# What is the furthest star for which we can measure the diameter?

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If a star is close enough that we can measure its parallax and that it doesn't appear point-like, then we can deduce its diameter. Can we measure the diameter of all the stars for which we can measure the parallax, or is the angular diameter a limiting factor? How far out is the diameter of a star measurable?

It's neither the angular diameter or prallax precision that is the limiting factor, but the fact that it is difficult to get the interferometric measurements for faint stars.

State-of-the-art angular diameters are measured by infrared interferometry (e.g. with the CHARA array - Gordon et al. 2019). The most precise measurements of angular diameters have an uncertainty of about 17 microarcseconds in that paper. That means the smallest angular diameters that can yield a solid detection of the disc of a star are about $$heta = 50$$ microarcseconds. But such measurements can only be achieved for stars down to about $$V sim 6$$.

We can write down an equation for the relationship between $$heta$$, the radius of the star $$R$$ and its distance $$D$$. $$D = frac{2R}{ heta} = 186left(frac{R}{R_{odot}} ight) left(frac{ heta}{50 mu{ m as}} ight)^{-1} { m pc}$$

This means that if a $$1R_{odot}$$ star has its distance known more precisely than the $$sim 30$$% error in the angular diameter, then it's radius can be measured out to distances of about 200 pc (with a precision of about 30%). This is easily achieved - a $$1R_{odot}$$ solar-type star at a distance of 200 pc has a brightness of $$V sim 11$$ and has a parallax uncertainty in the Gaia catalogue of only about 0.1 mas (mostly a systematic error at present), meaning the distance is known to about 2%. Thus, parallax uncertainty is not the limiting factor. However, a 6th magnitude solar-type star would be at $$D sim 20$$ pc. Much further away than this and the source is just not bright enough to get an angular diameter measurement with CHARA.

On the other hand, a large red giant star, with $$R sim 200 R_{odot}$$ could in principle have its radius measured to 30% out to a distance of 37 kpc with the angular resolution available. Such a star would not however have a measurable parallax at present. The parallax would be 27 microarcseconds, which should just be resolvable in the final Gaia data reduction in a few years time. However, the star would also have $$V sim 14$$ and is way too faint for the current interferometric capabilities of CHARA. A 6th magnitude red giant is at $$D sim 500$$ pc, a distance at which both its angular diameter and parallax are precisely measurable.

So the answer is that the limitation is not the angular resolution, or the parallax. It is the brightness limit for the interferometers that are used to do the measurements. This limitation means that you can measure the diamters of solar-type dwarfs to a few 10s of pc, and the diameters of the largest red giants to about 1 kpc.

Update: There are other interferometers that can be used. The VLTI/Amber interferometer in Chile utilises larger telescopes than CHARA and can in principle work to fainter magnitudes. A paper by Chesneau et al. (2014) measures an angular diameter for HR5171A (a red hypergiant) to be $$3.39 pm 0.02$$ mas, at a distance of 3.6 kpc. https://www.aanda.org/articles/aa/full_html/2014/03/aa22421-13/aa22421-13.html

Although much further away, this huge star is bright enough ($$V sim 6.5$$) that an angular diameter measurement can still be achieved. Note though that the limitation here is still the brightness of the object, not its distance or angular size.

The parallax is always easier to measure than the angular size of any planet. This is also true for most stars, excluding hypergiants and some supergiants. The parallax is given by

$$displaystyle heta_{p} = frac{d_e}{d}$$

where $$d_e$$ is the Earth-Sun distance and is $$d$$ the distance of the star. Instead, the angular size of an object with radius $$r$$ is

$$displaystyle heta_{r} = frac{r}{d}$$

Let $$heta_{min,p}$$ and $$heta_{min, r}$$ be the best angular resolutions we can obtain for a measurement of parallax and angular size respectively. Clearly, the angular size will be the limiting factor if

$$displaystyle frac{r}{ heta_{min,r}} < frac{d_e}{ heta_{min,p}}$$

Now, let us take $$heta_{min,p} approx 10^{-5}$$ as (precision of Gaia mission) and $$heta_{min,r} approx 2 cdot 10^{-6}$$ as (The Navy Prototype Optical Interferometer). Therefore, the condition for the angular size to be the limiting factor is (to the closest order of magnitude):

$$r < d_e/10$$

Of course there can be no planet with a radius larger than a tenth the Earth-Sun distance, therefore the angular size is always the limiting factor for planets.

The largest main sequence stars (O-type stars) have radii of about $$20 R_{⊙}$$, while the Earth-Sun distance is just over $$200 R_{⊙}$$. Therefore, O-type stars sit across the boundary, whereas the angular size is the limiting factor for all other main sequence stars.

Now, let's examine stars outside the main sequence. The largest stars are the Hypergiants, with radii greater than $$1000 R_{⊙}$$. The largest known star to date is VY Canis Majoris, with a radius of $$1,420 R_{⊙}$$. Supergiants usually range from $$50 R_{⊙}$$ to $$500 R_{⊙}$$, so in this case the distance is the limiting factor. Using the best interferometers to date, we would be able to measure diameters up to 200 $$mu$$as with a precision of 1%. This corresponds to measuring the diameter of a typical supergiant up to a distance of 2kpc. The first star to have its angular diameter measured was Betelgeuse.

## Methuselah's Star is not older than the Universe after all. But it's still pretty frakking old.

There are times in science when you get a result that you know is wrong, but you don't know why. This can lead to some interesting suppositions and many times to very interesting science.

Like, for example, when you find a star that appears to be, um, older than the Universe.

Such is HD 140283, a star similar to the Sun that lies 200 light years from Earth — pretty close as stars go. It's been known for some time to be a special kind of star, one we say is low metallicity.

