Abstract
Gravity is the weakest of all the fundamental forces, but for all of its want of strength its reach is very long, and it dictates the large-scale structure of the galaxy. Indeed, it is the action of gravity that controls the fate of matter contained within molecular clouds, and it is the long reach of gravity that makes star formation possible. Before looking at the details of star formation, however, let us first briefly summarize what has been deduced about the interstellar medium in earlier chapters, and to again contradict Walt Whitman let us do it in numerical and tabular form. Table 5.1 presents the characteristic numbers and identifiers of the various phases of the interstellar medium.
The stars, from whence? – Ask Chaos – he can tell.
These bright temptations to idolatry,
From darkness, and confusion, took their birth.
—Edward Young, the poem “Night Thoughts,” written between 1742 and 1745
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Notes
- 1.
The coldest known natural object is that of the Boomerang Nebula (NGC 40). Observations with the Atacama Large Millimeter/submillimeter Array (ALMA) telescope in 2013 reveal that this protoplanetary nebula has regions where the gas temperature is ~1 K (about 1° above absolute zero). These regions are even colder than the 3 K cosmic microwave background (CMB) radiation.
- 2.
These are the various “fiddle” factors that are typically required to make a theory actually work.
- 3.
Payne’s thesis advisors were Harlow Shapley and Arthur Eddington [4].
- 4.
Indeed, the emission lines of helium were observed in the spectrum of the Sun’s corona before chemists were able to isolate it from the mineral cleveite (in which helium builds up through uranium decay) and thereafter study it in the laboratory.
- 5.
The idea that stars, and specifically the Sun, might be liquid bodies was largely based on the enumeration of their density. For the Sun the average density is 1400 kg/m3, which, in fact, is about the same as the density as liquid honey. Since most liquids are incompressible, which essentially means that their density is constant throughout, it seemed quite reasonable to astronomers in the late nineteenth century that stars were composed of some hot, incompressible liquid. Additionally, at that time, the continuum part of stellar spectra could, via Kirchoff’s laws (recall Chap. 2), reasonably be interpreted as that expected from a hot solid or hot liquid body.
- 6.
Formally, the column density is the product: n = 1 m2 × (ρ /m H) × 2R, where R is the radius of the assumed spherical structure, ρ is the density, and m H is the mass of the hydrogen atom. The factor of 2 enters in since it is the diameter of the spherical region that is required in the calculation of n.
- 7.
At the time that Eddington was developing his ideas the structure of atomic nuclei was not well understood, and it was assumed that nuclei were entirely composed of protons. James Chadwick discovered the neutron, which has essentially the same mass as the proton, in 1932, and this provided the present-day picture of atomic nuclei being composed of both neutrons and protons.
- 8.
By way of comparisons the Astrophysical Data System indicates that Stephen Hawking’s A Brief History of Time (published in 1988) has been reference 108 times. Hawking’s first paper on black hole radiation, however, has been referenced 5369 times. The author’s best citation count is 77, which speaks volumes!
- 9.
Tinsely was, in fact, the very first female professor of astronomy at Yale, taking up the post in 1978.
- 10.
Prior to re-ionization the entire structure and evolution of the universe was determined by its dark matter content alone.
- 11.
This is the so-called Methuselah (or Genesis) planet located in the globular cluster Messier 4. This 2.5 Jovian-mass planet (PSR B1620-26b) is actually on a circumbinary orbit about a binary pulsar and white dwarf system, and is estimated to be 12.7 billion years old, the oldest exoplanet currently known.
References
The dynamic collapse time is the free-fall time under gravity. The gravitational force acting upon a small blob of material, of mass m, at the surface of a star will be F grav = G m M / R 2, where M and R are the mass and radius of the gas cloud or star, and where G is the universal gravitational constant. The acceleration of the surface blob, a blob will then, by Newton’s second law of motion, be a blob = F grav / m = G M / R 2. After a time t the blobs displacement S from its initial location will be S = ½ a blob t 2. The collapse time corresponds to the time required for the blob to move through a distance corresponding to the initial radius of the gas cloud or star R. Accordingly, τ dyn = (R 3 / G M)½. This expression is usually simplified by noting that the average density of a star is < ρ > = M / (4/3 π R 3), and accordingly, τ dyn = (3 / 4 π G < ρ >)½. If the imbalance between gravity and the pressure gradient is some fraction ε of the gravitational term, then a blob = ε F grav / m and the collapse time will be t col = τ dyn / ε ½, and accordingly, for the Sun, given t col > 4.5 billion years, so ε < 10−28.
