l'Heure Bleue

Well, it took a while but I finally got around to taking my new telescope out last weekend (Friday, August 1). After all, it's not easy finding a nice, pleasant, clear evening in Minnesota on a weekend night -- when one can stay up until the wee hours without having to worry about work the next day. ...and what a night it was -- perfectly clear, cool, and without any trace of moon.

I was immediately impressed with the views through this telescope -- looking at any random point in the sky was like looking at salt spilled on black paper, and when trained on the Andromeda Galaxy (M31), the disk was as beautiful as I've ever seen it. Jupiter showed nice bands, even though I was limited to about 75x (12mm eyepiece at 910mm focal length). The contrast was incredible -- blacks were black, and bright areas were bright, and there was a clear, sharp line between them. No color, no stray light, no distractions -- nothing. The real strength of this telescope will be in photography, however -- as a visual instrument, 5" is just a bit too small, even if they are a nearly perfect 5 inches.

I am even more excited now, as a friend of mine has loaned me a CCD camera. This particular model was originally designed for microscopy, but should serve nicely as a detector at the end of the TMB. The sensor isn't huge and the pixels are on the small-side (6-point-something microns), but this is about a year earlier than I thought I'd be able to get my hands on such a nice camera -- very exciting! More posts will certainly be forthcoming when the first images through the telescope have been obtained.

-Jon

This is just a short note regarding a very fascinating web site by Christian Buil. He has apparantly constructed a number of spectrographs, including an Echelle spectrograph, specifically for astronomical use.

The projects on this site are essentially a catalog of what I would like to do if only I had more time and money...

The spectroscope that I've mentioned in previous entries was purchased off of Ebay several years ago, but it has never had a working slit assembly. The slit is a critical part of a spectroscope -- it controls the amount of light allowed into the instrument, and it's the illuminated slit whose image is viewed through the eyepiece.

I was able to purchase an adjustable slit assembly from the Surplus Shed online, however this assembly did not connect to my spectroscope. Thankfully, a friend of mine has a lathe in his basement & was kind enough to adjust the diameter of a critical component to the point where it would fit.

This is the result:



...and it works well too (I may post spectra in the future). I have no idea how the original slit assembly mounted to the spectroscope because there does not appear to be any threads or set screws, and this is the only real flaw in the current situation. While a friction fit works to a degree, it is loose & the slit is prone to rotating or sliding out & loosing focus. It the future i may have to drill and thread a hole in the telescope assembly to add a set screw. For now, however, the arrangement works well enough.

Well, my TMB 130 signature series refractor arrived last week (Thursday, May 15, 2008) after having been on order since late July, 2007. It would seem that it cannot reach focus without a diagonal mirror, and I do not own one, so I haven't had a chance to try it out yet. It looks great though.



I've got a MaxBright diagonal on order from Astro-Physics, so hopefully I'll get a chance to test it out when I get back from California in early June.

Something there is more immortal even than the stars,
(Many the burials, many the days and nights, passing away,)
Something that shall endure longer even than lustrous Jupiter
Longer than sun or any revolving satellite,
Or the radiant sisters the Pleiades.

-- Walt Whitman, On the Beach at Night

My plan is for this to be my final entry in the "Sirius B & White Dwarfs" series of blog entries.

In my last entry I discussed the nature and origin of electron degeneracy pressure. In this entry I will (briefly) discuss a program that I wrote to solve for the structure of a white dwarf star -- that is, to calculate the mass and density vs. radius profiles of such a star, starting from the core and working towards the surface.



Some of the most critical work in this area was done by Chandrasekhar and is addressed in his book (which I mentioned in my last entry) or in his 1935 paper The Highly Collapsed Configurations of a Stellar Mass, Monthly Notices of the Royal Astronomical Society, Vol. 95, p.207-225.