In a nutshell, the early Universe was mostly hydrogen and helium, and heavier elements (what astronomers call metals for historical reasons) didn't start turning up until massive stars made them in their cores, exploded, and sent them sleeting out into space to get incorporated into newer stars.

What this means is that older stars tend to have fewer heavy elements in them, and younger stars have more. We use the Sun as a standard, because why not. By that measure, HD 140283 has a metallicity 1/250 th (0.004) that of the Sun!

That's extremely old, implying a very old age for this star. Other measurements seem to bolster that, which is why it was nicknamed Methuselah's Star. One measurement, using Hubble in 2013, put its age at about 14.5 ± 0.8 billion years old.

That's a bit of a problem. The Universe is only 13.77 ± 0.04 billion years old. Is this star older than Universe?

HD 140283, Methuselah’s Star, seen in a sky survey. Credit: NASA/GSFC/SkyView/DSS2

Well, pump the brakes here. The uncertainties in those numbers (the ± part) are carrying a lot of weight. 14.5 – 0.8 = 13.7, so it's entirely possible the age of the star is less than thought. There's no astronomer on the planet, I'd wager, who thinks the star is actually older than the Universe itself! Of course, if you look you'll find a lot of articles overstating its age, and some from less-then-reputable sites saying HD 140283 comes from an earlier Universe that survived through the Big Bang to ours. How that works exactly isn't clear (because it's flat-Earth level science).

The problem is narrowing down the uncertainties. One thing that helps is getting the distance, so that we know how bright it really is (we can measure the brightness here on Earth, and then correcting for distance know how much light it's actually emitting, called the luminosity). Recent advances have helped here, so a group of astronomers tackled this star once again. Using the best measurements to date, they used software that creates models of how stars change over time given basic properties of them like mass, luminosity, elemental content (that one's critical here, due to the low metal content of the star), and so on.

What they found is that the mass of the star is 0.81 ± 0.05 times that of the Sun, and its age is (drumroll please). 12.01 ± 0.05 billion years.

Ahhh, that's better. Comfortably younger than the cosmos, and therefore formed after the Big Bang.

But yeah, it's still really frakking old. The Sun is 4.6 billion years old, for comparison. The Milky Way galaxy itself is about 12 billion years old meaning this was one of the very first stars to be born in it, maybe even as the galaxy itself was forming. That's amazing.

Given its advanced age, it's not a surprise that the star is now dying. It's been running out of hydrogen fuel in its core for some time now, and is starting to expand into a red giant. At the moment that process is just starting, so it's what we call a subgiant. It's already about twice the Sun's diameter, and will eventually swell up quite a bit more. After that it will undergo a long process of losing mass through a subatomic particle wind, shrink, expand again, and eventually blow off all its outer layers to become a low mass white dwarf. At that point it will cool, fading away over more billions of years to become a black dwarf. The time that will take is actually much longer than it's already been alive, so in that sense it has a very, very long future ahead of it.

So it seems the cosmic age paradox of Methuselah's Star is now resolved.

What Methuselah's Star may look like up close (artist's depiction). Credit: NASA, SDO, and the AIA team. Modified by Phil Plait

But, I'll note, it's unlikely to be the very oldest star in the galaxy. The Milky Way is 120,000 light years across, so what are the odds the oldest star happens to be 200 light years away? It just happens to be close enough to study well, and the distance implies there are many more stars like it, and maybe older, in the galaxy, one of the earliest generations of stars in the Universe. Finding and analyzing them will add to our understanding of how these ancient stellar denizens behave.

The stars are old, but the day is yet young. Plenty more out there to find and study.

## How Was It Seen?

A group of astronomers had been monitoring a far-off supernova—the explosive death of a giant star—using the Hubble Space Telescope when they saw a new speck of light.

Its glow was given a boost thanks to what's called gravitational lensing. This is when gravity from a massive celestial object acts like a magnifying glass, bending and amplifying the light from objects behind it.

Five billion light-years from Earth, a galaxy cluster sits between our planet and Icarus. According to a model outlined this week in Nature Astronomy, Icarus was magnified when a star in that galaxy cluster moved in front of the more distant star, boosting it to 2,000 times its actual brightness.

“The source isn’t getting hotter it’s not exploding. The light is just being magnified. And that’s what you expect from gravitational lensing,” study leader Patrick Kelly of the University of Minnesota, Twin Cities says in a press release.

He added that Icarus is at least a hundred times farther than the next nearest star, at least among those that have not yet died blindingly explosive deaths. Scientists have observed galaxies at greater distances but have been unable to pick out their individual stars.

## What Is the Farthest Star From Earth?

Scientists will never know the farthest star from Earth, as the star is so far away that its light has not, nor will ever, have enough time to reach Earth. Even the stars within the visible universe are far too numerous to count, but the farthest one that humans have ever detected is about 55 million light years away. This incredibly distant star is called SDSS J 122952.66 +112227.8.

The farthest object that scientists have discovered is a galaxy by the name of UDFj-39546284. This is a very early galaxy that is about 13.2 billion light years away. However, scientists are constantly improving their visualization technology, and they hope to find even more distant objects in the near future.

The star Deneb is the farthest star that can be easily seen with the naked eye. It is thought to be between 1,400 and 3,000 light years from Earth. Deneb is one of the brightest in the northern sky and is part the constellation Cygnus, the Swan.

The Sun is the closest star to the planet Earth. The Earth orbits the Sun at a distance of about 93 million miles. Scientists use this distance as a convenient unit of measure for astronomical distances. Scientists call this distance one &ldquoastronomical unit,&rdquo often abbreviated AU.

## 18.3 Diameters of Stars

It is easy to measure the diameter of the Sun. Its angular diameter—that is, its apparent size on the sky—is about 1/2°. If we know the angle the Sun takes up in the sky and how far away it is, we can calculate its true (linear) diameter, which is 1.39 million kilometers, or about 109 times the diameter of Earth.

