# When will the number of stars be a maximum?

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There are very roughly a "mole" of stars in the universe. Wikipedia quotes an estimate of $$3 imes 10^{23}$$ though the number is associated with some debate and uncertainly.

I'd like to know if there are estimates of when the number of stars in the universe will maximize. Is it expected to increase asymptotically to some maximum, or will it peak and then decrease.

I suppose this could depend on what the definition of "star" is taken to be, if brown or black dwarf objects are counted or not. I don't want to pre-specify, it's more likely that a good, well-informed answer will include this information.

TL; DR Somewhere between now and a few hundred billion years time. (For a co-moving volume) Now read on.

If stellar remnants are included, then the answer is very far in the future indeed, if and when the constituents of baryons begin to decay. So let's assume that "stars" means those things that are undergoing nuclear fusion reactions to power their luminosity. Let's further assume that the stellar mass function, ($$N(m)$$ is the number of stars per unit mass) we see in the neighbourhood of the Sun is representative of populations in all galaxies at all times (difficult to make a start, without assuming this).

The number of stars that have been born is equal to the sum over time (the integral) and over mass of $$N(m)$$ multiplied by the rate at which mass is turned into stars in a comoving volume of the universe $$Phi(t)$$.

We then need to subtract a sum over time and mass of the rate of stellar death in the same comoving volume. The rate of stellar death is the rate of stellar birth at a time $$t- au(m)$$, where $$au(m)$$ is the mass-dependent stellar lifetime. We ignore mass transfer in binary systems and assume that multiples can be treated as independent stellar components.

Thus the number of stars at time $$t$$ is approximately $$N_*(t) = int_0^{t} int_m N(m) Phi(t') - N(m)Phi(t'- au(m)) dm dt' .$$ To find where this is a maximum, we differentiate with respect to time and then equate to zero. i.e. We look for the time when the stellar birth and death rates are the same.

I was going to (and possibly still will) attempt some sort of analytic approximation, but Madau & Dickinson (2014) have done it better and taken into account the metallicity dependence of stellar lifetimes and the chemical evolution of galaxies. The star formation rate peaked about 10 billion years ago, is more than an order of magnitude lower now and is exponentially decreasing with a time constant of 3.9 billion years.

The integrated stellar mass is shown in their Fig 11 (shown below). It is still increasing today, but at a very low rate and has not passed through a maximum. The reason for this is that most stars have masses of 0.2-0.3 solar masses and lifetimes much longer than the age of the universe. Even if these stars are added at a very slow rate, their death rate is zero at present.

If star formation did continue at a low-level then the number of stars would only begin to significantly diminish once the stars near the peak of the stellar mass function, that were born at the earliest times, start to die. The lifetime of a 0.25 solar mass star is around a trillion years (Laughlin et al. 1997).

On the other hand if star formation ceased now then the number of stars would immediately begin to diminish.

Perhaps we could argue that the current exponential decline will continue and the peak will come in another few billion years when stars of 0.8-0.9 solar masses begin dying off. However, that is futurology given that we have no first principles theory that explains the time-dependence of star formation, so I believe the best answer that can be given is somewhere between now and a few hundred billion years time.

Note that this answer assumes a co-moving volume. If the question asked is phrased in terms of the observable universe then because the number of stars has nearly reached a plateau, then the answer becomes close to whatever age the volume of the observable universe is maximised. I say "close to" because you have to factor in that the observable universe includes stars in distance slices at all cosmic epochs. I am unwilling to undertake this horrendous calculation, but note that the current concordance cosmological model has our observable universe slowly increasing from around a radius of 45 billion light years now, to about 60 billion light years in the far future Davis & Lineweaver 2005, and this may compensate for a slow decline in the number of stars in a co-moving volume.

## Explain to me, what is FWHM?

I am puzzled. What does Full Width, Half Measure mean? How do I apply that to my imaging, when, for instance, I use SharpCap and have the option of applying FWHM to my stacking options?

### #2 Bigdan

I believe it relates to focusing. When I am imaging, I have two mechanical methods for focusing: Bahtinov mask, and a low FWHM number displayed by my camera control software. Consistently, when I focus my CCD with Bahtinov mask, I notice when I come to focus, the FWHM is at its lowest value. I haven't really checked it. I use Deep Sky Stacker for stacking, and register my images first. It gives a score, and I use the image with the highest score as the reference image. I haven't checked to see if there is a correlation between highest score and lowest FWHM, but I think there are other factors taken into account, also.

### #3 syscore

I am puzzled. What does Full Width, Half Measure mean? How do I apply that to my imaging, when, for instance, I use SharpCap and have the option of applying FWHM to my stacking options?

### #4 tolgagumus

A stars 3D profile somewhat looks like a bell. FWHM is the width of the bell at its midway point hence "half". For all practical purposes you want this number as small as possible. As the star goes out of focus, the shape of the bell gets fatter and the mid point gets larger in diameter.

Sent from my SAMSUNG-SM-G870A using Tapatalk

### #5 Zebenelgenubi

I am puzzled. What does Full Width, Half Measure mean? How do I apply that to my imaging, when, for instance, I use SharpCap and have the option of applying FWHM to my stacking options?

Wow that's an excellent discussion of what is going on in an astro image. Thanks for the reference.

### #6 Jon Rista

FWHM is actually Full Width at Half Maximum, which indicates it is the width of the star profile half way to the peak. Here is an example cross-section plot of a large star (and you can see much smaller stars as the spikes in the plot around the large bell curve), and I've marked where the FWHM is at half the height of the star:

The star peaks at 4000 ADU, the mean background level is about 700 ADU, which puts the half maximum at 2350. The "full width" is the width between the two red arrows, at that half maximum level. That is a standard measure of the size of a star.

### #7 freestar8n

And for people who are learning about FWHM for the first time - the next question is - how does the fwhm vary with star intensity? You can look at stars in an image and the bright ones are much "bigger." But in fact - as long as the stars are not saturated, the fwhm is about the same - regardless of brightness. That's because a brighter star will be taller and "wider" overall - but it still maintains the same profile shape.

As the profile gets taller, the "waist" where you measure the width of the star (which is half way up to the max intensity) also rises - and it stays about the same.