The labels red and white applied to red giants and white dwarfs are set according to Wien’s law: λ max T = constant. Using the notion that the wavelength of maximum intensity λ max is an approximate measure of the dominant color of a star, cool stars will have a value of λ max placed towards the red end of the spectrum – hence such stars will appear red to the human eye. Likewise, hot stars will have a value of λ max placed towards the shorter wavelength, blue end of the visual spectrum, and hence such stars are white to blue in color.
The question of solution uniqueness is still, technically, unsolved with respect to the equations of stellar structure. The so-called Vogt-Russell theorem does correctly state that once the mass and chemical composition at each point inside a star has been specified, then its internal structure is uniquely determined. What this theorem, which has never actually been proven in the general case, should really say is that a solution to the equations probably exists. As to whether there is only one solution is another issue. Part of the problem is that the four differential equations have four boundary conditions two of which apply at the center and two of which apply at the surface. Arbitrarily integrating the equations from say the center of a stellar model outwards will not in general produce the correct surface convergence conditions. In a 1966 review of the theory of stellar structure, Thomas Cowling (University of Leeds) recalled asking Douglas Hartree (Manchester University) in 1938 if he had thought of trying to solve the equations of stellar structure on his then newly invented (mechanical) differential analyzer. Hartree apparently replied that, “we thought of having a go at the equations of stellar structure; but they are horrible equations”. Indeed, the elegance and beauty of equations seen by the pure mathematician are often the horrendous nightmares of the applied mathematician. In the modern era, where rapid numerical computation is possible, it has occasionally been found that multiple solutions to the equations of stellar structure can be obtained. In general, however, it is probably fare to say that the uniqueness problem is one that the pure mathematician, rather than astrophysicist, specifically worries about. A good introduction to this topic is given in the (highly recommended) text by Rudolf Kippenhahn and Alfred Weigert, Stellar Structure and Evolution (Springer, 1990).
Payne’s thesis, Stellar Atmospheres; a Contribution to the Observational Study of High Temperature in the Reversing Layers of Stars, was the first doctoral thesis to be awarded in astronomy by Radcliffe College (now part of Harvard University), Cambridge MA. The life story of Payne-Gaposchkin is remarkable and reveals much about the male dominated world of early 20th century academia. Payne studied botany, physics and chemistry at Cambridge University in England, but graduated without a degree since women could not receive a degree certification from the University at that time. Inspired, however, by a lecture presented by Arthur Eddington on the 1919 eclipse expedition which proved the correctness of Einstein’s general relativity, Payne found a way to begin astronomical research in the new graduate program initiated by Harlow Shapley at Harvard College Observatory. Technically, while Payne did reveal in her thesis that stars must be mostly made of hydrogen, Henry Russell (the doyen of American astronomers at that time) persuaded her that the result must be wrong, and she accordingly did not push the conclusion. Payne was denied a doctoral degree from Harvard because of her gender (this is why the thesis was presented to Radcliffe College) although Otto Struve (Yerkes Observatory) was to later write that Payne had produced, “undoubtedly the most brilliant Ph.D. thesis ever written in astronomy”. Ironically, Russell was eventually converted to the idea that stars must be mostly composed of hydrogen, and he is often given credit for this discovery.
Perhaps the last (heroic) hurrah with respect to solving for the equations of stellar structure with a mechanical desk-top calculator is that by Fred Hoyle and Martin Schwarzschild in 1955 – On the Evolution of Type II stars, Astrophysical Journal Supplement Series, 2, 1–40 (1955). Indeed, the determination of just one stellar model would take several days worth of hand-calculation to complete. Hoyle and Schwarzschild conclude their length paper by noting that future progress in the field of stellar evolution will only be made with the, “fully automatic representation [of the equations], using large electrical machines”. And this is exactly what Haselgrove and Hoyle did in a 1956 publication in the Monthly Notices of the Royal Astronomical Society (116, 515–526). It is amusing from a modern perspective to read the somewhat euphoric comment at the end of this paper that, “the machine [an EDSAC 1] is theoretically capable of obtaining a sufficiently accurate solution to the equations in a few hours”. With the introduction of the electronic computer, Louis Henyey (University of California, Berkeley) introduced in 1959 the matrix inversion technique that is almost universally used to solve the equations of structure in stellar evolution codes to this day.