Unfortunately, the Sun is the only star whose angular diameter is easily measured. All the other stars are so far away that they look like pinpoints of light through even the largest ground-based telescopes. (They often seem to be bigger, but that is merely distortion introduced by turbulence in Earth’s atmosphere.) Luckily, there are several techniques that astronomers can use to estimate the sizes of stars.

### Stars Blocked by the Moon

One technique, which gives very precise diameters but can be used for only a few stars, is to observe the dimming of light that occurs when the Moon passes in front of a star. What astronomers measure (with great precision) is the time required for the star’s brightness to drop to zero as the edge of the Moon moves across the star’s disk. Since we know how rapidly the Moon moves in its orbit around Earth, it is possible to calculate the angular diameter of the star. If the distance to the star is also known, we can calculate its diameter in kilometers. This method works only for fairly bright stars that happen to lie along the zodiac, where the Moon (or, much more rarely, a planet) can pass in front of them as seen from Earth.

### Eclipsing Binary Stars

Accurate sizes for a large number of stars come from measurements of eclipsing binary star systems, and so we must make a brief detour from our main story to examine this type of star system. Some binary stars are lined up in such a way that, when viewed from Earth, each star passes in front of the other during every revolution (Figure 18.10). When one star blocks the light of the other, preventing it from reaching Earth, the luminosity of the system decreases, and astronomers say that an eclipse has occurred.

The discovery of the first eclipsing binary helped solve a long-standing puzzle in astronomy. The star Algol, in the constellation of Perseus, changes its brightness in an odd but regular way. Normally, Algol is a fairly bright star, but at intervals of 2 days, 20 hours, 49 minutes, it fades to one-third of its regular brightness. After a few hours, it brightens to normal again. This effect is easily seen, even without a telescope, if you know what to look for.

In 1783, a young English astronomer named John Goodricke (1764–1786) made a careful study of Algol (see the feature on John Goodricke for a discussion of his life and work). Even though Goodricke could neither hear nor speak, he made a number of major discoveries in the 21 years of his brief life. He suggested that Algol’s unusual brightness variations might be due to an invisible companion that regularly passes in front of the brighter star and blocks its light. Unfortunately, Goodricke had no way to test this idea, since it was not until about a century later that equipment became good enough to measure Algol’s spectrum.

In 1889, the German astronomer Hermann Vogel (1841–1907) demonstrated that, like Mizar, Algol is a spectroscopic binary. The spectral lines of Algol were not observed to be double because the fainter star of the pair gives off too-little light compared with the brighter star for its lines to be conspicuous in the composite spectrum. Nevertheless, the periodic shifting back and forth of the brighter star’s lines gave evidence that it was revolving about an unseen companion. (The lines of both components need not be visible for a star to be recognized as a spectroscopic binary.)

The discovery that Algol is a spectroscopic binary verified Goodricke’s hypothesis. The plane in which the stars revolve is turned nearly edgewise to our line of sight, and each star passes in front of the other during every revolution. (The eclipse of the fainter star in the Algol system is not very noticeable because the part of it that is covered contributes little to the total light of the system. This second eclipse can, however, be detected by careful measurements.)

Any binary star produces eclipses if viewed from the proper direction, near the plane of its orbit, so that one star passes in front of the other (see Figure 18.10). But from our vantage point on Earth, only a few binary star systems are oriented in this way.

### Making Connections

#### Astronomy and Mythology: Algol the Demon Star and Perseus the Hero

The name Algol comes from the Arabic Ras al Ghul, meaning “the demon’s head.” 3 The word “ghoul” in English has the same derivation. As discussed in Observing the Sky: The Birth of Astronomy, many of the bright stars have Arabic names because during the long dark ages in medieval Europe, it was Arabic astronomers who preserved and expanded the Greek and Roman knowledge of the skies. The reference to the demon is part of the ancient Greek legend of the hero Perseus, who is commemorated by the constellation in which we find Algol and whose adventures involve many of the characters associated with the northern constellations.

Perseus was one of the many half-god heroes fathered by Zeus (Jupiter in the Roman version), the king of the gods in Greek mythology. Zeus had, to put it delicately, a roving eye and was always fathering somebody or other with a human maiden who caught his fancy. (Perseus derives from Per Zeus, meaning “fathered by Zeus.”) Set adrift with his mother by an (understandably) upset stepfather, Perseus grew up on an island in the Aegean Sea. The king there, taking an interest in Perseus’ mother, tried to get rid of the young man by assigning him an extremely difficult task.

In a moment of overarching pride, a beautiful young woman named Medusa had compared her golden hair to that of the goddess Athena (Minerva for the Romans). The Greek gods did not take kindly to being compared to mere mortals, and Athena turned Medusa into a gorgon: a hideous, evil creature with writhing snakes for hair and a face that turned anyone who looked at it into stone. Perseus was given the task of slaying this demon, which seemed like a pretty sure way to get him out of the way forever.

But because Perseus had a god for a father, some of the other gods gave him tools for the job, including Athena’s reflective shield and the winged sandals of Hermes (Mercury in the Roman story). By flying over her and looking only at her reflection, Perseus was able to cut off Medusa’s head without ever looking at her directly. Taking her head (which, conveniently, could still turn onlookers to stone even without being attached to her body) with him, Perseus continued on to other adventures.

He next came to a rocky seashore, where boasting had gotten another family into serious trouble with the gods. Queen Cassiopeia had dared to compare her own beauty to that of the Nereids, sea nymphs who were daughters of Poseidon (Neptune in Roman mythology), the god of the sea. Poseidon was so offended that he created a sea-monster named Cetus to devastate the kingdom. King Cepheus, Cassiopeia’s beleaguered husband, consulted the oracle, who told him that he must sacrifice his beautiful daughter Andromeda to the monster.

