## Wednesday, 25 June 2008

### Space Math IV Educator Guide

Audience: Educators

These activities comprise a series of 31 practical mathematics applications in space science. This collection of activities is based on a weekly series of problems distributed to teachers during the 2007-2008 school year. The problems in this booklet investigate space phenomena and math applications such as black holes, sunspots, the moon's atmosphere, scientific notation, evaluating functions, Benford's law and geometry. The problems are authentic glimpses of modern science and engineering issues, often involving actual research data. Each word problem includes background information. The one-page assignments are accompanied by one-page teachers answer keys.

Space Math IV  [11MB PDF file]

More booklets in this series:
Space Math I
Space Math II
Space Math III
Space Math V
Space Math VI
Space Math VII
Astrobiology Math
Black Hole Math
Earth Math
Electromagnetic Math
Image Scale Math
Lunar Math
Magnetic Math
Remote Sensing Math
Solar Math
Space Weather Math
Transit Math

Source: NASA

## Friday, 20 June 2008

### Proposal Pembentukan Club Astro Fisika Kota Banjar

Proposal Pembentukan Club Astro Fisika
Kota Banjar

BAB I

PENDAHULUAN

A. Latar Belakang

1. Dalam rangka mewujudkan tujuan pendidikan secara maksimal maka diperlukan proses pembelajaran yang kondusif dengan melibatkan semua komponen pembelajaran secara optimal. Salah satu komponen penting yang menjadikan proses pembelajaran menjadi lancar dan kondusif adalah multi media pendidikan .

2. Klub atau Himpunan belajar sebagai tempat rombongan pelajar melakukan aktivitas pembelajaran memiliki peranan yang strategis dalam rangka menciptakan suasana dan rasa belajar yang inovatif serta kreatif bagi para siswa. Keberadaannya membawa dampak yang lebih luas seperti, kebersamaan, rasa aman, rasa memiliki, ketenangan dan hal-hal positif lainnya dalam proses KBM bidang Sains Astro Fisika.

3. Kota Banjar dalam hal ini SMAN 1 Banjar (Manbaul Ullum) sebagai salah satu sekolah menengah atas di kota Banjar juga merasakan betapa pentingnya keberadaan Club Astro Fisika sebagai salah satu unsur penentu keberhasilan diraihnya Prestasi dalam Kompetisi Sains Astro Fisika di tingkat Provinsi, Nasional dan Internasional, Sekolah yang memiliki siswa.....orang yang terbagi dalam....kelas (rombongan belajar) ini saat ini memiliki....ruang kelas, sehingga idealnya membutuhkan 1 Club Sains Astro Fisika yang representative.

4. Bertolak dari pemikiran di atas maka SMAN 1 Banjar menganggap bahwa pembentukan Club Astro Fisika di SMA 1 Banjar Khusunya di kota Banjar adalah hal yang sangat penting dan mendesak untuk diwujudkan. Untuk itulah kami mengajukan PROPOSAL IMBAL SWADAYA PENINGKATAN AKSES SISWA KOTA BANJAR TERHADAP BIDANG SAINS ASTRO FISIKA DI KOTA BANJAR.

B. Visi dan Misi

Visi
Motto:
“Bringing Astrophysics to Our Society, Let’s Observe and Research Our Universe”
“Kota Banjar Bisa!”

Menjadi Pusat Pelatihan dan pengenalan (Centre of Introduction and Training) Astro Fisika yang unggul dalam mewujudkan Siswa-Siswi Kota Banjar yang Profesional, Adaptabel, Responsif dan Bijaksana terhadap bidang sains Astro Fisika.