So fwhm should be constant regardless of star intensity - and that's why it's a good measure of sharpness and focus/seeing/guiding quality.

### #8 John Miele

That is a great explanation and answers that nagging question I have had about whether I should worry about the brightness when evaluating fwhm!

### #9 Jon Rista

It should probably be pointed out, since the example I shared deviates from Franks description. that very large stars, even if they are not saturated (the star in my example above is the large, VERY BRIGHT star Gamma Cas), will often have significantly larger FWHMs than the "average" star in the frame. You can see many other spikes in the plot in my screenshot above. those are smaller more average sized stars around Gamma Cas. You can see that at their FWHM, they are all about the same, which does line up with what Frank said.

Usually, when measuring FWHMs, the tool you use to do it will have some means of configuring what level of brightness to cut the calculation off at. Some also have a black point setting. These can be used to tune the algorithm to look a the right information to give you a useful result.

### #10 freestar8n

Jon - those look like noise spikes rather than stars.

### #11 Jon Rista

Jon - those look like noise spikes rather than stars.

I'm pretty sure they are stars, as there are a number of very small stars scattered about very close to Gamma Cas itself in the image. This is data that has been calibrated, cosmetically cleaned to eliminate any remnant hot pixels, and integrated with winsorized sigma clipping. The noise profile is pretty clean. so the only spikes left should be the stars. This was also from pretty short subs, 90 seconds, and most of the stars except Gamma Gas are pretty small. few pixels across.

Edited by Jon Rista, 21 November 2016 - 04:30 PM.

### #12 schmeah

So do you guys get somewhat lower FWHMs with Ha starfields than with luminance, even when taken on the same night with the same conditions and optimal focus? And if so, what explains that? To be honest I'm not certain that I trust the reliability of FWHM, eccentricity/roundness measurements nor the correlation between these measurements and RMS guiding error. Or at least not with the software that I use (CCD stack and Maxim LE).

### #13 Jon Rista

Derek, I think the main reason NB stars tend to be smaller is they are a narrow sliver of the spectrum, so they can actually be focused better than the entire spectrum (i.e. an L filter). The L filter is focusing all the light from 700nm to 390nm, and even a very good scope is still going to have some dispersion at one wavelength or another. So even with the best possible L focus, the stars are still going to have larger FWHM than say from an Ha filter. With the Ha filter, you are focusing a very specific and narrow range of wavelengths, so you don't have to worry about the dispersion in other wavelengths.

As for FWHM measurements and how they correlate with guide RMS. Guide RMS is one factor out of many that affect blur. I usually try to calculate what my best possible FWHMs are using this formula:

FWHM = SQRT(Seeing^2 + Dawes^2 + GuideRMS^2 + ImageScale^2)

There are additional blur factors, and they can come from a variety of things. The low pass filter on a DSLR can introduce significant blur. Wind can introduce blur. It may be better to separate GuideRMS from TrackingError as well, although I tend to roll them into one. Filters will introduce their own blur. And, optical aberrations will add some blur. I usually just go with Dawes as I don't know of any simple way to calculate what blur the various aberrations may be adding (although it could be non-trivial, and can be considered a key source of error in this calculation, when it tells you your FWHM should be 2" and you measure 3".)

In my case, my seeing seems to be around 1" or so on average, the Dawes limit of my scope is 0.76", my guide RMS is usually around 0.65" on average (although it's been edging closer to 0.55" more these days, so I may have to change my calculation), and my image scale is 1.3". With these numbers, I calcuate a best-possible FWHM of 1.92". I measure FWHMs ranging from about 2.1" to 2.5" from most of my data, and in general they seem to fall around 2.3-2.4". I know that my lens has some spherical aberration, and it seems to have a little bit of CA, although not much. Those undoubtedly account for some of the extra blur in my actual measurements. The rest could be that seeing is worse than I think, and I am sure that wind plays a role, as I always have a light breeze at the very least.

Measuring distances to objects within our Galaxy is not always a straightforward task – we cannot simply stretch out a measuring tape between two objects and read off the distance. Instead, a number of techniques have been developed that enable us to measure distances to stars without needing to leave the Solar System. One such method is trigonometric parallax, which depends on the apparent motion of nearby stars compared to more distant stars, using observations made six months apart.

A nearby object viewed from two different positions will appear to move with respect to a more distant background. This change is called parallax. A simple demonstration is to hold your finger up in front of your face and look at it with your left eye closed and then your right eye. The position of your finger will appear move compared to more distant objects.

By measuring the amount of the shift of the object’s position (relative to a fixed background, such as the very distant stars) with observations made from the ends of a known baseline, the distance to the object can be calculated.

A conveniently long baseline for measuring the parallax of stars (stellar parallax) is the diameter of the Earth’s orbit, where observations are made 6 months apart. The definition of the parallax angle may be determined from the diagram below:

If the parallax angle, p, is measured in arcseconds (arcsec), then the distance to the star, d in parsecs (pc) is given by:

It is important to note that in this example we assume that both the Sun and star are not moving with a transverse velocity with respect to each other. If they were this would complicate the picture as presented here. In practice stars with significant proper motions require at least three epochs of observation to accurately separate their proper motions from their parallax. Stars that are members of binaries further complicate the picture.

The only star with a parallax greater than 1 arcsec as seen from the Earth is the Sun – all other known stars are at distances greater than 1 pc and parallax angles less than 1 arcsec. When measuring the parallax of a star, it is important to account for the star’s proper motion, and the parallax of any of the ‘fixed’ stars used as references.

Over a 4 year period from 1989 to 1993, the Hipparcos Space Astrometry Mission measured the trigonometric parallax of nearly 120,000 stars with an accuracy of 0.002 arcsec. The GAIA mission, to be launched in 2010, will be able to measure parallaxes to an accuracy of 10 -6 arcsec, allowing distances to be determined for more than 200 million stars.

Study Astronomy Online at Swinburne University
All material is © Swinburne University of Technology except where indicated.

## When will the number of stars be a maximum? - Astronomy

The following table shows the number of stars in each magnitude range, the cumulative number of stars from -1 magnitude to the current magnitude of the row, and the precentage increase in stars with an increase of one magnitude.