When Perseus came along and found Andromeda chained to a rock near the sea, awaiting her fate, he rescued her by turning the monster to stone. (Scholars of mythology actually trace the essence of this story back to far-older legends from ancient Mesopotamia, in which the god-hero Marduk vanquishes a monster named Tiamat. Symbolically, a hero like Perseus or Marduk is usually associated with the Sun, the monster with the power of night, and the beautiful maiden with the fragile beauty of dawn, which the Sun releases after its nightly struggle with darkness.)

Many of the characters in these Greek legends can be found as constellations in the sky, not necessarily resembling their namesakes but serving as reminders of the story. For example, vain Cassiopeia is sentenced to be very close to the celestial pole, rotating perpetually around the sky and hanging upside down every winter. The ancients imagined Andromeda still chained to her rock (it is much easier to see the chain of stars than to recognize the beautiful maiden in this star grouping). Perseus is next to her with the head of Medusa swinging from his belt. Algol represents this gorgon head and has long been associated with evil and bad fortune in such tales. Some commentators have speculated that the star’s change in brightness (which can be observed with the unaided eye) may have contributed to its unpleasant reputation, with the ancients regarding such a change as a sort of evil “wink.”

### Diameters of Eclipsing Binary Stars

We now turn back to the main thread of our story to discuss how all this can be used to measure the sizes of stars. The technique involves making a light curve of an eclipsing binary, a graph that plots how the brightness changes with time. Let us consider a hypothetical binary system in which the stars are very different in size, like those illustrated in Figure 18.11. To make life easy, we will assume that the orbit is viewed exactly edge-on.

Even though we cannot see the two stars separately in such a system, the light curve can tell us what is happening. When the smaller star just starts to pass behind the larger star (a point we call first contact), the brightness begins to drop. The eclipse becomes total (the smaller star is completely hidden) at the point called second contact. At the end of the total eclipse (third contact), the smaller star begins to emerge. When the smaller star has reached last contact, the eclipse is completely over.

To see how this allows us to measure diameters, look carefully at Figure 18.11. During the time interval between the first and second contacts, the smaller star has moved a distance equal to its own diameter. During the time interval from the first to third contacts, the smaller star has moved a distance equal to the diameter of the larger star. If the spectral lines of both stars are visible in the spectrum of the binary, then the speed of the smaller star with respect to the larger one can be measured from the Doppler shift. But knowing the speed with which the smaller star is moving and how long it took to cover some distance can tell the span of that distance—in this case, the diameters of the stars. The speed multiplied by the time interval from the first to second contact gives the diameter of the smaller star. We multiply the speed by the time between the first and third contacts to get the diameter of the larger star.

In actuality, the situation with eclipsing binaries is often a bit more complicated: orbits are generally not seen exactly edge-on, and the light from each star may be only partially blocked by the other. Furthermore, binary star orbits, just like the orbits of the planets, are ellipses, not circles. However, all these effects can be sorted out from very careful measurements of the light curve.

The Eclipsing Binary Simulator allows you to explore how the eclipse timing can be used to determine the size of stars in a binary pair. Other traits can be explored as well, such as their mass, separation, and surface temperatures.

### Using the Radiation Law to Get the Diameter

Another method for measuring star diameters makes use of the Stefan-Boltzmann law for the relationship between energy radiated and temperature (see Radiation and Spectra). In this method, the energy flux (energy emitted per second per square meter by a blackbody, like the Sun) is given by

where σ is a constant and T is the temperature. The surface area of a sphere (like a star) is given by

The luminosity (L) of a star is then given by its surface area in square meters times the energy flux:

Previously, we determined the masses of the two stars in the Sirius binary system. Sirius gives off 8200 times more energy than its fainter companion star, although both stars have nearly identical temperatures. The extremely large difference in luminosity is due to the difference in radius, since the temperatures and hence the energy fluxes for the two stars are nearly the same. To determine the relative sizes of the two stars, we take the ratio of the corresponding luminosities:

Therefore, the relative sizes of the two stars can be found by taking the square root of the relative luminosity. Since 8200 = 91 8200 = 91 , the radius of Sirius is 91 times larger than the radium of its faint companion.

The method for determining the radius shown here requires both stars be visible, which is not always the case.

Use the Stellar Luminosity Simulato to explore the relationship between a star’s surface temperature, luminosity, and radius. Move the sliders to see what happens. Try to make two stars with the same luminosity but different surface temperatures.

### Stellar Diameters

The results of many stellar size measurements over the years have shown that most nearby stars are roughly the size of the Sun, with typical diameters of a million kilometers or so. Faint stars, as we might have expected, are generally smaller than more luminous stars. However, there are some dramatic exceptions to this simple generalization.

A few of the very luminous stars, those that are also red (indicating relatively low surface temperatures), turn out to be truly enormous. These stars are called, appropriately enough, giant stars or supergiant stars . An example is Betelgeuse , the second brightest star in the constellation of Orion and one of the dozen brightest stars in our sky. Its diameter, remarkably, is greater than 10 AU (1.5 billion kilometers!), large enough to fill the entire inner solar system almost as far out as Jupiter. In Stars from Adolescence to Old Age, we will look in detail at the evolutionary process that leads to the formation of such giant and supergiant stars.

Watch this star size comparison video for a striking visual that highlights the size of stars versus planets and the range of sizes among stars.

## This star is the farthest ever seen. It’s 9 billion light-years away.

Astronomers using the Hubble Space Telescope have found the farthest star ever observed, a bright dot 9 billion light-years away. Forget thanking their lucky stars: This discovery required the fortuitous alignment of a massive galactic cluster. The cluster warped the starlight, bending it toward Earth while magnifying the star 2,000 times.