Deriving the equation of state (i.e., pressure / density relationship) for a degenerate electron gas is the first and probably most theoretically challenging step towards solving for the structure of a white dwarf. This equation of state is represented parametrically by Chandrasekhar in equation 3 of his paper. After deriving the equation of state, solving for the structure of the star is conceptually simple. One simply has to solve a pair of first-order ordinary differential equations simultaneously (Equation 6 in Chandrasekhar's paper). Those equations are the equation of hydrostatic equilibrium:



...and a second differential equation that is used to simply add up all the mass interior to a given radius:



In a "normal" model of stellar structure (i.e., a model of a star where fusion is still actively occuring), there are additional differential equations that describe the rate of energy generation and transport to the surface, but since a white dwarf is a dead star -- i.e., there is no fusion going on -- these terms do not apply. Furthermore, since the equation of state for a degenerate electron gas does not depend on temperature (since the electrons are already moving faster -- due to quantum-mechanical considerations -- than they should be going based on the range of temperatures encountered in white dwarfs), there is no differential equation to account for temperature either. Even if temperature were considered, the degenerate electron gas is such a good conductor of heat that the star is essentially isothermal anyway. All things considered, this is a very simple model.

After defining the problem with the equation of state (which I have not presented, but is available in the paper that I linked to above) and the two differential equations, Chandrasekhar performs some substitutions and combines all these equations into a single equation -- Equation 16 in his paper:



This equation is written with dimensionless variables, but eta is related to the radius and phi is related to the density. For exact definitions, please see Chandrasekhar's paper (it's free online -- just click the link above). As this is a second-order differential equation, it requires two boundary conditions for unique solution. These are given immediately after Equation 16, but basically amount to "density goes to zero at the surface" and "d-phi/d-eta = 0 at the core due to symmetry".

From my standpoint, I wished to solve Equation 16 above in order to have a complete density vs. radius and mass vs. radius model of a white dwarf star. I accomplished this numerically in Fortran 95, and put the results online here. By supplying the two parameters that describe the star and two additional parameters that control the integration, one can create profiles of white dwarfs of any given mass. If all goes well, a plot similar to the one above should appear after submitting the form. Please see the bottom of the front page for some additional technical details regarding the code. Also, the actual code can be downloaded from the bottom of that page. Compiling it will require a Fortran 95 compiler, and g95 should work (it's what I used).

As one tries various values for 1/y0^2 (a quantity that is related to the central density), it should become apparent that as this quantity approaches zero, the size of the star also approaches zero while the overall mass approaches a finite limit -- about 1.4 solar masses. This, of course, is the Chandrasekhar limit -- the maximum possible mass for a white dwarf.

I will not go into further details here, but suffice to say that this will conclude my series on the physical nature of white dwarfs. There is quite a lot of additional theory, of course. The results that my program can generate are 70+ years old, and the field has progressed a great deal in that time.

One point that does deserve some mention is what happens to white dwarfs that approach the Chandrasekhar limit. The equation of state used in this model does not account for the possibility of inverse beta decay. As the mass of the star (and hence the centeral density) becomes very large, electrons and protons combine to form neutrons (inverse beta decay). In a star close to the Chandrasekhar limit, this process can lead to a star with fewer electrons than might otherwise be expected, so the model becomes increasingly inaccurate as one approaches the 1/y0^2=0 limit. If this process progresses to completion, the entire star is converted to neutrons leading to a neutron star rather than a white dwarf. Such an entity requires a different equation of state and is not dealt with in this model.

After arranging a mirror to direct light into my spectroscope and arranging a camera to take macro-photographs of the vernier, I had a much easier time measuring spectral lines. Below is an image that I took today, along with an annotated diagram created with my Matlab script.

The wavelengths that I observed are:

486.1 (hydrogen-beta -- used as a reference)
516.6 (unresolved doublet, Mg and Fe)
517.1 (Mg)
518.0 (Mg)
527.3 (Fe -- not shown on these diagrams)

The annotations in the second image are literature data (from Wikipedia), not observed data. All wavelength values are in nanometers.

Matlab is a wonderful tool -- one that I use every day in my work as a graduate student. Since I have been playing with spectra lately, I thought it would be useful to have some reference spectra. The spectral line data is easily available from the NIST. Also, other authors have created online (java & javascript) programs that generate images of line spectra from this data, such as this one by Joachim Köppen (based on original work by John Talbot).