M i s i
Untuk mencapai visi tersebut di atas, maka misi yang telah ditetapkan dan hendak dituju oleh Klub Astro Fisika Kota Banjar adalah :

• Mengembangkan Keunggulan melalui peningkatan Pengetahuan, Kecakapan, Ketelatenan dan Kebijaksanaan dengan mengedepankan Kebenaran, Kemandirian, Kreatifitas serta menumbuhkan rasa Kejujuran dan Kepedulian terhadap bidang Sains Astro Fisika, sehingga Siswa-Siswi Kota Banjar dapat menggapai prestasi puncak dalam kompetisi-kompetisi Astro Fisika di tingkat Provinsi, Nasional dan International. 3G Kota Banjar –“Go Get Gold”-
• Meningkatkan pemahaman warga (khususnya kaum Akademisi) Kota Banjar terhadap bidang Sains Astro Fisika
• Meningkatkan kompetensi siswa dan guru sesuai dengan bidang Sains Astro Fisika
• Meningkatkan fasilitas Multi Media dalam bidang Sains Astro Fisika

C. Tujuan dan Sasaran

PembentukanClub Astro Fisika di SMAN 1 Banjar di Kota Banjar bertujuan:
a. Memberikan tempat belajar siswa dengan segala perlengkapannya dalam bidang Sains Astro Fisika.
b. Mengantarkan Siswa-Siswi Kota Banjar agar menggapai prestasi puncak dalam kompetisi-kompetisi Astro Fisika di tingkat Provinsi, Nasional dan International. 3G Kota Cimahi –“Go Get Gold”-.
c. Agar pelaksanaan teori dan praktik (learning by doing) sesuai dengan tingkat kemampuan siswa tanpa adanya alasan keterbatasan fasilitas di Kota Banjar.

a. Pembentukan Organisasi Club Astro Fisika Kota Banjar yang representative.
b. Mengusahakan dan memproduksi Multi Media Learning yang lengkap dalam bidang Sains Astro Fisika.

BAB II
PROGRAM PEMBENTUKAN PRASARANA KLUB
ASTRO FISIKA KOTA BANJAR

A. Tahap perencanaan

a. Melakukan pendataan dan analisis (SWOT Analysis) kebutuhan terhadap Objek
b. Membuat gambaran sesuai kebutuhan rencana pembentukan Club, terdiri dari:
• Struktur Organisasi dan AD & ART
• Daftar fasilitas yang dibutuhkan
• Anggaran yang dibutuhkan
• Gambaran detail meliputi antara lain : Program-program, teknis pelaksanaan, laporan Akhir dll.
d. Membuat rencana waktu pelaksanaan pembentukan dan pelaksanaan;
e. Menyusun rencana kebutuhan tenaga pembimbing.

B. Tahap pelaksanaan

a. Mengarahkan dan membimbing secara periodik kepada pelaksana selama pekerjaan pembentukan berlangsung.
b. Memeriksa dan membuat laporan kemajuan pekerjaan terhadap hasil pelaksanaan pekerjaan pembentukan yang dilakukan oleh pelaksana.
c. Memantau dan membuat laporan mingguan, bulanan dan tahunan terhadap pelaksanaan pembentukan dan berjalannya program kepada pihak yang mempunyai kewenangan.
d. Membuat Dokumnetasi perkembangan fisik pekerjaan pembentukan Club yang menunjukan kondisi awal (0%), menengah (50%) dan akhir (100%).

C. Prospek di Masa yang Akan Datang

1. Menjadi Pusat Pelatihan dan Seminar dalam bidang sains Astro Fisika di Provinsi Jawa Barat, Banten dan Sebagian Jawa Tengah untuk Sektor Pendidikan dasar dan menengah. Sehingga Kota Banjar melahirkan Siswa-Siswi Berprestasi dalam kompetisi Sains Astro Fisika di level Provinsi, Nasional dan International
2. Memproduksi Multi Media Pendidikan Sains Astro Fisika yang beromset Jutaan Rupiah perbulan, (Edupreneurship) seperti:
*Produksi Software yang berkaitan (Jaringan Web di JABAR BANTEN dan Sebagian Jawa Tengah)
*Alat-alat sederhana dan pemesanan peralatan Astro Fisika.
*Asistensi Pembelajaran Astro Fisika yang mapan.

D. Objek Pembentukan Klub Astro Fisika Kota Banjar

Objek dari pembentukan Klub Astro Fisika Kota Banjar ini adalah Masyarakat Akademisi Kota Banjar khususnya Siswa dan Guru dalam tingkat pendidikan dasar dan menengah (SD, SMP, SMA), dan Masyarakat Kota Banjar pada umumnya.