Officially, the star's name is MACS J1149+2223 Lensed Star 1. But the astronomers call it Icarus. Icarus was a hundredfold more distant than any other lone star previously detected, according to Patrick L. Kelly, an astrophysicist at the University of Minnesota. Normally only phenomena such as supernovas — catastrophic stellar explosions — or entire galaxies are detectable at such vast distances.

Kelly had not set out to spy on record-breaking stars. No other individual star had been seen in such a distance. The astrophysicist and his colleagues were studying Hubble images of a supernova called SN Refsdal. But in 2016, they found a blip that appeared in the same galaxy that housed the supernova. This blip, as Kelly and his co-authors wrote in a report published recently in the journal Nature Astronomy, was not another supernova but a blue supergiant star.

About halfway between Icarus's spiral galaxy and our own was a third object, a massive cluster of galaxies that acted as the lens of the magnifying glass. It just so happened that Icarus passed “along the critical curve” of the cluster, Kelly said, which warped the starlight in our direction — a process called gravitational lensing. The effect was like “a natural telescope, much more powerful than anything we could build.”

The authors of the new study identified Icarus as a stable star and not an explosion because its temperature did not appear to fluctuate. Supernovas “start out very hot and cool down. We saw no evidence of a change in temperature,” Kelly said. Still, Icarus was massive, hotter than the Sun, and possibly thousands of times as luminous.

It also no longer exists, Kelly said. Blue supergiant stars do not have life spans of nine billion years. He suspected Icarus either collapsed into a black hole or wound up as a neutron star.

A little sphere at which many small mirrors are glued can be used to observe the earth's rotation, to measure the angular diameter of the sun and the eccentricity of the elliptic orbit of the earth. When the sun is shining through a window the mirrors project the images onto a wall or onto a piece of white paper in the shade or into the far end of a darkened hallway through an opened door.

Knowing the radius of Earth's orbit and by measuring the apparent size of the sun we can easily determine the diameter of the sun and the eccentricity of the orbit of the Sun using trigonometry or simple geometry.

#### 1. Introduction

The Sun's distance and its diameter was not known exactly until the British Captain James Cook observed the transit of Venus in 1769. A reasonable accurate value was derived from data in 1835 by the astronomer Enke. The actual distance between Earth and Sun varies from a minimum of 147 097 000 km to a maximum of 152 086 000 km because of Earth's elliptical orbit. We use 150 000 000 km (=1 AU) as the distance.

#### 2. Projection-Methods for Sun observation

You can use binoculars mounted on a tripod pointing to the sun, one glass can be closed or you can hold a 2-liter bottle in a stand and glue a small piece of a mirror (less than 0,5 cm × 0,5 cm) near the middle of the bottle along the length of the cylinder. Fill the bottle with water before using it. Trace the image of the sun on a viewing screen, which is placed several meters away from the mirror. Never look directly at the sun.

The easiest way is to buy at Christmas time a sphere with many tiny hot glued mirrors, used as decoration for the tree. Fix the sphere in a stand near the window in a stand. If the sun is shining at the ball images of the sun appear onto the wall behind. The important points are that the mirror must be small compared to the distance of projection which should be at least 2 m, and you must get an image bright enough to measure. The projection from the mirror should be flat on to the surface of projection. If the image is round, then it is being projected correctly. Keep in mind that the projected image of the sun will be faint, so it needs to be projected into a darkened area. If the distance is too short, the image will to be faint. If it is too large, the edges of the image will be fuzzy. Look for an image which is round, sharp enough, and bright enough to measure.

#### 3. Measuring the apparent angular diameter

Onto a piece of paper draw a circle bigger than the images of the sun. Trace the image in the middle of the circle. Using a stopwatch start timing at the first contact: circle / edge of the image. Stop timing when the image of the sun has moved completely out of the circle you have traced. See Fig. 1. Try this several times. Make at least 5 measurements.

#### 4. Explanations

The rotation of the Earth causes the image of the Sun to appear to move across the screen. The angular velocity of the daily rotation is easily determined. If the earth spins 360 degrees per 24 hours, then it spins 15 degrees per hour or 1/4 degrees per minutes. The time, it takes for the sun's image to move "one Sun diameter", is about 2 minutes. For larger or smaller images, the time will be constant, as it is a measure of an other constant, the spin of the earth. The apparent angular diameter of the sun can be found using the proportion:

15 degrees / 60 minutes = x degrees / 2 minutes

The small mirror acts like the hole of a big pinhole camera. The development of a normal pinhole camera projecting the image of the sun on a screen to a model reflecting the sun's light using a tiny mirror is shown in Fig. 2.

Locally you see the pictures of the rectangular mirrors.

Only at the equator the angular velocity of the sun is 1°/4 min.

If the sun stands above or below the celestial equator, its velocity is smaller. Polaris does not move, but the stars of Orion near the celestial equator move a lot concerning the spin of the earth.

As far as the declination of the sun d is concerned the path of the sun is given by s = 1°/4 min × cos δ × t . Measuring t minutes for the passage of the sun you can determine the its diameter.

Because -23,3º < δ < 23,5º the mistake ignoring the position of the Sun is at maximum 9%. During the equinox the mistake is minimal.

The tiny mirror is reflecting light from the sun and is producing an image of the sun on the screen. There are two similar triangles in this experiment. One is an triangle whose base is the diameter of the sun, and whose congruent sides are rays from each side of the sun to your mirror. The base of the second triangle is the diameter of the sun's image on the screen. Its congruent sides are rays coming from the mirror. Since these are similar triangles, the angular diameter of the image on the screen is the same as the angular diameter of the sun in the sky.