I decided to write a similar program in Matlab rather than Javascript, so it must be run locally on a machine equipped with Matlab. This program takes data consisting of line wavelengths and intensities and produces an image of the resulting spectrum. It treats each line as a Gaussian distribution in wavelength and at each wavelength it sums up the contributions from all lines. The resulting spectrum is then rendered in color to produce a realistic-looking display. The spectrum of calcium is visible below.



If anyone wishes to examine and/or run the code at home, it is available here.

Twinkle, twinkle, little star,
How I wonder what you are!
Up above the world so high,
Like a diamond in the sky!

-- Jane Taylor, "The Star", 1806

Today I happened to notice that the sun was shining through the window and directly onto a spectroscope that was sitting on my bench. I purchased this piece of equipment on eBay several years ago and am now in the process of returning it to working condition. The slit is still a jerry-rigged assembly of masking tape and aluminum foil, but I thought I'd take advantage of the favorable position of the sun to see if I could observe any spectral lines. This is something that I've tried before, but have never successfully accomplished.

After aligning the first telescope with the sun, I positioned the 2nd telescope for viewing and the lines could not have been more obvious. To record the event I grabbed my digital camera (just a little Casio -- nothing particularly fancy) and held it to the eyepiece. The following are some of the spectra that I observed:





I am fairly sure that the doublet in the orange portion of the last (bottom) spectrum is due to sodium (the D lines). Unfortunately there was not enough time to measure the wavelengths of these lines due to the position of the sun (it was only minutes from setting), so I cannot be absolutely sure of their identity. Still, after looking at a diagram I believe that the middle figure (the one that is entirely green) shows some lines from Magnesium (b1 and b2) and Iron (the E line). I am not entirely sure which is which, however. Also, the top figure probably shows the hydrogen-beta line (Fraunhofer F line -- one of the Balmer series lines).

More information on these lines can be found here. There are sites out there with far more detailed databases of solar lines (thousands are known), but this diagram is simple enough for those of us with very basic equipment. A very high resolution spectrum can be obtained from the NOAO (National Optical Astronomy Observatory).

When conditions are more favorable and this instrument is in better condition I will post a more detailed set of observations and some photos of the instrument itself.

Last summer I was trying to locate a site away from town with dark skies. Of course, just having dark skies isn't really enough -- it also has to be accessible by car. To satisfy these dual requirements I wrote a page that couples Google Maps to data supplied by the International Dark-Sky Association to produce the "Dark Sky Finder" utility:



Take a look and enjoy! I would like to see this utility improved & more widely used, but I don't have the time to do it justice. I have contacted the IDA in an attempt to transfer it to them for better publicity & coding support. If any astronomically-oriented web developers are reading this and would like to continue with this project, please let me know! I am always willing to just GPL what I've done so far & put it on SourceForge (or some such thing) if there is interest, but I don't want to bother setting up a SourceForge project if it would not be used.

It is my quasi long-term goal to put together a system for astrophotography. This consists of three major units: a telescope, a telescope mount, and a CCD camera. Unfortunately, this equipment is very expensive. The mount that I selected -- and the first piece of equipment in this set -- was a Takahashi NJP – about $7500 retail, but on July 22, 2007 I found one that included a Meade Giant Field Tripod on Astromart for about $5500. After corresponding with the seller, I sent a check on July 31.

Needless to say, I was very anxious as I waited at home for the UPS driver to deliver it on August 9. After unpacking the boxes (it was shipped in 6 packages) I was not at all pleased with what I saw. The Temma (electronics) control box was severely dented and, although it was not obvious at the time, the right ascension motor was also destroyed.



The seller (who will remain anonymous) was very helpful in contacting UPS and filing a claim, and on August 14 UPS was supposed to contact me to schedule an inspection visit to photograph the damage. They eventually did call and visited on August 16. The inspector was noncommittal when asked about whether or not the claim would be paid – for the most part she just took photos. After this, there were no developments for quite some time, and what correspondence did occur was between the seller and UPS.

In any case, I was now left with a non-functional and very expensive mount, so repairing it was the only real option. I had it crated in wood and shipped (via FedEx this time) back to Texas Nautical Repair on August 21 – it arrived on August 24 and repairs began.