BAB III
P E N D A N A A N

A. Rencana Pembiayaan Pembentukan Club Astro Fisika secara Keseluruhan

Dana yang dibutuhkan untuk pembentukan Club Astro Fisika secara keseluruhan adalah sebesar Rp 28.000.000,00 (Dua puluh Delapan juta rupiah) dengan rincian anggaran biaya terlampir.

B. Pembiayaan yang Dibebankan kepada Pemerintah

Rp.18.000.000,00 (Delapan belas juta rupiah).

C. Pembiayaan yang Dibebankan kepada pihak lain.
a. Setiap Sekolah yang Berminat menjadi Anggota Jaringan Club ini.
* @ Sekolah Rp.500.000,00 jika dalam tahap pertama, ada sekitar 10 sekolah yang berminat mengikuti program ini maka:
Dana yang kita dapatkan adalah, sekitar Rp.5.000.000,00.
b. Pihak Sponsor. Kami mengestimasikan dana yang diperoleh dari sektor ini sebesar Rp.4.000.000,00
c. Iuran kami sendiri sebagai anggota dan pengurus serta sebagian keuntungan dari proses Produksi multi media dalam bidang Sains Astro Fisika. Rp.1.000.000,00 di peroleh dari Advertising di Web Blog kami (Google Adsense), Pembuatan Modul-modul, Software, dan lainnya yang berkaitan dengan Sains Astro Fisika.

BAB V

P E N U T U P

Akhirnya kami berharap bahwa proposal ini mendapat persetujuan sehingga Pembentukan Club Astro Fisika di Kota Banjar yang berkedudukan di SMAN 1 Banjar dapat direalisasikan guna menciptakan suasana proses pembelajaran yang kondusif dan kreatif dalam rangka pencapaian tujuan pendidikan secara maksimal, Amien.

Banjar, 18 Agustus 2009
Menyetujui,

DAFTAR LAMPIRAN

1. Analisis Tingkat Kebutuhan Siswa Kota Cimahi terhadap Sains Astro Fisika
2. Foto – foto kegiatan kami
3. Rencana Anggaran Biaya (RAB)
4. Surat Pernyataan Dana Pendamping
5. Surat Pernyataan Pelaksanaan dengan Swakelola
6. SK Tim Pelatih
7. Data Siswa
8. Foto Copy Rekening Club Astro Fisika

Oleh:

Arip Nurahman

Pendidikan Fisika, FPMIPA, Universitas Pendidikan Indonesia
&
Follower Open Course Ware at MIT-Harvard University

## Wednesday, 18 June 2008

### Indonesian Space Technology Innovation Research & Development

Indonesia Space Technology Innovation Research & Development
Arip Nurahman
Department of Physics
Faculty of Sciences and Mathematics, Indonesia University of Education

and

Follower Open Course Ware at Massachusetts Institute of Technology
Cambridge, USA
Department of Physics
http://web.mit.edu/physics/
http://ocw.mit.edu/OcwWeb/Physics/index.htm
&
Aeronautics and Astronautics Engineering
http://web.mit.edu/aeroastro/www/
http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/index.htm

Space technology is technology that is related to entering space, maintaining and using systems during spaceflight and returning people and things from space.

"Every day" technologies such as weather forecasting, remote sensing, GPS systems, satellite television, and some long distance communications systems critically rely on space infrastructure. Of sciences astronomy and Earth sciences (via remote sensing) most notably benefit from space technology.

Computers and telemetry were once leading edge technologies that might have been considered "space technology" because of their criticality to boosters and spacecraft. They existed prior to the Space Race of the Cold War but their development was vastly accelerated to meet the needs of the two major superpowers' space programs.

While still used today in spacecraft and missiles, the more prosaic applications such as remote monitoring (via telemetry) of patients, water plants, highway conditions, etc. and the widespread use of computers far surpasses their space applications in quantity and variety of application.

Space is such an alien environment that attempting to work in it requires new techniques and knowledge. New technologies originating with or accelerated by space-related endeavors are often subsequently exploited in other economic activities.

This has been widely pointed to as beneficial by space advocates and enthusiasts favoring the investment of public funds in space activities and programs. Political opponents counter that it would be far cheaper to develop specific technologies directly if they are beneficial and scoff at this justification for public expenditures on space-related research.