#### 5. Sun's Diameter in km

Younger students who are not yet able to use trigonometric functions can use similar triangles and proportions to determine the diameter in km. They draw a symmetric triangle: basis angle d = 10° at the top, length of the height (distance: basis - top of the triangle) d(E,S) = h = 15 cm. The basis b of the drawn triangle is then measured as 2,6 cm. See Fig. 3.

Because the distance D from the Earth to the Sun is nearly 150 Mil km, h is easily stretched to that scale. The goal is to change the value of h and also the triangle by calculation to the real distance D. First the angle has to be diminished to 1/2 degree by calculation. The steps are shown below.

Using trigonometry we get the same result: sin δ/2 = tan d/2 = δ/2 / D(E,S).

The distance to the sun varies through the year, ranging from a minimum of 147 097 000 km (0,983 28 AU) to a maximum of 152 086 000 km (1,016 67 AU). You could find out the exact distance on the day of your observation, but the level of accuracy of this observation doesn't warrant it.

The semi-diameter has been observed since 1985 with the Tokyo Photoelectric Meridian Circle (Tokyo PMC) at Mitika. The mean value of the apparent diameter of 1 919,66 arcsec leads to an actual diameter of the sun of 1 391 000 km. The diameter variations are found to have amplitude larger than 0,08 arcsec. Most of them have a period longer than 130 days.

Although the Sun is spinning around its axis the difference between the radius from the equator to the centre and from the pole to the centre varies at maximum only some 10 kilometers. The sun is therefore nearly a perfect sphere. The sun's diameter is about 109 times the diameter of the earth.

#### 6. The eccentricity of the Earth

The actual distances of planets from the sun are continually changing, because their orbits are ellipses governed by Kepler's laws. The first states that planets move in ellipses with the Sun in one focus.

It is convenient to start with the construction of ellipses by the gardeners. With two fixpoints and a string you can easily trace some ellipses on the blackboard. The dimensions and shape of an ellipse are described by the semimajor axis a, the eccentricity e and the numerical eccentricity e. Another useful quantity is the closest distance to the sun called perihelion distance. Figure 4 shows these quantities. In detail e = e/a, rP + rA = 2a.

The changes in the distance to sun can be determined by the different angular diameters of the sun. For good results series of continuous observations are to be made during one year. In the easiest case two observation weeks are needed: the first week in January and first week in July.

Remember the sun stands in general above or below the celestial equator. The fact is described by the declination of the sun d. Therefore t real = t × cos δ / 4 min. (In January d = -23° and in July d = +23°). The mistake ignoring the position of the Sun is max 9%.

Figure 4 shows measurements the apparent radius of Sun ( closed line astronomical reference book and crosses observations of the students during one year time using a telescope).

1 - The orbit of the Earth is no circle with the Sun in the middle. Because the apparent angle diameter f is very small, the relation is fA × rA = fP × rP.

rA: aphel distance, rP: perihel distance, fA and fP are the angle diameter at aphel and perihel.

2 - If the earth's orbit is elliptic the Sun can not stand in the centre of the ellipse (different diameters in January and July).

3 - By comparing the maximum and minimum diameter the numeric eccentricity can be estimated.

The exact value is e (Earth) = 0,0167

#### Proposals for didactical activities: Sun's Diameter

Measure the apparent diameter of the sun by measuring the transit time of the sun's image.

Calculate the apparent diameter f in arcsec.

Determine the diameter in km by using trigonometric relations.

Determine the eccentricity of the earth using the given data of the sun in the year 1999.

Diagram: month versa apparent diameter f

#### Bibliography

[1] Dieter Vornholz, Astronomie auf Klassenfahrten, Westermann 1992.

[2] Peter Kniesel, Physikalische Experimente in der Astronomie, Päd Zentrum Berlin.

## Scale models

One way to understand size scales is by considering a model. Notice that it is not possible to view the relative size of the sun and Earth and also their distance apart in this small diagram. There is just too big of a difference in the relative length scales. One thing we can do to better understand the real sizes and distances of objects is to build a scale model.

Let's build a generic scale model as an illustration.

In this diagram, we have a real system and a scale model system. Let's think about what needs to be true for our scale model to be accurate.

In the real system, the blue ball is three times as big as the diameter of the red ball. So, in our scale model, the diameter of the blue model ball needs to be three times as big as the diameter of the red model ball.

In the real system, the Distance between the blue and red ball is four times the diameter of the blue ball. So, in our scale model, the Distance between the model balls needs to be four times the diameter of the model blue ball.

We could build a smaller scale model as well, as long as we kept the relative sizes the same between the diameters and distances.

How do we figure out how big to make the balls and distances in our model, to keep it accurate? We need to start by choosing one model object's size and compare it to the real object. This will determine the other sizes for our model.

We need to measure the diameter of the real red ball and the distance between the real blue ball and real red ball to figure out how big to make these sizes in our model.

Now we can calculate the sizes we need for our model.

The ratio (fraction) of the model red ball to the real red ball is the same as the ratio of the model blue ball to the real blue ball.

We solve for the diameter of the model red ball, and then insert values to find the diameter we need for the red ball in our model.

We can find the Distance between the blue and red balls in our model the same way. This was easy, since we could just use a very similar equation. Once we set the scale using the blue model ball compared to the blue real ball, we have the scale for anything in the model compared to the real system. If we had nine balls, all different distances apart, we could use the same scale to find the correct sizes and distances in our model.

Now, let's build a scale model of our Solar system.

Let's assume that in our model, the sun is the size of a bowling ball. Comparing the size of a bowling ball will allow us to calculate the other relative distances in our scale model.

Deciding to use a bowling ball, with a diameter of about 22 cm (2.2 x 10-1 m) sets the scale for our scale model.