On the average when you can increase the faintest stars you can observer by one magnitude fainter you can observer about three times (3X) more stars. For example, if can observer magnitude 2 stars in the city and can observer magnitude 3 at your home you should be able to see three time more stars at your home. If you go to a star party where you can see magnitude 5 stars you should see about 27 times more stars at the star party as compared to observing in the city (magnitude 2 to 3 is about 3x, magnitude 3 to 4 is about 3x, and magnitude 4 to 5 is about 3x for total of 3x3x3=27).

The number of stars in the table are for the complete sky. Under ideal conditions an observer can only see one half of the sky at any time. Also the stars are not evenly distriubuted across the sky. Some parts of the sky have more stars per unit sky area than others parts of the sky.

Data is based on the Tycho Catalogue which was obtained from page VII of the Millennium Star Atlas, Volume I, Sky Publishing Corporation and European Space Agency. The Tycho Catalog is believed to be 99.9 percent complete to magnitude 10.0 and 90 percent complete to magnitude 10.5. Table data for magnitudes 11 to 20 are projected on the average increased of 291%. 291 % is the average increase of stars between magnitudes 6 to 7, 7 to 8, 8 to 9, and 9 to 10.

Magnitude Range Number of Stars
per Range
Cumulative
Stars
% Increase in
Stars Seen
-1 -1.50 to -0.51 2 2
0 -0.50 to +0.49 6 8 400%
1 +0.50 to +1.49 14 22 275%
2 +1.50 to +2.49 71 93 423%
3 +2.50 to +3.49 190 283 304%
4 +3.50 to +4.49 610 893 316%
5 +4.50 to +5.49 1,929 2,822 316%
6 +5.50 to +6.49 5,946 8,768 311%
7 +6.50 to +7.49 17,765 26,533 303%
8 +7.50 to +8.49 51,094 77,627 293%
9 +8.50 to +9.49 140,062 217,689 280%
10 +9.50 to +10.49 409,194 626,883 288%
11 +10.50 to +11.49 1,196,690 1,823,573 291%
12 +11.50 to +12.49 3,481,113 5,304,685 291%
13 +12.50 to +13.49 10,126,390 15,431,076 291%
14 +13.50 to +14.49 29,457,184 44,888,260 291%
15 +14.50 to +15.49 85,689,537 130,577,797 291%
16 +15.50 to +16.49 249,266,759 379,844,556 291%
17 +16.50 to +17.49 725,105,060 1,104,949,615 291%
18 +17.50 to +18.49 2,109,295,881 3,214,245,496 291%
19 +18.50 to +19.49 6,135,840,666 9,350,086,162 291%
20 +19.50 to +20.49 17,848,866,544 27,198,952,706 291%

The dimmest star that can be seen without optical aid in a dark sky is around 6 magnitude depending upon the observer's eyesight and sky conditions. The below Telescope Limiting Magnitude table gives a rough idea of the faintest star magnitude that can been seen through a different aperture telescopes. The magnitude values are not precise because many factors affect the magnitude values such as optics, sky conditions, etc. Also the 2 inch row can be use for a 10 x 50 binoculars which is very close to a 2 inch (51 mm) telescope.

Telescope Limiting Magnitude Table is from page 5 in the excellent book Star Ware, Second Edition by Philip S. Harrington, John Wiley & Sons, Inc.

Telescope Limiting Magnitude

Aperture
Inches
Aperture
mm
Faintest
Magnitude
2 51 10.3
3 76 11.2
4 102 11.8
6 152 12.7
8 203 13.3
10 254 13.8
12.5 318 14.3
14 356 14.5
16 406 14.8
18 457 15.1
20 508 15.3
24 610 15.7
30 762 16.2

Aperture and Limiting Magnitude Table from The Guide to Amteur Astronomy by Jack Newton and Philip Teece, Second edition, page 33.

## When will the number of stars be a maximum? - Astronomy

I am confused about the number of stars in our Milky Way galaxy. Some sources say that the Milky Way consists of 100 billion stars. Other say that the MASS of our galaxy is roughly 100 billion times the mass of the Sun. So because most of the galaxy mass is in the interstellar gaseous and dust nebulae there must be less than a 100 billion stars in it or the total mass must be greater. Which of these is correct? And did someone estimate the number of ACTUAL stars in our Galaxy?

Most of the mass in the galaxy is NOT in interstellar gaseous and dust nebulae. Most of the luminous matter is in stars and not nebulae. Now, the mass of the galaxy is mostly dominated by dark matter, which is something that is not detected by any telescope, or anything except through its gravity. But as far as the luminous matter goes, most of it is stars.

About the number of stars: People have studied the mass distribution of stars in the galaxy. Further, one also knows the amount of light put out by each type of star. So, by measuring the total amount of light in the galaxy (called luminosity), and knowing the mass, one can estimate the number of stars that are there in the galaxy. So, even though we cannot actually count the number of stars in the galaxy, we can estimate the number of stars in the galaxy as roughly 100 billion (100,000,000,000). It turns out that there are many more stars with mass less than the mass of the Sun than with mass more than the mass of the Sun. So, it all works out right.

Jagadheep built a new receiver for the Arecibo radio telescope that works between 6 and 8 GHz. He studies 6.7 GHz methanol masers in our Galaxy. These masers occur at sites where massive stars are being born. He got his Ph.D from Cornell in January 2007 and was a postdoctoral fellow at the Max Planck Insitute for Radio Astronomy in Germany. After that, he worked at the Institute for Astronomy at the University of Hawaii as the Submillimeter Postdoctoral Fellow. Jagadheep is currently at the Indian Institute of Space Scence and Technology.

## When will the number of stars be a maximum? - Astronomy

Overview of Radio Emission from Astronomical Objects

When we look at the sky at night with our unaided eyes, we see about 2000 stars of various levels of brightness, and if we are far from city lights we may see the faint band of the Milky Way, which is the light from billions of stars making up our galaxy. But if our eyes were able to see radiowaves, the sky might look like the image below.

(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

It may appear similar to the starry sky, but in fact most of the point-like objects are not stars, but luminous radio galaxies billions of light years away. The larger sources are ionized clouds of hydrogen, or supernova remnants.