## Contents

Specific space technologies
Future space technologies

## Sunday, 15 June 2008

### Oscilloscope

A physics lab demo of the oscilloscope.

## Wednesday, 11 June 2008

### Space Exploration Nuclear Engine

Design of a Sodium-cooled Epithermal Long-term Exploration Nuclear Engine

Author:
P. Yarsky, A.C. Kadak, and M.J. Drisco

Arip Nurahman
Department of Physics
Faculty of Sciences and Mathematics, Indonesia University of Education

and

Follower Open Course Ware at Massachusetts Institute of Technology
Cambridge, USA
Department of Physics
http://web.mit.edu/physics/
http://ocw.mit.edu/OcwWeb/Physics/index.htm
&
Aeronautics and Astronautics Engineering
http://web.mit.edu/aeroastro/www/
http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/index.htm

Abstract

To facilitate the mission to Mars initiative, the current work has focused on conceptual designs for transformational and enabling space nuclear reactor technologies. A matrix of design alternatives for both the reactor core and the power conversion unit were considered. Based on a preliminary screening of technologies using simplified analyses, a conceptual design was established for more detailed design work.

The boiling sodium Rankine cycle was selected for the power conversion unit, and the reactor core is an ultra high power density core with highly enriched uranium fuel. The sodium Rankine cycle has many advantages, lending to a highly efficient radiator and compact reactor core. The sodium-cooled, epithermal long-term exploration nuclear engine (SELENE) is designed to be scalable to meet many differing mission requirements.

The SELENE core is a comprised of Nb-1Zr clad, lead bonded, UC plates in a honeycomb pattern. The fuel plates are arranged into a rectangular grid, roughly 25cm on each end. The fuel is in two zones, one is 97 a/o enriched in 235U and the other is 70 a/o enriched in 235U. The core is a fast spectrum reactor, BeO reflected, and ex-core controlled. Three designs are proposed, the first is for a 10 MWth /1.0 MWe Low Temperature (1300 K) system (SELENE-10-LT) and the second for a 10 MWth /1.2 MWe High Temperature (1500 K) system (SELENE-10-HT) and the third for a 27 MWth 12.6 MWe system (SELENE-30).

All of these designs utilize essentially the same system architecture. Three designs are proposed so that low power variants can be used to verify the technology and develop experience. The reactor systems may then by uprated to a higher power level. The system lifetime is 9 effective full power months, corresponding roughly to a single trip from Earth to Mars and back

## Sunday, 8 June 2008

### How The Sun Shines

How the Sun Shines
by John N. Bahcall
29 June2000

Arip Nurahman
Department of Physics, Faculty of sciences and Mathematics
Indonesia University of Education

What makes the sun shine? How does the sun produce the vast amount of energy necessary to support life on earth? These questions challenged scientists for a hundred and fifty years, beginning in the middle of the nineteenth century. Theoretical physicists battled geologists and evolutionary biologists in a heated controversy over who had the correct answer.

Why was there so much fuss about this scientific puzzle? The nineteenth-century astronomer John Herschel described eloquently the fundamental role of sunshine in all of human life in his 1833 Treatise on Astronomy:

The sun's rays are the ultimate source of almost every motion which takes place on the surface of the earth. By its heat are produced all winds,...By their vivifying action vegetables are elaborated from inorganic matter, and become, in their turn, the support of animals and of man, and the sources of those great deposits of dynamical efficiency which are laid up for human use in our coal strata.

Sunshine makes life possible on earth.

In this essay, we shall review from an historical perspective the development of our understanding of how the sun (the nearest star) shines, beginning in the following section with the nineteenth-century controversy over the age of the sun. In later sections, we shall see how seemingly unrelated discoveries in fundamental physics led to a theory of nuclear energy generation in stars that resolved the controversy over the age of the sun and explained the origin of solar radiation. In the section just before the summary, we shall discuss how experiments that were designed to test the theory of nuclear energy generation in stars revealed a new mystery, the Mystery of the Missing Neutrinos.

The Age of the Sun
How old is the sun? How does the sun shine? These questions are two sides of the same coin, as we shall see.

