous waves actually be detected with enough confidence to infer the density of different layers of the Sun? As to whether the lensing signal could be detectable at all, the authors of today’s article find that such a detection could be made with a high degree of confidence using known pulsars with about one year’s worth of observation time given a signal-to-noise ratio of around 100 or greater. The signal-to-noise ratio can depend on many factors including the loudness of a given source, the sensitivity of the detector, and the timespan of data collection, but to put this number in perspective, current signal-to-noise ratio upper limits for continuous wave detections from LIGO searches are estimated to be around 10.
Unfortunately, accurately measuring the solar density at several different solar depths is even trickier, as the accuracy of each measurement depends on a variety of factors including how many total layers of the Sun one attempts to measure and how one sets the distance between each layer (Figure 3). For one year of continuous wave observation of the three best pulsar candidates, the signal-to-noise ratio needed to accurately measure the solar density at two different layers is found to be ~104, which is an order of magnitude higher than is even expected from the next-generation gravitational wave detectors Cosmic Explorer and the Einstein Telescope. To measure densities across 6 or 10 different layers of the Sun’s interior, the signal-to-noise ratio requirements grow to ~106 and ~107, respectively (Figure 4), well beyond the capabilities of planned detectors unless more rapidly spinning pulsars passing behind the Sun can be found and loudly heard in these future detectors.
Figure 3: Estimates of the solar density profile will have to be made at different chosen slices called “annuli” (depicted here as the numbered concentric circles). Choices for how to pick these annuli affect how accurately their densities can be recovered for a given continuous wave pass. Credit: Takahashi et al. 2023
Figure 4: This plot depicts the uncertainty in solar density measurements at various solar depths using the expected lensing signatures of three known pulsars as they pass behind the Sun. The solid black line depicts the expected solar density as a function of solar radius (measured here as the angular position on the face of the Sun relative to its center). In the case where one attempts to measure six distinct densities with a very high signal-to-noise ratio of 106 over one year of observation (left plot), uncertainties can become fairly low close to or far away from the Sun’s center. To achieve similar uncertainties across 10 annuli, a comparative signal-to-noise ratio of 107 is necessary.Credit: Takahashi et al. 2023
Prospects for Gravitational Wave Vision
In recent years, combining gravitational waves with gravitational lensing has been proposed as a way to learn all sorts of new things about our universe, from constraints on the Hubble constant to independent measurements of the masses of distant stars. While it may take many more years for this sort of analysis to mature to a point where it can give us useful information about the Sun’s density profile, the fact that it may be possible at all is remarkable. For many decades the detection of gravitational waves was thought to be impossible. Now, not only are we detecting them with regularity, but we are finding all sorts of new ways to learn about our universe with each passing day — and that’s something to be excited about, even if you won’t be seeing gravitational wave vision goggles in stores anytime soon!
Original astrobite edited by Jessie Thwaites.
About the author, Lucas Brown:
I’m a current master’s student at Tufts University interested in
cosmology, relativity, and gravitational physics. I am currently doing
research on the stochastic gravitational wave background and pulsar
timing arrays. Outside of physics I love playing piano, climbing, and
spending time with my dog.