Looking toward the center of our galaxy, our radio eyes would see a large variety of strange features, most of which are not visible in other wavelengths.
The Galactic Center - First Light from MeerKAT Radio Telescope
Credit: https://www.gizmodo.com.au/2018/07/new-south-african-telescope-releases-epic-image-of-the-galactic-centre/

Owens Valley Long Wavelength Array Movie

The Electromagnetic Spectrum

• Radio waves reach the ground
• Can observe objects or phenomena that are difficult or impossible to detect in other wavelength ranges
• Can use radio emission for quantatitive physical diagnostics of object parameters
Note that the window closes at the long-wavelength end of the spectrum--not because of the atmosphere, which remains transparent to long-wavelength radio waves--but rather due to the ionosphere, which reflects the radiation.

A second reason is that some objects and phenomena are invisible or hard to detect in other wavelengths, and can only be seen, or can be seen with greater sensitivity, in the radio. Here are a few of many many examples from which we could choose:

Neutral hydrogen traces interactions among galaxies in the M81 group.
(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

Centaurus A -- peculiar galaxy with radio lobes. From HST web site.

Jupiter's Radiation Belt The Sun .
(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

The third important reason to explore astronomical objects in radio wavelengths is that the emission properties provide quantitative physical information about conditions in the source. We will see that radio emission is produced in a large number of ways. The low-energy radio photons are relatively easy to produce, which makes radio emission sensitive to a great many parameters. However, the number of mechanisms is itself a problem. Before one can use the emission to give information, one must first determine which radio emission mechanism is responsible for the emission. In practice, the most accurate way to determine the emission mechanism is to have spectral information, since different emission mechanisms have different characteristic spectral properties. In addition to helping to determine the emission mechanism, quantifying spectral properties such as peak brightness, peak frequency, spectral slopes, etc., also provides quantitative diagnostic parameters.

For all of these reasons and more, the radio range of wavelengths is as essential as gamma ray, X-ray, UV, optical, and IR for providing a complete picture of the physical nature of astronomical sources.

The term "single element" means either single parabolic dishes, or in some cases single dipole elements. Here are a few pictures:

Arecibo: The largest single dish in the world, 306 m
(c) Cornell University / National Science Foundation

Green Bank Telescope (GBT): The largest fully steerable single dish in the world, 100 x 110 m
(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

RATAN 600: Diameter 600 m, part of a "dish" reflecting surface Metsahovi: Large mm dish

1.22 l /D , where q is the angular diameter of the Airy Disk at the half-power point (the full-width-half-maximum, or FWHM) in radians. At a frequency of 5 GHz, even the Arecibo dish has an angular resolution of only about 50 arcseconds. The fully-steerable GBT has a resolution at this frequency of only 150 arcseconds.

Because of the limited spatial resolution of single element telescopes, sophisticated techniques have been developed to combine single elements into multiple-element arrays, which work together to form a single telescope. In such arrays, the spatial resolution is determined not by the size of the individual elements, but rather by the maximum separation between elements, which is referred to as the baseline length, B . With an interferometer, the diffraction limit is q

l /B , where B can extend to many (even thousands of) km.

We now show some examples of interferometer arrays:

Close-up of VLA (Very Large Array)
(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

Aerial view of VLA in its most compact configuration.
(c) National Radio Astronomy Observatory / Associated Universities, Inc. / National Science Foundation

Ten antennas of NJIT's 13-antenna Expanded Owens Valley Solar Array (EOVSA)

• Low Frequency Array (LOFAR)
• Atacama Large Millimeter Array (ALMA)
• Frequency Agile Solar Radiotelescope (FASR)
• Square Kilometer Array (MeerKAT (South Africa))
• the peak wavelength shifts proportional to temperature l maxT = 2.898x10 - 3 m K . (Wien Displacement Law)
• the intensity increases as the square of the frequency at low frequencies (Rayleigh-Jeans Law)
• the intensity decreases exponentially at high frequencies (Wien Law)
• the flux of radiation emitted by a blackbody increases as the fourth-power of the temperature (Stefan-Boltzmann Law)
 2hc 2 / l 5 B l (T) = (wavelength form) (1) e hc/ l kT - 1

 2h n 3 /c 2 B n (T) = (frequency form) (2) e h n/ kT - 1
• To derive the Wien displacement law, find the maximum of the function by setting dB l (T) / d l = 0 , to get l maxT = hc/5k = 2.898 x 10 - 3 m
• To derive the Rayleigh-Jeans law, expand e h n/ kT in Bn (T) for h n << kT to get Bn (T) = 2kT n 2 /c2
• To derive the Wien Law, expand e h n/ kT in Bn (T) for h n >>kT to get Bn (T) = (2h n 3 /c2 ) e - h n/ kT
• To derive the Stefan-Boltzmann Law, integrate Bl (T) over all wavelengths--hint: use the relation

kT ). We have T = h n /k = (6.63 x 10 - 34 J s)(1 x 10 11 s - 1 )/1.38 x 10 - 23 J/K = 4.8 K !

So even very cold sources at high frequencies still meet the Rayleigh-Jeans criterion. This turns out to be especially useful for radio astronomy, which we will discuss in a moment. But first, let's look at another plot of the Planck function, with axes suitable for a visual appreciation of the Rayleigh-Jeans limit.

By plotting B n (T) on a log-log plot, the part of the curve that obeys the Rayleigh-Jeans Law,

which is just the power per unit area. In radio astronomy, we often discuss a related quantity called the flux density, which is the monochromatic intensity (or the Planck function) integrated over solid angle:
 S = I( n ) d W ( units: W m - 2 Hz - 1 ) (4)

In fact, the flux density is a fundamental quantity measured by radio telescopes, and is the basis for two different units: 1 Jansky (Jy) = 10 - 26 W m - 2 Hz - 1
1 Solar Flux Unit (sfu) = 10 - 22 W m - 2 Hz - 1 = 10000 Jy.