In Jeans formula we can insert, Egravity = k GM 2 / R, where k is a constant related to the distribution of matter inside of a cloud of gas having mass M and radius R, and G is the universal gravitational constant, and E thermal = 3/2 (R g / μ) T M, where R g is the gas constant, μ is the molecular weight of the gas in atomic mass units (μ = 2.016 for H2). With these two terms the collapse condition becomes |E gravity| > E thermal, and this requires R < R J = (2 k G / 3) (μ / R g) (M / T). If we assume that the gas cloud is spherical and that the gas has a uniform density ρ, then by substituting for the mass term, the collapse condition becomes: R > [(R g / μ G) (T / ρ)] ½. The collapse criterion is often expressed in terms of the Jeans mass, which from the foregoing can be written as M J = [(3/4π)(3R g/2kGμ)3] ½ (T 3 / ρ) ½.Constants aside, Jeans criterion is essentially determined by the ratio of the temperature of the molecular cloud to its density, and this makes sense if we perform a little more algebra. Indeed, given that the speed of sound c S in a gas of temperature T is c S = (R g T / μ)½ this indicates that for collapse to begin R ≈ c S t coll – in other words collapse occurs if the time for a pressure wave to cross the cloud (R / c S) is larger than the gravitational collapse time.
This follows from the scaling argument which gives the accretion rate as M acc ~ M c / t coll, where M c is the mass of the core growing at the center of the collapsing cloud. Given the core has density ρ c and radius r c, so ignoring constant terms M c ~ ρ c r c 3, and since for an isothermal configuration the density varies as ρ ~ r −2, so, to order of magnitude, the mass of the core increases linearly with radius: M c ~ r c. Now, going back to the relation for accretion, we find: M acc ~ r c / t coll and this must be approximately constant since the collapse time is proportional to the radius: t coll ~ r. The argument just presented is perhaps not overly convincing, but detailed numerical simulations indicate that it essentially holds true and that the proto-stellar core, at the center of a collapsing molecular cloud, grows by accreting material at a relatively steady rate. In this manner the mass of the proto-stellar core at the center of the collapsing cloud grows at a steady rate, with the core mass at a time t + Δt being: M c(t + Δt) = M c(t) + Δt M acc. The typical accretion rate onto a newly formed proto-stellar core is generally taken to be of order 2x10−6 M⨀/yr, with the in-fall velocity being about 500 m/s.
The angular momentum per unit mass of material is given by the quantity L = h v t, where h is the distance from the cloud’s spin-axis and v t is the velocity perpendicular to the direction towards the cloud center. According to the angle ϕ at which the in-falling material is located with respect to the spin-axis, so h = r sin ϕ, where r is the radial distance from the cloud center. In this manner material that is initially located close to the spin axis (where h is small and ϕ ≈ 0°) has only a small amount of angular momentum, while material further away from the spin axis (where h is of order the cloud radius R and ϕ ≈ 90°) has a large amount of angular momentum. The critical point about angular momentum is that it is a conserved quantity, and this means that for any given blob of material the angular momentum L must remain constant at all times. What this means for our collapsing cloud matter is that as it approaches the central proto-star, so the value of h becomes smaller and to compensate for this the velocity v r must increase.
S. Vogt et al, 2007. Power Output during the Tour De France. International Journal of Sports Medicine, 28(9), 756. While Lance Armstrong was the winner of the 2005 Tour, this title along with his six other Tour titles were stripped from the record books, due to doping allegations, in 2012.
The discrepancy between the age estimates for the Sun and Earth derived by geologists, evolutionary biologists and physicists came to a head in the late 1800s. The key antagonist was William Thomson (who was the first scientist to be awarded a peerage and is possibly better known today under his title of Lord Kelvin of Largs), who estimated in 1862 that the Sun could be no more than a million years old. To this, in a second publication in 1863, he added that the Earth had an age of order 98 million years. Kelvin’s analysis was based upon classical thermodynamics. Assuming that the Earth had formed as a molten mass he calculated how long it would take for it to cool to its present temperature, repeating, in fact, a calculation and experimental result that had been presented much earlier by Isaac Newton. The logic of Kelvin’s calculation and methodology was impeccable, and in spite of later criticisms he stuck dogmatically to his numbers right to the end of his life (in 1907). The problem for Kelvin was not that his calculations were wrong, far from it in fact, the problem was that his assumptions were wrong. Yes the Earth did form as a molten sphere, but the surface temperature is not governed according to a simple whole Earth cooling-off process. Likewise, Kelvin’s calculations assumed that the Earth had no internal energy source – it is now known that the Earth’s interior is heated by the decay of radioactive elements. As late as 1909, Mark Twain was to write in his Letters from the Earth, that “as Lord Kelvin is the highest authority in science now living [not actually so when the letters were published], I think we must yield to him and accept his views”. This, in a nutshell, captures the problem inherent to many contentious scientific debates – authority should always be questioned and only occasionally listened to with any degree of confidence.