The rate at which the sun is radiating energy is easily computed by using the measured rate at which energy reaches the earth's surface and the distance between the earth and the sun. The total energy that the sun has radiated away over its lifetime is approximately the product of the current rate at which energy is being emitted, which is called the solar luminosity, times the age of the sun.

The older the sun is, the greater the total amount of radiated solar energy. The greater the radiated energy, or the larger the age of the sun, the more difficult it is to find an explanation of the source of solar energy.

To better appreciate how difficult it is to find an explanation, let us consider a specific illustration of the enormous rate at which the sun radiates energy. Suppose we put a cubic centimeter of ice outside on a summer day in such a way that all of the sunshine is absorbed by the ice. Even at the great distance between the earth and the sun, sunshine will melt the ice cube in about 40 minutes. Since this would happen anywhere in space at the earth's distance from the sun, a huge spherical shell of ice centered on the sun and 300 million km (200 million miles) in diameter would be melted at the same time. Or, shrinking the same amount of ice down to the surface of the sun, we can calculate that an area ten thousand times the area of the earth's surface and about half a kilometer (0.3 mile) thick would also be melted in 40 minutes by the energy pouring out of the sun.

In this section, we shall discuss how nineteenth-century scientists tried to determine the source of solar energy, using the solar age as a clue.

Conflicting Estimates of the Solar Age
The energy source for solar radiation was believed by nineteenth-century physicists to be gravitation. In an influential lecture in 1854, Hermann von Helmholtz, a German professor of physiology who became a distinguished researcher and physics professor, proposed that the origin of the sun's enormous radiated energy is the gravitational contraction of a large mass. Somewhat earlier, in the 1840s, J.R. Mayer (another German physician) and J.J. Waterson had also suggested that the origin of solar radiation is the conversion of gravitational energy into heat.1

Biologists and geologists considered the effects of solar radiation, while physicists concentrated on the origin of the radiated energy. In 1859, Charles Darwin, in the first edition of On The Origin of the Species by Natural Selection, made a crude calculation of the age of the earth by estimating how long it would take erosion occurring at the current observed rate to wash away the Weald, a great valley that stretches between the North and South Downs across the south of England. He obtained a number for the "denudation of the Weald'' in the range of 300 million years, apparently long enough for natural selection to have produced the astounding range of species that exist on earth.

As Herschel stressed, the sun's heat is responsible for life and for most geological evolution on earth. Hence, Darwin's estimate of a minimum age for geological activity on the earth implied a minimum estimate for the amount of energy that the sun has radiated.

Firmly opposed to Darwinian natural selection, William Thompson, later Lord Kelvin, was a professor at the University of Glasgow and one of the great physicists of the nineteenth century. In addition to his many contributions to applied science and to engineering, Thompson formulated the second law of thermodynamics and set up the absolute temperature scale, which was subsequently named the Kelvin scale in his honor. The second law of thermodynamics states that heat naturally flows from a hotter to a colder body, not the opposite. Thompson therefore realized that the sun and the earth must get colder unless there is an external energy source and that eventually the earth will become too cold to support life.

Kelvin, like Helmholtz, was convinced that the sun's luminosity was produced by the conversion of gravitational energy into heat. In an early (1854) version of this idea, Kelvin suggested that the sun's heat might be produced continually by the impact of meteors falling onto its surface. Kelvin was forced by astronomical evidence to modify his hypothesis and he then argued that the primary source of the energy available to the sun was the gravitational energy of the primordial meteors from which it was formed.

Thus, with great authority and eloquence Lord Kelvin declared in 1862:

That some form of the meteoric theory is certainly the true and complete explanation of solar heat can scarcely be doubted, when the following reasons are considered: (1) No other natural explanation, except by chemical action, can be conceived. (2) The chemical theory is quite insufficient, because the most energetic chemical action we know, taking place between substances amounting to the whole sun's mass, would only generate about 3,000 years' heat. (3) There is no difficulty in accounting for 20,000,000 years' heat by the meteoric theory.

Kelvin continued by attacking Darwin's estimate directly, asking rhetorically:

What then are we to think of such geological estimates as [Darwin's] 300,000,000 years for the "denudation of the Weald''?

Believing Darwin was wrong in his estimate of the age of the earth, Kelvin also believed that Darwin was wrong about the time available for natural selection to operate.