On September 4 I received an email from the seller indicating that UPS had initially denied the claim due to insufficient packaging (apparently having it double-boxed with the inner “box” consisting of a $400 Pelican case is considered “insufficient”), but that he intended to fight that determination. I do not know exactly what happened next, but on September 10 I received another email from the seller indicating that the claim would be paid by UPS. On October 17 the seller mailed me a check to reimburse me for the cost of shipping the mount back to Texas for repair (which UPS reimbursed him for, I understand).

On October 22 the repairs were finished (costing about $1250, which the seller paid and UPS reimbursed) and Art at Texas Nautical Repair shipped the mount back to me via UPS. I was not particularly happy with this choice of companies, but it arrived intact on October 25. Due to a 3-week business trip to Japan on November 3, I did not do any extensive testing other than to verify that the mount was physically intact.

So, after 4 months of waiting, I have a working NJP mount. The seller in this case was incredibly helpful, and the whole experience didn’t cost me anything extra in the end. Still, I will always have a deep-seated distrust of UPS due to this incident, and I intend to avoid doing business with them whenever possible. Had the seller not been willing to stand up to their initial denial of the claim, I would have been out $1250 due to their gross mishandling of such a valuable item.

In this entry I’ll be referring to the textbooks Principles of Stellar Evolution and Nucleosynthesis (1st Ed.) by Donald Clayton. It’s a good textbook on the physical nature of stars, and it’s available on Amazon for a good price (though I can’t guarantee that the version they sell will have identical section & equation numbering as my 1970’s hardcover version). Another classic text is Chandrasekhar’s An Introduction to the Study of Stellar Structure. This is not quite as modern or as thorough in discussing the computation of main-sequence stellar models, but the material on degenerate matter & white dwarfs is very well done (by the man who won a Nobel Prize for this work).

As I described previously, a white dwarf is supported by a force known as “degeneracy pressure”. Where does this pressure come from, and what is it? Before touching on those questions, it might be helpful to understand where “normal” pressure comes from. That is, when one fills a balloon, why does it push back? Normal, or ideal gas pressure, comes from the collisions of many particles with each other or with the walls of a container. The effect is similar to being hit with a tennis ball – when the ball bounces off, you feel a momentary force exerted on wherever it hit. That is, when one places more and more air molecules in a balloon, there are more collisions with the walls and the pressure increases. In the case of normal temperatures & densities, gas particles move with a distribution of speeds that follows the Maxwell-Boltzmann distribution. This is discussed with more mathematical rigor in section 2-1 of Clayton’s book. If the particles in question are photons, rather than molecules, atoms, or electrons, then the pressure is called radiation pressure.

Degeneracy pressure is fundamentally different, and much less intuitive. It is a consequence of the laws of quantum mechanics as they apply to electrons. One should note that in a white dwarf, degeneracy pressure is due entirely to the electrons. Because of the high temperatures & pressures, all of the atoms in the star are assumed to be totally ionized, so the electrons & nuclei move separately. The nuclei provide mass to the star and do participate in ideal-gas type collisions, but they will not become degenerate. That would require much higher densities – i.e., neutron stars. Therefore, the matter in a white dwarf will be referred to as a degenerate electron gas, as the electrons are the only degenerate component.

Degeneracy pressure is difficult to explain in a way that is intuitive, but I find that the easiest way to approach it is from the Heisenberg Uncertainty Principle:



In other words, the product of the uncertainty in position (?x) and the uncertainty in momentum (?p) must be greater than some numerical constant involving Planck’s constant. When a gas is very, very dense, the positions of the particles become “fixed” – that is, we know that if there are 10 particles in 1 unit of volume, that each particle is confined to 1/10 of a volume unit. Therefore, as the density increases, the uncertainty in position (?x) decreases. Eventually, if this trend continues, we will bump up against Heisenberg’s limit. In order to compensate, we will require that the uncertainty in momentum become higher & higher. This means that at very high densities, some particles may be traveling slowly, but some particles will also be traveling very fast – regardless of thermal temperature. The distribution of speeds will no longer follow the Maxwell-Boltzmann distribution, with many particles forced to travel faster than they “should” be going, even if the gas may be any temperature – even near absolute zero. In an extreme case, where some of the particles are traveling extremely fast, they can actually approach the speed of light. Such a "relativistic" degenerate gas requires additional mathematical considerations, as special relativity imposes an upper bound (c) on the velocity, even though the maximum momentum may be arbitrarily high, as required by the density.