We are now ready to show a great conceptual simplification that the Rayleigh-Jeans limit gives to the discipline of radio astronomy. We have so far been talking about blackbodies, which are by definition optically thick and in thermal equilibrium. What if a source is not optically thick? In that case, its emission will appear weaker (lower intensity) than if it were optically thick. Whether or not a source is optically thick is a function of frequency. As it turns out, many radio-emitting plasmas are optically thick at low frequencies, but optically thin at high frequencies. In this case, the brightness follows the Planck function up to some frequency, then begins to fall away as it becomes more and more optically thin with frequency. Schematically, it looks something like this:
Radio spectrum for a 10 6 K plasma that is optically thick below about 10 GHz, and optically thin
at higher frequencies. The brightness below 10 GHz corresponds to a million degree blackbody.

We will discuss optical depth in more detail in two weeks, when we discuss radiative transfer. For now, we just want to develop the idea of brightness temperature.

In the Rayleigh-Jeans limit, a blackbody has a temperature given by the Rayleigh-Jeans Law, eq (3), i.e. T = B n (T)c 2 /2k n 2

so as long as the plasma in the above figure is optically thick, we can use the brightness of the emission to determine the plasma temperature. But when it is optically thin, the brightness, or intensity, is less than the Planck function. Nevertheless, we can still talk about a brightness temperature, or the equivalent temperature that a blackbody would have in order to be as bright. The brightness temperature is the same as the true temperature only for an optically thick blackbody. We designate the brightness temperature as T b . Using this notation, the flux density measured by a radio telescope becomes:
 S = 2kT b n 2 /c 2 d W = 2k n 2 /c 2 T b d W (5)

where we have substituted B n for I( n ) in (4), and used (3). So the flux density measured by a radio telescope is just the brightness temperature integrated over the source, times some fundamental constants and frequency-squared.

So far, eq (5) pertains only to thermal emission, but we can extend it to all radio emission simply by considering non-thermal sources as having an effective temperature T eff . For a single electron of energy E , its effective temperature is just its kinetic temperature T eff = E/k .

To summarize, then, the brightness temperature is the equivalent temperature a black body would have in order to be as bright as the observed brightness. It is important to realize that this is a useful concept only for radiation that obeys the Rayleigh-Jeans Law.

One last point to make is the limit of the integral in eq (5). We earlier mentioned the resolution of a single dish antenna of diameter D , as q

1.22 l /D . This is also the width of the field of view of the antenna--only a source in an area of the sky within this angular distance can be seen. The field of view is also called the beam. Let's look at some consequences of this.

## Circumpolar stars never rise or set

Star trails image via Yuri Beletsky Nightscapes.

Circumpolar stars always reside above the horizon, and for that reason, never rise or set. All the stars at the Earth’s North and South Poles are circumpolar. Meanwhile, no star is circumpolar at the equator.

Anyplace else has some circumpolar stars, and some stars that rise and set daily. The closer you are to either the North or South Pole, the greater the circle of circumpolar stars, and the closer you are to the equator, the smaller.

From the Northern Hemisphere, all the stars in the sky go full circle around the north celestial pole once a day – or more precisely, go full circle every 23 hours and 56 minutes. And from the Southern Hemisphere, all the stars in the sky go full circle around the south celestial pole in 23 hours and 56 minutes.

The Big Dipper and the W-shaped constellation Cassiopeia circle around Polaris, the North Star, in a period of 23 hours and 56 minutes. The Big Dipper is circumpolar at 41 o N. latitude, and all latitudes farther north.

We in the Northern Hemisphere are particularly lucky to have Polaris, a moderately-bright star, closely marking the north celestial pole – the point in the starry sky that’s at zenith (directly overhead) at the Earth’s North Pole.

At the equator (0 o latitude) the star Polaris – the stellar hub – sits right on the northern horizon, so no star can be circumpolar at the Earth’s equator. But at the North Pole (90 o ) Polaris shines at zenith (directly overhead), so from the North Pole every star in the sky stays above the horizon all day long every day of the year.

The circle of circumpolar stars in your sky is determined by your latitude. For instance, at 30 o North latitude, the circle of stars within a radius of 30 o from Polaris is circumpolar. In the same vein, at 45 o or 60 o N. latitude, the circle of stars within 45 o or 60 o , respectively, of Polaris would be circumpolar. Finally, at the North Pole, the circle of stars all the way to the horizon is circumpolar.

View larger. The stars revolve around the North Star, which serves as the center of the great celestial clock. Star trails produced by long time exposure photograph.

At 41 o North Latitude (the latitude of New York City), and all latitudes farther north, the famous Big Dipper asterism is circumpolar. That’s because the southernmost star of the Big Dipper, Alkaid – the star marking the end of the Big Dipper handle – is 41 o south of the north celestial pole (or 49 o north of the celestial equator).

If you’re in the northern U.S., Canada or at a similar latitude, the Big Dipper is circumpolar for you – always above the horizon. These images show the Dipper’s location at around midnight in these seasons. Just remember “spring up and fall down” for the Dipper’s appearance in our northern sky. It ascends in the northeast on spring evenings, and descends in the northwest on fall evenings. Image via burro.astr.cwru.edu

Bottom line: Every star rises and sets as seen from the Earth’s equator, but no star rises or sets at the Earth’s North and South Poles. Instead, as viewed from the poles, every star is circumpolar. Between the equator and the poles … you’ll see some circumpolar stars and some stars that rise and set daily.

## Astronomy and Astrophysics (ASTRO)

ASTRO 1 Astronomical Universe (3) (GN)(BA) This course meets the Bachelor of Arts degree requirements. Students who have passed ASTRO 5, ASTRO 6, ASTRO 7N or ASTRO 10 may not take this course for credit. Overview of modern understanding of the astronomical universe. ASTRO 1 is an introductory course for non-science majors. It provides a broad introduction to Astronomy with qualitative descriptions of the dazzling and varied contents of the universe including planets, the Sun and other stars, exoplanets, red giants, white dwarfs, neutron stars, black holes, supernovae, galaxies, dark matter, and more. The course will explore how these objects form and change and interact, how the whole whole universe formed and changes (cosmology), and where Earth fits in the vast scheme of things. Students will learn how our relative place, orientation, and motion in space dictate our changing view of the sky (daily and yearly sky motions, phases of the moon) and conditions on Earth (arctic, tropics, and seasonal changes). Descriptions will build upon the basic physics of gravity, light, and atoms, and will be discussed in the context of the process of science as a robust and self-correcting way of learning and knowing that relies on making and testing predictions by gathering evidence. The goal of this course is to cover most of the areas of modern astronomy at a level which requires only basic mathematics.