Lord Kelvin estimated the lifetime of the sun, and by implication the earth, as follows. He calculated the gravitational energy of an object with a mass equal to the sun's mass and a radius equal to the sun's radius and divided the result by the rate at which the sun radiates away energy. This calculation yielded a lifetime of only 30 million years. The corresponding estimate for the lifetime sustainable by chemical energy was much smaller because chemical processes release very little energy.

Who Was Right?
As we have just seen, in the nineteenth century you could get very different estimates for the age of the sun, depending upon whom you asked. Prominent theoretical physicists argued, based upon the sources of energy that were known at that time, that the sun was at most a few tens of million years old. Many geologists and biologists concluded that the sun must have been shining for at least several hundreds of millions of years in order to account for geological changes and the evolution of living things, both of which depend critically upon energy from the sun. Thus the age of the sun, and the origin of solar energy, were important questions not only for physics and astronomy, but also for geology and biology.

Darwin was so shaken by the power of Kelvin's analysis and by the authority of his theoretical expertise that in the last editions of On The Origin of the Species he eliminated all mention of specific time scales. He wrote in 1869 to Alfred Russel Wallace, the codiscoverer of natural selection, complaining about Lord Kelvin:

Thompson's views on the recent age of the world have been for some time one of my sorest troubles.

Today we know that Lord Kelvin was wrong and the geologists and evolutionary biologists were right. Radioactive dating of meteorites shows that the sun is 4.6 billion years old.

What was wrong with Kelvin's analysis? An analogy may help. Suppose a friend observed you using your computer and tried to figure out how long the computer had been operating. A plausible estimate might be no more than a few hours, since that is the maximum length of time over which a battery could supply the required amount of power. The flaw in this analysis is the assumption that your computer is necessarily powered by a battery. The estimate of a few hours could be wrong if you computer were operated from an electrical power outlet in the wall. The assumption that a battery supplies the power for your computer is analogous to Lord Kelvin's assumption that gravitational energy powers the sun.

Since nineteenth century theoretical physicists did not know about the possibility of transforming nuclear mass into energy, they calculated a maximum age for the sun that was too short. Nevertheless, Kelvin and his colleagues made a lasting contribution to the sciences of astronomy, geology, and biology by insisting on the principle that valid inferences in all fields of research must be consistent with the fundamental laws of physics.

We will now discuss some of the landmark developments in the understanding of how nuclear mass is used as the fuel for stars.

A Glimpse of the Solution
The turning point in the battle between theoretical physicists and empirical geologists and biologists occurred in 1896. In the course of an experiment designed to study x-rays discovered the previous year by Wilhelm Röntgen, Henri Becquerel stored some uranium-covered plates in a desk drawer next to photographic plates wrapped in dark paper. Because it was cloudy in Paris for a couple of days, Becquerel was not able to "energize'' his photographic plates by exposing them to sunlight as he had intended. On developing the photographic plates, he found to his surprise strong images of his uranium crystals. He had discovered natural radioactivity, due to nuclear transformations of uranium.

The significance of Becquerel's discovery became apparent in 1903, when Pierre Curie and his young assistant, Albert Laborde, announced that radium salts constantly release heat. The most extraordinary aspect of this new discovery was that radium radiated heat without cooling down to the temperature of its surroundings. The radiation from radium revealed a previously unknown source of energy. William Wilson and George Darwin almost immediately proposed that radioactivity might be the source of the sun's radiated energy.

The young prince of experimental physics, Ernest Rutherford, then a professor of physics at McGill University in Montreal, discovered the enormous energy released by alpha particle radiation from radioactive substances. In 1904, he announced:

The discovery of the radio-active elements, which in their disintegration liberate enormous amounts of energy, thus increases the possible limit of the duration of life on this planet, and allows the time claimed by the geologist and biologist for the process of evolution.

The discovery of radioactivity opened up the possibility that nuclear energy might be the origin of solar radiation. This development freed theorists from relying in their calculations on gravitational energy. However, subsequent astronomical observations showed that the sun does not contain a lot of radioactive materials, but instead is mostly hydrogen in gaseous form. Moreover, the rate at which radioactivity delivers energy does not depend on the stellar temperature, while observations of stars suggested that the amount of energy radiated by a star does depend sensitively upon the star's interior temperature. Something other than radioactivity is required to release nuclear energy within a star.