Jeans built his energy generation scheme upon ideas and physical models that are no longer held to be true. Indeed, as early as 1904 Jeans invoked the idea that energy might be liberated as a consequence of the, “annihilation of two either strains of opposite kinds”. Even by 1904 this idea was somewhat outdated since the existence of an all-space-pervading ether had been ruled-out through the pivotal experiments of Albert Michelson and Edward Morley in 1887. While Jeans invoked the idea of converting matter into energy, indeed, at a time prior to Einstein’s 1905 publication of special relativity in which the famous E = mc 2 formula appeared, the idea of antimatter was essentially solidified in its modern form in the pioneering publications of Paul Dirac in 1928. The first antimatter particle, the positron, was discovered in 1932 via cloud chamber experiment conducted by Carl Anderson – in this case the positron (the antimatter particle to the electron) was produced and observed during a cosmic ray shower event. The annihilation of electron-positron pairs, to liberate 2m e c 2 = 1.64x10−13 joules worth of energy, is in fact one of the steps in the proton-proton chain by which stars do generate internal energy via fusion reactions: the positron being produced in the first step of the process when two hydrogen atoms combine to produce a deuterium nucleus.
It is interesting to note that while Lockyer’s meteoritic impact hypothesis was never largely popular, as late as 1927, one finds Harlow Shapley and Cecilia Payne (later and more famously, Payne-Gaposchkin) publishing the results of a study looking for the, Spectroscopic Evidence of the Fall of Meteors [sic] into Stars (Harvard College Observatory, Circular 317 (1927). The idea and methodology behind the study by Shapley and Payne is certainly sound, but remarkably they conclude, “the question of the alternative hypothesis that the diffuse nebulae (a) are wholly gaseous, or (b) are mainly meteoric, seems to be decided in favor of the latter”. Well, of course, hind-sight for us is 20–20, and it should be pointed out that while it is now clear that stars are not powered and/or even adversely affected by meteorite, asteroid or comet impacts, such impacts will most definitely occur. The observations provided by Sun-monitoring spacecraft reveal, for example, that it is occasionally directly-struck by a wayward comet and that sungrazing comets skim through its outer corona every few months. Most cometary encounters with the Sun (and by inference other stars) end in the cometary nucleus being harmlessly destroyed, John Brown (Astronomer Royal for Scotland, and University of Glasgow) and co-workers, however, have recently looked at what happens when a particularly large cometary nucleus crashes into the Sun head-on. Writing in The Astrophysical Journal for 9 July, 2015, Brown along with Robert Carlson (JPL) and Mark Toner, argue that in some cases the nucleus will undergo a powerful airburst explosion and accordingly produce observable flare-like phenomena. Described by Brown as being, “supersonic snowballs from hell”, even these massive cometary impacts will have no discernable affect upon the Sun’s overall energy budget. Such a collision with a planet, however, is an entirely different story, as was revealed by the dramatic atmospheric scaring of Jupiter’s upper cloud deck in the wake of the multiple comet Shoemaker-Levy 9 impacts in July of 1994.
The Coulomb energy between two protons a distance r apart is U = κ e 2/r, where κ is the Coulomb constant and e is the proton charge. The average temperature T of the gas that is required to overcome such a Coulomb barrier is 3kT /2, where k is the Boltzmann constant. Equating these two expressions for a separation corresponding to the classical radius of the nucleus r nuc = 10−15 meters, gives T ~ 9 billion Kelvin. This (classical physics) gas temperature for which the constituent protons will be moving with sufficient kinetic energy that they might approach each other as close as the size of a typical atomic nucleus, is some 900 times larger than the actual temperature required for fusion reactions to begin, which is T ~ 107 Kelvin. It is entirely the effect of quantum tunneling that allows the protons to circumvent the otherwise insurmountable effect of the Coulomb barrier at lower temperature.
The elements deuterium, lithium, beryllium and boron are not actually produced by stars – rather, in fact they are rapidly destroyed in the interiors of stars. The origin of these elements was a mystery to B2FH and they invoked an unspecified x-process to explain their origin. It is now known that these elements along with hydrogen and helium are produced during the Big Bang, primordial nucleosynthesis – indeed, models of the very early universe are fine-tuned so as to replicated the observed abundances of these elements
In the modern era there is the irony that making, or explaining the existence of our universe is not considered to be the main problem issue, but rather it is explaining the existence of a universe that has all the very special features that enable the existence of observers like us. A wonderful introduction to this latter topic is given by Fred Adams (University if Michigan) in his article, Stars in other universes: stellar structure with different fundamental constants (arxiv.org/pdf/0807.3697v1.pdf).