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

ASTRO 1H Astronomical Universe (3) (GN)(BA) This Honors course meets the Bachelor of Arts degree requirements. Students who have passed ASTRO 5, ASTRO 6, ASTRO 7N or ASTRO 10 may not take this course for credit. Overview of modern understanding of the astronomical universe. ASTRO 1H is an introductory course for non-science majors. It provides a broad introduction to Astronomy with qualitative descriptions of the dazzling and varied contents of the universe including planets, the Sun and other stars, exoplanets, red giants, white dwarfs, neutron stars, black holes, supernovae, galaxies, dark matter, and more. The course will explore how these objects form and change and interact, how the whole whole universe formed and changes (cosmology), and where Earth fits in the vast scheme of things. Students will learn how our relative place, orientation, and motion in space dictate our changing view of the sky (daily and yearly sky motions, phases of the moon) and conditions on Earth (arctic, tropics, and seasonal changes). Descriptions will build upon the basic physics of gravity, light, and atoms, and will be discussed in the context of the process of science as a robust and self-correcting way of learning and knowing that relies on making and testing predictions by gathering evidence. The goal of this course is to cover most of the areas of modern astronomy at a level which requires only basic mathematics.

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

The development of our modern understanding of the visible sky and planetary systems. Students who have passed ASTRO 1, ASTRO 7N, or ASTRO 10 may not take this course for credit. ASTRO 5 The Sky and Planets (3) (GN) will introduce students to the wonders of the universe and help them to understand how the universe works through the laws of physics. During the semester, they will learn about the different observed motions of objects in our sky, how astronomical objects influence our concepts of time, the nature of light and spectra, how planetary systems are formed and comparative details about our solar system and other planetary systems. Many colorful images and movies of the solar system have been collected by robotic satellite missions like Voyagers I & II, the Magellan mission to Venus, Mars rovers and orbiters, the Galileo and Juno missions to Jupiter, the Cassini and Huygens missions to Saturn, and the New Horizons mission to Pluto and the Kuiper Belt. These and other images will be used to convey the excitement of discovery and nature of astronomical study of the Solar System to our students.

Prerequisite: Students who have passed ASTRO 001 or ASTRO 010 may not take this course.

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

ASTRO 6 Astronomical Universe (3) (GN) This course meets the Bachelor of Arts degree requirements. Students who have passed ASTRO 1, ASTRO 7N, or ASTRO 10 may not take this course for credit. Overview of modern understanding of stars, galaxies, and cosmology. ASTRO 6 is an introductory course for non-science majors. It provides a broad introduction to many areas of Astronomy with qualitative descriptions of the dazzling and varied contents of the universe including the Sun and other stars, red giants, white dwarfs, neutron stars, black holes, supernovae, galaxies, dark matter, and more. The course will explore how these objects form and change and interact, how the whole whole universe formed and changes (cosmology), and where Earth fits in the vast scheme of things. Descriptions will build upon the basic physics of gravity, light, and atoms, and will be discussed in the context of the process of science as a robust and self-correcting way of learning and knowing that relies on making and testing predictions by gathering evidence. The goal of this course is to cover most of the areas of modern astronomy at a level which requires only basic mathematics.

Prerequisite: Students who have passed ASTRO 001 and ASTRO 010 may not take this course.

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

ASTRO 7N (GA/GN) is both an introductory course in astronomy for non-science majors and a creative space for those with science backgrounds interested in visual arts it provides students the opportunity to demonstrate understanding and develop a personal connection to the subject by designing four art projects. Students will learn the broad concepts of astronomy by playing an immersive video game, which allows them to 1) explore seasons, phases of the Moon, light, gravity, and telescopes from a virtual colony on Mars 2) fly from planet to planet in the Solar System and learn about their properties and formation 3) visit the Sun and other stars, learn how they produce energy, and about their life cycles 4) fly through the cosmos and construct their own universe, particle by particle. Students will also learn about the relationships and exchanges between arts and sciences, and explore inspiration and perspective on these topics by designing themed art projects using traditional and digital media. These projects include assembling a photo- journal of astronomically-relevant subjects, constructing their own video-game-like scene, interpreting data to inform a plausible depiction of an alien world, and producing three- color images using methods like those employed by astronomers to compose and display Hubble Space Telescope images. Students who have passed ASTRO 1, ASTRO 5, ASTRO 6 or ASTRO 10 may not take this course for credit.

General Education: Arts (GA)

General Education: Natural Sciences (GN)

General Education - Integrative: Interdomain

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Integrative Thinking

ASTRO 10 Elementary Astronomy) (GN) (BA) This course meets the Bachelor of Arts degree requirements. Students who have passed ASTRO, 1, ASTRO 5, ASTRO 6, or ASTRO 7N may not take this course for credit. Students may not receive General Education credit for ASTRO 10 unless they also take ASTRO 11. Overview of modern understanding of the astronomical universe. ASTRO 10 is an introductory course for non-science majors. It provides a broad introduction to Astronomy with qualitative descriptions of the dazzling and varied contents of the universe including planets, the Sun and other stars, exoplanets, red giants, white dwarfs, neutron stars, black holes, supernovae, galaxies, dark matter, and more. The course will explore how these objects form and change and interact, how the whole whole universe formed and changes (cosmology), and where Earth fits in the vast scheme of things. Students will learn how our relative place, orientation, and motion in space dictate our changing view of the sky (daily and yearly sky motions, phases of the moon) and conditions on Earth (arctic, tropics, and seasonal changes). Descriptions will build upon the basic physics of gravity, light, and atoms, and will be discussed in the context of the process of science as a robust and self-correcting way of learning and knowing that relies on making and testing predictions by gathering evidence. The goal of this course is to cover most of the areas of modern astronomy at a level which requires only basic mathematics.