In the next sections, we shall trace the steps that led to what we now believe is the correct understanding of how stars shine.

The Direction Established
The next fundamental advance came once again from an unexpected direction. In 1905, Albert Einstein derived his famous relation between mass and energy, E = mc2, as a consequence of the special theory of relativity. Einstein's equation showed that a tiny amount of mass could, in principle, be converted into a tremendous amount of energy. His relation generalized and extended the nineteenth century law of conservation of energy of von Helmholtz and Mayer to include the conversion of mass into energy.

What was the connection between Einstein's equation and the energy source of the sun? The answer was not obvious. Astronomers did their part by defining the constraints that observations of stars imposed on possible explanations of stellar energy generation. In 1919, Henry Norris Russell, the leading theoretical astronomer in the United States, summarized in a concise form the astronomical hints on the nature of the stellar energy source. Russell stressed that the most important clue was the high temperature in the interiors of stars.

Aston showed in 1920 that four hydrogen nuclei are heavier than a helium nucleus.

F.W. Aston discovered in 1920 the key experimental element in the puzzle. He made precise measurements of the masses of many different atoms, among them hydrogen and helium. Aston found that four hydrogen nuclei were heavier than a helium nucleus. This was not the principal goal of the experiments he performed, which were motivated in large part by looking for isotopes of neon.

The importance of Aston's measurements was immediately recognized by Sir Arthur Eddington, the brilliant English astrophysicist. Eddington argued in his 1920 presidential address to the British Association for the Advancement of Science that Aston's measurement of the mass difference between hydrogen and helium meant that the sun could shine by converting hydrogen atoms to helium. This burning of hydrogen into helium would (according to Einstein's relation between mass and energy) release about 0.7% of the mass equivalent of the energy. In principle, this could allow the sun to shine for about a 100 billion years.

In a frighteningly prescient insight, Eddington went on to remark about the connection between stellar energy generation and the future of humanity:

If, indeed, the sub-atomic energy in the stars is being freely used to maintain their great furnaces, it seems to bring a little nearer to fulfillment our dream of controlling this latent power for the well-being of the human race---or for its suicide.

Understanding the Process
The next major step in understanding how stars produce energy from nuclear burning, resulted from applying quantum mechanics to the explanation of nuclear radioactivity. This application was made without any reference to what happens in stars. According to classical physics, two particles with the same sign of electrical charge will repel each other, as if they were repulsed by a mutual recognition of 'bad breath'. Classically, the probability that two positively charged particles get very close together is zero. But, some things that cannot happen in classical physics can occur in the real world which is described on a microscopic scale by quantum mechanics.

In 1928, George Gamow, the great Russian-American theoretical physicist, derived a quantum-mechanical formula that gave a non-zero probability of two charged particles overcoming their mutual electrostatic repulsion and coming very close together. This quantum mechanical probability is now universally known as the "Gamow factor.'' It is widely used to explain the measured rates of certain radioactive decays.

In the decade that followed Gamow's epochal work, Atkinson and Houtermans and later Gamow and Teller used the Gamow factor to derive the rate at which nuclear reactions would proceed at the high temperatures believed to exist in the interiors of stars. The Gamow factor was needed in order to estimate how often two nuclei with the same sign of electrical charge would get close enough together to fuse and thereby generate energy according to Einstein's relation between excess mass and energy release.

In 1938, C.F. von Weizsäcker came close to solving the problem of how some stars shine. He discovered a nuclear cycle, now known as the carbon-nitrogen-oxygen (CNO) cycle, in which hydrogen nuclei could be burned using carbon as a catalyst. However, von Weizsäcker did not investigate the rate at which energy would be produced in a star by the CNO cycle nor did he study the crucial dependence upon stellar temperature.

The CNO Cycle

For stars heavier than the sun, theoretical models show that the CNO (carbon-nitrogen-oxygen) cycle of nuclear fusion is the dominant source of energy generation. The cycle results in the fusion of four hydrogen nuclei (1H, protons) into a single helium nucleus (4He, alpha particle), which supplies energy to the star in accordance with Einstein's equation. Ordinary carbon,12C, serves as a catalyst in this set of reactions and is regenerated. Only relatively low energy neutrinos ()are produced in this cycle.