The Eddington Luminosity is determined according to a balance being achieved between the radiation pressure force acting outward and the gravitational force acting inward. If we consider a small mass m of material at the surface of a star of mass M, radius R and Luminosity L, the inward gravitational force on the small mass will be F grav = G M m / R 2. The outward radiation pressure force acting on the mass m, on the other hand, will be F rad = [(L / c) / 4π R 2] κ m, where the term in square brackets is the radiation pressure, and κ is the opacity. The latter term is a measure of the ability of the stellar gas to absorb radiation, with the typical travel distance of a photon between successive absorption events being expressed as l = 1 / κρ, where ρ is the density. In the Sun the distance between absorption events (the so-called mean free path) is typically just a few millimeters. The dimensional units for opacity (which is usually a complex function of composition, temperature and density) are m2/kg. A dynamic balance is achieved when F rad = F grav and this condition yields the Eddington luminosity: L = L Edd = 4 π G M c / κ, which is a result dependent upon the mass of the stars, but is independent of its radius.
The first gravitational waves from a black hole merger event were recorded by the LIGO collaboration on 14 September 2015 (recall figure 2.16). This event has been interpreted as the coalescence of two black holes with initial masses of 36 and 29 solar masses respectively. While such black hole mergers have long been anticipated, the problem now is that there is no clear pathway by which stars can actually produce black holes with the masses (apparently) required. Perhaps the best studied (stellar mass) black hole system is that of Cygnus X-1, and in this case, the black hole has an estimated mass of 15 M⨀. A second event was detected by LIGO on 26 December 2015; this time due to the coalescence of two black holes with masses of 14 and an 8 M⨀. The existence of black holes with masses of many tens of solar masses hints at the possibility that the upper stellar mass limit might range much higher than the presently accepted value of 150 solar masses. There is a certain irony, and indeed surprise, that the first events detected by LIGO are the result of black hole coalescences events, since it was expected that closer, that is within the Milky Way, neutron-star coalescence events would provide stronger and more common signals - as ever the Universe continues to challenge the expectations of human calculation.
Energy generation within Population III stars runs in a different manner to that of Population I or II stars. The latter stars contain some heavy elements, especially 12C, and can accordingly generate energy via the CNO cycle (recall section 5.7). Population III stars have only hydrogen and helium within their interiors and so must initially generate energy via the proton-proton chain (PPC). Since the PPC runs at a much lower temperature than the CNO cycle it acts as a poor internal thermostat and the central core becomes inordinately dense and hot. This latter situation enables the early onset of the triple-α reaction and the star begins to form 12C from helium nuclei. The newly formed carbon then enables the onset of energy generation through the CNO cycle. With the various energy generation cycles switching on-and-off, a Population III star is highly prone to the development of pulsation cycles, and this has consequences for the final end phase. Between 10 and 100 M⨀, the end phase will be that of a core-collapse supernova. For masses between 100 and ~250 M⨀, the end phase will be that of a pair-instability supernova. In this latter case the temperature inside the star is so high after helium burning ends that electron-positron pairs begin to form, and this robs energy from the star and its pressure gradient can no longer support the weight of overlying layers – the star begins to collapse. This collapse increases the temperature and density of the core to such an extent that explosive oxygen burning occurs and this triggers the disruption of the entire star. For pair-instability supernovae no central core remnant survives. Black holes as an end phase of stellar evolution will only appear again once the initial mass exceeds ~ 250 M⨀ and this time they will form by direct gravitational collapse.
See the research paper: N. Mashian and A. Loeb: CEMP stars: possible hosts to carbon planets in the early universe (arxiv.org/pdf/1603.06943v2.pdf). Details of the observational criteria for distinguishing between the various CEMP star types are reviewed by Jimni Yoon et al., in the article, Observational constraints on first-star nucleosynthesis. 1. Evidence for multiple progenitors of CEMP-no stars (arxiv.org/pdf/1607.06336v1.pdf).
This calculation is based upon the arguments and analysis presented by Fred Adams and Gregory Laughlin in their article, A Dying Universe: the long term fate and evolution of astrophysical objects (arxiv.org/pdf/9701131.v1.pdf).
The author has previously discussed the details of this star and its companions in: Alpha Centauri: unveiling the secrets of our nearest stellar neighbor (Springer, New York, 2015).
State of the art planetary models, with a range of compositions, have been described in the research paper: Mass-radius relationships for solid planets, by Seager, Kuchner, Hier-Majumder and Militzer (The Astrophysical Journal, 669, 1279–1297, 2007).
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Beech, M. (2017). In the Grip of Gravity. In: The Pillars of Creation. Springer Praxis Books(). Springer, Cham. https://doi.org/10.1007/978-3-319-48775-5_5
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