We also need to know the other real distances involved. Now we can use the scale set by the bowling ball to find the diameter of our model earth and how far away from our model sun to place it.

Using the scale set by the bowling ball compared to the real size of the Sun, we calculated the size of Earth in our model. In our model, Earth is the size of a small bead, a couple of millimeters in diameter.

We can use the same formula to find out how far apart the bead would be from the bowling ball in our model. Here, we calculate that the distance in our model would be 24 meters.

Now, if the true distance to the nearest star, Alpha Centauri, is 4.0 x 10 16 m, how far away would it be in our model?

## How can you measure a star's size? Wait for an asteroid to block it.

One of the things I like about science is how — if you know your tools well and are clever enough — it allows you to solve a problem that would otherwise seem impossible. Even more fun is when the method itself feels like a Rube Goldberg machine * .

Astronomers really want to be able to know how big, physically, stars are. Our understanding of how stars behave (like how they shine, how hot they are, how bright they are, and even (especially) how big planets are that are orbiting them) depends a lot on how big they are.

Now, you can figure this out if you know how far away they are and how big they appear to be. A little high school trig solves that problem easily. But how do you measure their apparent size, how big they seem to be in the sky?

Stars are big objects, millions of kilometers wide, but (except for the Sun) they’re inconveniently located trillions or quadrillions of kilometers away. That makes them look very small. For example, one of the stars in the Alpha Centauri system (the closest star system to us) is about the same physical size of the Sun. But it’s also nearly 270,000 times farther from Earth than the Sun is, so it only looks 1/270000 times the size. That makes it only about a millionth of a degree across — far too small to resolve even using Hubble. And that’s the closest star.

But there’s another way. Instead of measuring the star’s size directly, you can wait for some object to pass in front of it, and see how long it takes the star’s light to fade. If you know how fast the object moves, you use that to figure out the star’s size.

Artwork depicting an asteroid occulting a distant star, with the diffraction pattern of the star exaggerated for effect. Credit: DESY, Lucid Berlin

You can do this with the edge of the Moon, but there are lots of issues with that that limit the accuracy. However, there are other solid objects in the solar system that do this too: asteroids!

The problem there is that asteroids are small and relatively far away from us, so the odds of seeing one passing in front of a star aren’t that high. Worse, if you want to use a big, unmovable telescope, that reduces the chance even more.

But it is possible, and some astronomers did just that. I think my favorite part of all this, too, is that they used a telescope designed to do something completely different!

The four VERITAS telescopes, each made up of 350 smaller mirrors, watch the skies for flashes of light from gamm and cosmic rays blasting down into Earth’s atmosphere. Credit: VERITAS

In southern Arizona there’s an observatory called VERITAS: the Very Energetic Radiation Imaging Telescope Array System. It consists of four separate telescopes, and each of those telescopes is itself made up of a collection of 350 smaller mirrors, each about 60 centimeters across.

The telescopes are designed to look for brief flashes of light in the sky when very energetic gamma rays and cosmic rays from space slam into our atmosphere. When these hit an atom or molecule of air, they create a shower of subatomic particles that move faster than light in air, creating a sort-of optical shock wave called Cherenkov radiation. This creates a detectable flash of light that can be used to learn more about such interesting objects as black holes, exploding stars, and so forth.

The thing is, that makes VERITAS very sensitive to faint sources, which means it can take really short exposures of stars. That’s important, because an asteroid passing in front of a star only takes about a tenth of a second to block it! So if you can take very rapid exposures you can actually measure dimming of starlight accurately.

Ocean waves diffract and spread out as they enter the narrow opening, spreading out and interfering with each other, similar to how light behaves. Credit: Google Earth via Exploring Our Fluid Earth

But there’s a catch. A star doesn’t just fade away, it actually fluctuates in brightness, up and down, brightening and dimming rapidly as it gets blocked out. That’s because of a phenomenon called diffraction. Under these circumstances, light behaves like a wave, and when it passes by something with a sharp edge the waves interfere with each other and change the way they behave (not to get into an inception of analogies, but this is like when you move back and forth in a bathtub, and the waves can add to each other causing water to slosh out of the tub).

But that’s OK! The physics of how the light behaves depends on the apparent size of the star, so by measuring that behavior very carefully, the math can be turned around to calculate just how big the star is in the sky.

Knowing this, a team of astronomers determined that, in 2018, two different asteroids would block (or as astronomers call it, occult) two different stars. On February 22 asteroid (1165) Imprinetta occulted the star TYC 5517-227-1, and then on May 22 (201) Penelope occulted TYC 278-748-1. Both stars were bright enough to allow VERITAS to take very rapid exposures: 3 milliseconds each for the first occultation, and 0.4 ms for the second.

Here’s what they saw for Imprinetta occulting TYC 5517-227-1:

The brightness of the star TYC 5517-227-1 goes up and down as the asteroid Imprinetta passes in front of it (left) and then leaves (right), the diffraction pattern causing the rapid fluctuations. Credit: Benbow et al.

Because there are four telescopes there are four measurements (labeled T1 – T4). As you can see, the star whipped up and down in brightness before dropping to zero. By fitting that measurement with known physics, they were able to get an apparent size of the star: 0.125 milliarcseconds (with about a 15% uncertainty). That’s an incredibly small angle there are 3600 arcseconds in a degree, and the Moon for example is about 1800 arcseconds in size on the sky. This star’s apparent disk is so small it’s like seeing a U.S. quarter from 40,000 kilometers away!

The star’s distance is known to be 2,670 light years, so that makes its actual, physical diameter about 11 times that of the Sun. That’s a big star! In fact, follow up observations showed it to be an orange star, slightly cooler than the Sun. That means this must be a red giant, a star that was once like the Sun but is at the end of its life, now swollen and huge. This was not known before these observations no one knew if this was a normal star just churning away, or one nearer the end of its life. Now we know.