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

Selected experiments and explorations to illustrate major astronomical principles and techniques. Telescopic observations of planets, stars and nebulae. ASTRO 11 Elementary Astronomy Laboratory (1) (GN)(BA) This course meets the Bachelor of Arts degree requirements. ASTRO 11 is the 1 credit laboratory component of this overview of astronomy and is intended to be taken in conjuction with ASTRO 10. It covers material similar to the lecture component in ASTRO 10, but the selected topics are covered in more depth and are focused on active learning components. Weekly two-hour labs may include investigating the habitable zone of a variety of stars, investigating the phases of the moon, analysis of the properties of stars in a color-magnitude diagram, analysis of the colorful spectra of different chemical elements, and exploration of one of the deepest images of space ever obtained. In addition, students will complete a semester nighttime observing project that typically involves learning some constellations, tracing phases of the moon, and sketching images seen through telescopes or binoculars at the student observatory.

Enforced Prerequisite at Enrollment: or concurrent: ASTRO 1 or ASTRO 10

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

Being in the Universe" considers three fundamental questions of human existence from both humanistic and scientific perspectives: (1) What is the nature of our universe, and to what extent are creatures like ourselves a predictable consequence of it? (2) What is the nature of time, and what does it mean to be a conscious being living our lives through time? (3) What would it mean for humans to be alone in the Galaxy or the universe, or alternatively, not alone? "Being in the Universe" is an integrative GH+GN GenEd course. The course's three major units cover the following topics: (1) We discuss cosmology and religion as human enterprises, as well as the history of science (2) We study the basic scientific theory of the Big Bang universe, and consider its implications for human life (3) We address contemporary theories of the multiverse from scientific, philosophical, and literary perspectives (4) We consider the thermodynamic and relativistic theories of time, and the basic philosophical approaches to time, and discuss the implications of these for our ordinary human experience of the past, present, and future (5) We discuss the history of life in the universe, the possibility of life on other planets, and the social, religious, and imaginative reactions to those possibilities in literature and film.

Bachelor of Arts: Humanities

Bachelor of Arts: Natural Sciences

General Education: Humanities (GH)

General Education: Natural Sciences (GN)

General Education - Integrative: Interdomain

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Integrative Thinking

GenEd Learning Objective: Key Literacies

Introduction to the study of modern astronomy through discussions, activities, and writing.

The course is designed to provide first year undergraduate students in both the ASTRO and PASTR majors with necessary tools and techniques to perform research. Students will practice a variety of techniques on authentic astronomical data, which might include light curves from the Kepler mission, galaxy and stellar spectra from the Sloan Digital Sky Survey, or pulsar data from the Green Bank or Arecibo telescopes. An emphasis will be placed on using common tools for observational astronomy, such as viewing astronomical FITS images in SAOimage. Students will be introduced to the common programming languages and environments used by astronomers at the time the course is offered, which currently includes Python and IDL. Students will be given experience in calculating statistical information about a set of astronomical data using the R programming language and its built-in tools. Students will make plots to illustrate a pattern in their data using the tools in Python, IDL, or R, for example.

Formal courses given infrequently to explore, in depth, a comparatively narrow subject which may be topical or of special interest.

This course is designed to engage students with the big ideas of astronomy in ways that will help them understand both the content of astronomy, as well as the practices of science as carried out by astronomers. The course is designed for prospective elementary and middle school teachers (PK-4 and 4-8 majors), although it is available to other non-science majors. Throughout the course, students engage in a series of investigations that lead towards the development of evidence-based explanations for patterns observed in the current Solar System. Investigations will include computer-based simulations, night-sky observations, and use of simple laboratory equipment. These investigations lead students towards an understanding of how observations of the current Solar System can be explained by the model of its formation. The course is designed to build from students' own personal observations of the day and night sky towards developing increasingly sophisticated explanations for those phenomena and beyond. Conducting these astronomy investigations will help students understand fundamental aspects of physics, thus broadly preparing them for future science teaching in these domains. The course models evidence-based pedagogy, thus helping to prepare students for future teaching careers as they learn effective strategies for teaching science.

Exploration of Cosmology, Birth, and Ultimate Fate of the Universe Origin of Galaxies, Quasars, and Dark Matter. For non-science majors ASTRO 120 The Big Bang Universe (3) (GN)(BA) This course meets the Bachelor of Arts degree requirements. Astronomical observations made during the last 70 years, combined with mathematical physical theory (Einstein's General Relativity), has led to a dramatic new view of the history of the Universe. Ten to twenty billion years ago, all the material that is now contained in stars, planets, and galaxies was then compressed into a region, smaller than a pinhead, and so hot that atoms could not survive. This fiery cauldron cooled and expanded, forming hydrogen and helium, and eventually all the materials and structures that we know today. This course will discuss the evidence, theories and controversies of this new scientific cosmology, commonly known as 'the Big Bang'. This class is designed for the non-science students who, after learning the fundamentals of astronomy in ASTRO 1(GN), ASTRO 5 (GN) or ASTRO 10 (GN), want to pursue further the questions of cosmology. The great success of the Big Bang theory in explaining the expansion of the Universe, the synthesis of the chemical elements, and the relic radiation leftover from the first moments are reviewed. Some of the questions discussed are still debated in the scientific community. For example: Why do some galaxies have stunning spiral structures, while others are relatively featureless ellipticals? What is the "dark matter" that may have emerged from the Big Bang, and seems to make a larger contribution to the mass of the universe than all of the material we are familiar with? What can the most distant and oldest objects we know of, the quasars, tell us about how galaxies formed? In presenting the development of this subject, the empirical and conceptual methods of modern physical science are conveyed. Students are assigned problems that exercise the use of elementary mathematics and physics to address real issues, and will confront discussions of interpretation and meaning in essays. A final project allows them to explore individual interests.