In fact, this same condition will exist when the temperature of a gas is decreased – very low temperatures will have the same effect as very high densities. What really matters when considering degeneracy is the combination of the temperature & density, rather then either number taken in isolation. Even a very dense gas may “break” its degeneracy at a high-enough temperature (such that the normal thermal velocities of the particles become higher than the lower-limit imposed by the Heisenberg Uncertainty Principle). Additionally, the onset of degeneracy is not sudden as density is gradually increased. As long as the temperature is not zero, there will always be some contribution from ideal gas pressure. Therefore, degeneracy can be spoken of as partial vs. "total" (with the understanding that "total" is an approximation), or as non-relativistic vs. relativistic (with relativistic degeneracy occurring at very high densities). In fact, at conditions of moderate density & very high temperature, partial AND relativistic degeneracy could occur simultaneously. An interesting diagram can be found in Clayton's book, Figure 2-11.

To summarize, under "normal" conditions, particle velocities must obey a Maxwell-Boltzmann distribution, but Heisenberg's Uncertainty Principle says that there will be an uncertainty in the speed, and that uncertainty is related to the density (since density relates to uncertainty in particle position). If the required uncertainty in speed is much greater than the speed predicted by the Maxwell-Boltzmann distribution, then the matter may be called "degenerate", and some particles will be found to be traveling much faster than expected. These speedy particles will, in turn, exert a higher pressure when they collide with other particles or with a container wall.

It is not my intention to write a textbook here, so I will end the discussion without going into mathematical derivations. For those who are interested, I recommend chapter 2 of Clayton’s book, or Chapter 8 of Chandrasekhar’s book for a full, technical treatment.

At this time of year, the star Sirius is high in the southern sky (to Northern Hemisphere observers) after dark. It's easy to spot -- it's the brightest star visible in the evening sky at an apparent magnitude of about -1.44. Unlike the Sun, Sirius is actually two stars that are fairly close to the Earth. The larger & brighter of these stars, Sirius A, is a class A main sequence star that is about twice as massive as the sun and about 1.7 times as luminous. The other star, however, is truly remarkable.

The mass of Sirius B is about the same as the Sun, or slightly less, but all that mass is squeezed into a star that is about the size of the Earth. This means that its mean density is on the order of 2,000,000 times that of water -- 1 liter would weigh in excess of 2000 metric tons. Even though Sirius B is much hotter than its larger companion, it is much less luminous (by a factor of about 10,000) due to its small size. The two stars orbit their mutual center of gravity with an orbital period of about 50 years, but they never get very far apart from one another as seen from Earth. Even with a decent amateur telescope, "splitting" the pair is difficult due to their proximity and their huge difference in brightness. With the Hubble Space Telescope, the situation is different. The image below was taken by Hubble. The small star in the lower-left of the image is Sirius B, while the much larger star in the center is Sirius A.



Without the clear view provided by the vacuum of space, the light from Sirius A tends to spread out & overwhelm its tiny companion, so many amateur astronomers have never seen Sirius B. For those who wish to try, there is an interesting article and map in the February 2008 issue of Sky & Telescope (page 30).

What is it that makes Sirius B so remarkable though? Why is it the same mass as the Sun, but only about 1/100 of the Sun's diameter? Sirius B is a white dwarf. That is, it's a star that has exhausted its fuel supply and is now slowly cooling off, just as a hot stove burner or lamp filament may continue to glow for a few seconds after the power is removed. Unlike its companion, Sirius B is not generating any new energy through fusion, but simply radiating away what energy it has left-over. Long before humans started observing Sirius B, it was a normal main sequence star like the Sun or Sirius A -- although probably larger & brighter than either one. After running out of fuel, its core began to shrink & its outer layers most likely drifted out into space (although there is no planetary nebula visible today). Without the outward pressure provided by nuclear fusion in its core, the gravity of Sirius B squeezed it into its present size -- and then stopped.