The second star VERITAS observed turns out to have an apparent diameter of 0.094 milliarcseconds, and given its distances of 700 light years, it turns out to be about 2.2 times the size of the Sun. It has the same temperature as the Sun, so this means it must be a subgiant, a star nearing the end of its life, heading toward becoming a red giant in the next few million years or so.

Interestingly, the astronomers calculate that from a given site on Earth, roughly 5 stars bright enough to perform this measurement are occulted by asteroids every year. You can’t pick the stars, unfortunately — orbital mechanics does that for you — but this is still a decent number. And it’s not that hard to make the measurement if you have the right equipment. VERITAS has an advantage because it’s big, is used just to stare at big portions of the sky anyway, and has four telescopes to work with, giving you some redundancy. And while other observatories could do this, it’s a low priority for big ones compared to the other science they do.

Still, it’s pretty neat we can do this at all, and as time goes on I bet the technique will improve. I wonder what other observatories will give this a try… and what other interesting low-level but important science can be done when we think a little differently?

## The farthest galaxy in the universe

A team of astronomers used the Keck I telescope to measure the distance to an ancient galaxy. They deduced the target galaxy GN-z11 is not only the oldest galaxy but also the most distant. It's so distant it defines the very boundary of the observable universe itself. The team hopes this study can shed light on a period of cosmological history when the universe was only a few hundred million years old.

We've all asked ourselves the big questions at times: "How big is the universe?" or "How and when did galaxies form?" Astronomers take these questions very seriously, and use fantastic tools that push the boundaries of technology to try and answer them. Professor Nobunari Kashikawa from the Department of Astronomy at the University of Tokyo is driven by his curiosity about galaxies. In particular, he sought the most distant one we can observe in order to find out how and when it came to be.

"From previous studies, the galaxy GN-z11 seems to be the farthest detectable galaxy from us, at 13.4 billion light years, or 134 nonillion kilometers (that's 134 followed by 30 zeros)," said Kashikawa. "But measuring and verifying such a distance is not an easy task."

Kashikawa and his team measured what's known as the redshift of GN-z11 this refers to the way light stretches out, becomes redder, the farther it travels. Certain chemical signatures, called emission lines, imprint distinct patterns in the light from distant objects. By measuring how stretched these telltale signatures are, astronomers can deduce how far the light must have traveled, thus giving away the distance from the target galaxy.

"We looked at ultraviolet light specifically, as that is the area of the electromagnetic spectrum we expected to find the redshifted chemical signatures," said Kashikawa. "The Hubble Space Telescope detected the signature multiple times in the spectrum of GN-z11. However, even the Hubble cannot resolve ultraviolet emission lines to the degree we needed. So we turned to a more up-to-date ground-based spectrograph, an instrument to measure emission lines, called MOSFIRE, which is mounted to the Keck I telescope in Hawaii."

The MOSFIRE captured the emission lines from GN-z11 in detail, which allowed the team to make a much better estimation on its distance than was possible from previous data. When working with distances at these scales, it is not sensible to use our familiar units of kilometers or even multiples of them instead, astronomers use a value known as the redshift number denoted by z. Kashikawa and his team improved the accuracy of the galaxy's z value by a factor of 100. If subsequent observations can confirm this, then the astronomers can confidently say GN-z11 is the farthest galaxy ever detected in the universe.

In the first case, in regards to star measurement, I believe you're thinking of how the diameter of very large stars are measured using interferometry. Because light waves from the edges of these stars arrive at us in parallel, and they are waves, we can determine the diameter of the star by measuring the interference pattern between these light waves. (That's probably what you were thinking of — the "peaks" and "troughs" in the interference pattern.)

Stars like Betelgeux, Antares and Aldebaran have been measured in this way and the size agrees with the Stefan–Boltzmann law which can be used to calculate the radius of a spherical body if the luminosity and temperature are known.

I found this 1921 Popular Science article which describes it in detail.

The black hole pulsing thing is a completely different concept. I'm not sure what the logic of that is, perhaps it was talking of the doppler shifts of light rotating around the black hole or its interaction with a binary companion star?

I don't know what "ER peaks" means, but I think I get the idea. (Did you mean "EM", i.e., electromagnetic?)

Suppose you see a spiral galaxy that's 100,000 light-years across, and you're seeing it nearly edge-on. Suppose the entire galaxy's brightness increases and decreases significantly with a 1-year cycle.

It's not possible (or rather, it's vanishingly unlikely) that this is the result of all the stars pulsating in unison. The near edge of the galaxy is 50,000 light-years closer to us than the center, which is 50,000 light-years closer than the far edge. That means that the light that we're seeing now from the far edge must have originated 100,000 years sooner than the light we're seeing now from the near edge. Since nothing can travel faster than light, there is no natural phenomenon that could keep all the stars over that 100,000 light-year expanse synchronized with each other. And even if there were, they would appear to be synchronized only as seen from one particular direction.

If a 100,000 light-year galaxy appears to pulse with a 1-year cycle, then the pulses are coming from a much smaller body, probably at the galaxy's core. And that body is probably substantially smaller than 1 light-year, because the waves within it that cause it to pulsate are probably moving substantially slower than the speed of light.

If you see a light source pulsating with a 1-year cycle, then it must be less than 1 light-year across. If it's any bigger, then (a) whatever waves cause it to pulsate will take more than a year to cross it, so it can't stay synchronized with itself, and (b) even if it could, the pulses would be "blurred out" because we simultaneously see parts of the light source at different distances.

In your example, you got it backwards. If a black hole visibly pulses on a 10-minute cycle, then the body that's emitting the light must be smaller than 10 light-minutes.