Enforced Prerequisite at Enrollment: ASTRO 1 or ASTRO 6 or ASTRO 10

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

The predicted properties of black holes and the astronomical evidence for their existence are investigated in the context of modern ideas about space, time, and gravity. ASTRO 130 Black Holes in the Universe (3) (GN)(BA) This course meets the Bachelor of Arts degree requirements. Black Holes in the Universe introduces students to the predicted properties of black holes and the astronomical evidence for their existence. Modern ideas about the nature of space, time, and gravity are also covered. The key topics discussed in the course include Newton's and Einstein's theories of gravity, predicted properties of black holes, stars and their fates, how to detect a black hole, gamma-ray bursts, supermassive black holes in galactic nuclei, active galaxies, black hole spin, gravitational waves, Hawking radiation, singularities, and black hole child universes. The course is intended to be an attractive choice for students who are interested in enriching and broadening their understanding of modern physical science.The course is intended for students who have completed and enjoyed the one-semester survey of modern astronomy, ASTRO 1, 6, or 10. It has an interdisciplinary flavor, combining basic physical concepts, astronomical observations, and philosophical ideas to present a complete picture of the current understanding of black holes. Time is also devoted to provide historical insight into the development of our ideas about gravity from Kepler and Newton through Einstein and modern ideas about quantum gravity. Students use mathematics at the level of high school algebra.

Enforced Prerequisite at Enrollment: ASTRO 1 or ASTRO 6 or ASTRO 10

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

The problem of the existence of life beyond Earth is investigated, drawing from recent research in astronomy and other fields. ASTRO 140 Life in the Universe (3) (GN)(BA) This course meets the Bachelor of Arts degree requirements. The possibility of life beyond Earth is one of the great unsolved puzzles of human thought and has been debated for millennia. An answer would fundamentally change the relationship between the human race to the rest of the Universe. Advances in modern physics and astrophysics have dramatically changed and enriched the understanding of our cosmic surroundings, but have not yet produced an unambiguous evidence concerning the extraterrestrial life. Yet, significant progress has been made on certain aspects of the problem. Recent observations of protoplanetary disks around young stars, planets around solar-type stars and a rapidly spinning pulsar (a Penn State discovery), and pervasive organic molecules throughout the Galaxy give tantalizing, albeit indirect, hints in favor of the existence of nonterrestrial life. "Life in the Universe" is envisioned to be an attractive choice for students who are interested in enriching and broadening their understanding of modern science. The course is highly interdisciplinary, combining evidence from several fields of science to describe our chances to encounter life beyond Earth and the Solar System. Selecting this course would be a logical choice for students who completed and enjoyed ASTRO 1 (GN), ASTRO 5 (GN), or ASTRO 10 (GN). The students are expected to reach the following goals from this course: - learn to appreciate limitations of human experience and a role of the interdisciplinary approach in solving scientific problems - gain understanding of a relationship between the physical Earth, its biosphere, and the rest of the observable Universe - examine in some detail a contemporary problem of scientific investigation: the astrophysical evidence for planets around stars other than the Sun - assess the scientific significance of searches for extraterrestrial life including technological civilizations. Lectures systematically cover the topics listed in the course outline at a level appropriate for non-science students, although students from the Planetary Science & Astronomy major, as well as other science and engineering majors, can take the course. While general understanding of astronomy from the prerequisite course is expected, the necessary physical and astrophysical concepts are reintroduced to assure a logical and coherent flow of information throughout the course. Videos are used to illustrate a number of topics, such as the search for extraterrestrial intelligence, physical conditions on planets of the Solar System, the detection of planets around a neutron star, and to evaluate the scientific content of science fiction movies.

Enforced Prerequisite at Enrollment: ASTRO 1 or ASTRO 5 or ASTRO 10

Bachelor of Arts: Natural Sciences

General Education: Natural Sciences (GN)

GenEd Learning Objective: Crit and Analytical Think

GenEd Learning Objective: Key Literacies

The search for life beyond planet Earth has been the subject of much interdisciplinary scientific search and has stimulated human imagination. Scientific discoveries of exoplanets (outside of our solar system), of extremophiles (life which can survive in extreme conditions) and the discoveries of conditions on other bodies in our solar system which might be able to support life, has provided progress in answering the question of the existence of extraterrestrial life. Not only have a plethora of fictional work appeared in the film media to depict scenarios of life beyond Earth, but there has also been an abundance of video media created to present the scientific ideas to the wider audience beyond the scientific community. This course intends a critical evaluation of both nonfiction and fictional media works in the educational dissemination of scientific ideas and the effective presentation of concepts. We will analyze techniques in photography, mise en scene, editing, sound, dramatization and writing as they are applied to topics in astrobiology.

## Author information

### Affiliations

Instituut voor Sterrenkunde, KU Leuven, Leuven, Belgium

L. Decin, W. Homan, T. Danilovich, A. de Koter, C. Gielen & M. Van de Sande

School of Chemistry, University of Leeds, Leeds, UK

Astronomical Institute Anton Pannekoek, University of Amsterdam, Amsterdam, The Netherlands

A. de Koter & L. B. F. M. Waters

Hamburger Sternwarte, Hamburg, Germany

SRON Netherlands Institute for Space Research, Utrecht, The Netherlands

Onsala Space Observatory, Department of Space, Earth and Environment, Chalmers University of Technology, Onsala, Sweden

Instituto de Astrofísica de Canarias, La Laguna, Spain

Departamento de Astrofísica, Universidad de La Laguna (ULL), La Laguna, Spain

E. A. Milne Centre for Astrophysics, Department of Physics & Mathematics, University of Hull, Hull, UK

School of Physics and Astronomy, University of Birmingham, Birmingham, UK

I-BioStat, Universiteit Hasselt, Hasselt, Belgium

I-BioStat, KU Leuven, Leuven, Belgium

Department of Astrophysics, University of Vienna, Vienna, Austria

Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy, University of Manchester, Manchester, UK

Laboratory for Space Research, University of Hong Kong, Lung Fu Shan, Hong Kong

Centre for Mathematical Plasma-Astrophysics, KU Leuven, Leuven, Belgium

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### Contributions

L.D. identified the spiral structure in the ALMA data of OH 26.5 + 0.6 and OH 30.1 − 0.7, performed the full analysis and led the consortium, W.H., T.D. and A.d.K. contributed to the interpretation of the data, D.E., D.A.G.-H. and S.M. proposed the ALMA observations (ALMA proposals 2015.1.00054.S, 2016.1.00005.S and 2016.2.00088.S), S.M. reduced the ALMA data, D.E. did the sample analysis of the extreme OH/IR stars, G.M. gave advice on statistical matters, I.E.M. ran the ballistic simulations, C.G. made Fig. 4 and all authors contributed to the discussion.