In a normal star, there are nuclear reactions occurring in the core that liberate energy and raise the temperature to millions of degrees. This energy then flows outward -- from hot to cold -- until it reaches the surface. As a consequence of its high temperature, there is a high pressure in the core. That pressure is simply the ideal gas pressure, P = kT/v (where k is Boltzmann's constant and v is the specific volume -- the inverse of density). When fusion ceases, however, the core would tend to cool since there is no new energy being generated. Under other circumstances, this would result in a pressure drop due to the P = kT/v law. In the core of a star, however, this cannot be allowed to occur. The pressure in the core must be high enough to counter-act the gravitational force exerted by the mass of the outer layers of the star in order to keep the star in balance ("hydrostatic equilibrium"). In order to maintain the high central pressure, the outer layers squeeze in until a new equilibrium is reached. This causes the v (the specific volume) term in the ideal gas law to decrease & the temperature term to increase. The core is still at a very high pressure, but it has decreased in size. In shrinking, it has turned some of its gravitational potential energy into heat and thereby maintained a high enough core pressure to remain in equilibrium.

As this process continues, the core grows smaller and hotter, and its constituent particles become closer and closer together. In reality, this process occurs before the entire star is dead -- while fusion is still going on in a layer outside the core. There is a limit to how dense the core can become however. As its size decreases and its density increases, the particles in the core (particularly the electrons, as all atoms are completely ionized at the temperatures involved) begin to interact with each other in ways that are not accounted for by the ideal gas pressure. These interactions provide an additional pressure -- the pressure that supports Sirius B against its own gravity -- called "electron degeneracy pressure". I will discuss the nature of this pressure in a future entry.

When this process reaches completion -- when the core has ceased fusion and become as small and hot as it can, and when the outer layers have swelled so much that they have floated off into space -- a white dwarf is all that remains. Sirius B, like all white dwarfs, is the dead core of a former star -- incredibly dense and propped up against gravity by an entirely different kind of pressure than during its former life.

Sometimes there's so much beauty in the world I feel like I can't take it, like my heart's going to cave in.

-- Ricky Fitts, American Beauty, 1999

It is sometimes said that scientists are unromantic, that their passion to figure out robs the world of beauty and mystery. But is it not stirring to understand how the world actually works -- that white light is made of colors, that color is the way we perceive the wavelengths of light, that transparent air reflects light, that in so doing it discriminates among the waves, and that the sky is blue for the same reason that the sunset is red? It does no harm to the romance of the sunset to know a little bit about it.

-- Carl Sagan, Pale Blue Dot, 1994



Welcome to my new blog, l'Heure Bleue! I suppose that the first order of business is explaining the unusual name. The term is French for "Blue Hour" (no, I do not actually speak French, so I suppose that the name is slightly pretentious, but this is not my intent). The blue hour is (literally) a reference to the hour between daylight and darkness. This loosely connects to the primary theme of this blog -- Astronomy.

Professional astronomers will probably not be particularly interested in the site, as I am not a professional astronomer -- just a dilettante in the field. I am a (nearly) professional scientist, however: a 5th year PhD student in chemical engineering employed as a research assistant at the University of Minnesota in Minneapolis. In writing for this blog, my goal is to provide interested amateurs with some introduction to theoretical astrophysical concepts that are not usually within the realm of amateur astronomy. I'll also be describing my own adventures in the realm of amateur observational astronomy -- both visual & photographic (when I can afford a decent camera, that is).

It is the former goal, however, that inspired the name l'Heure Bleue. I am not a professional astronomer, but I do possess the mathematical & physical background of a PhD-level engineer. In this sense, the level of technical detail on this blog will (at times) be somewhere between the professional astronomical world and the non-technical amateur who is concerned only with pretty pictures. It is doubtful that anyone other than myself will find this combination interesting or helpful, but, well, c'est la vie...

-Jon


P.S. -- Also, I invite any professional astronomers or astrophysicists who spot an error in my ramblings to please correct me. The internet is too polluted with misinformation already, and the last thing I want is to contribute to the problem.