The figure is adapted from J.N. Bahcall, Neutrinos from the Sun, Scientific American, Volume 221, Number 1, July 1969, pp. 28-37.

By April 1938, it was almost as if the scientific stage had been intentionally set for the entry of Hans Bethe, the acknowledged master of nuclear physics. Professor Bethe had just completed a classic set of three papers in which he reviewed and analyzed all that was then known about nuclear physics. These works were known among his colleagues as "Bethe's bible.'' Gamow assembled a small conference of physicists and astrophysicists in Washington, D.C. to discuss the state of knowledge, and the unsolved problems, concerning the internal constitution of the stars.

In the course of the next six months or so, Bethe worked out the basic nuclear processes by which hydrogen is burned (fused) into helium in stellar interiors. Hydrogen is the most abundant constituent of the sun and similar stars, and indeed the most abundant element in the universe.

Bethe described the results of his calculations in a paper entitled "Energy Production in Stars,'' which is awesome to read. He authoritatively analyzed the different possibilities for reactions that burn nuclei and selected as most important the two processes that we now believe are responsible for sunshine. One process, the so-called p-p chain, builds helium out of hydrogen and is the dominant energy source in stars like the sun and less massive stars.

The p—p Chain Reaction

In theoretical models of the sun, the p—p chain of nuclear reactions illustrated here is the dominant source of energy production. Each reaction is labeled by a number in the upper left hand corner of the box in which it is contained. In reaction 1, two hydrogen nuclei (1H, protons) are fused to produce a heavy hydrogen nucleus (2H, a deuteron). This is the usual way nuclear burning gets started in the sun. On rare occasions, the process is started by reaction 2. Deuterons produced in reactions 1 and 2 fuse with protons to produce a light element of helium (3He). At this point, the p—p chain breaks into three branches, whose relative frequencies are indicated in the figure. The net result of this chain is the fusion of four protons into a single ordinary helium nucleus (4He) with energy being released to the star in accordance with Einstein's equation. Particles called 'neutrinos' () are emitted in these fusion processes. Their energies are shown in the figure in units of millions of electron volts (MeV). Reactions 2 and 4 were not discussed by Hans Bethe.

The figure is adapted from J.N. Bahcall, Neutrinos from the Sun, Scientific American, Volume 221, Number 1, July 1969, pp. 28-37.

The CNO cycle, the second process which was also considered by von Weizsäcker, is most important in stars that are more massive than the sun. Bethe used his results to estimate the central temperature of the sun and obtained a value that is within 20% of what we currently believe is the correct value (16 million degrees Kelvin).2 Moreover, he showed that his calculations led to a relation between stellar mass and stellar luminosity that was in satisfactory agreement with the available astronomical observations.

In the first two decades after the end of the second world war, many important details were added to Bethe's theory of nuclear burning in stars. Distinguished physicists and astrophysicists, especially A.G.W. Cameron, W.A. Fowler, F. Hoyle, E.E. Salpeter, M. Schwarzschild, and their experimental colleagues, returned eagerly to the question of how stars like the sun generate energy. From Bethe's work, the answer was known in principle: the sun produces the energy it radiates by burning hydrogen. According to this theory, the solar interior is a sort of controlled thermonuclear bomb on a giant scale.3 The theory leads to the successful calculation of the observed luminosities of stars similar to the sun and provides the basis for our current understanding of how stars shine and evolve over time. The idea that nuclear fusion powers stars is one of the cornerstones of modern astronomy and is used routinely by scientists in interpreting observations of stars and galaxies.

W.A. Fowler, Willy as he was universally known, led a team of colleagues in his Caltech Kellogg Laboratory and inspired physicists throughout the world to measure or calculate the most important details of the p-p chain and the CNO cycle. There was plenty of work to do and the experiments and the calculations were difficult. But, the work got done because understanding the specifics of solar energy generation was so interesting. Most of the efforts of Fowler and his colleagues M. Burbidge, G.R. Burbidge, F. Hoyle, and A.G.W. Cameron) soon shifted to the problem of how the heavy elements, which are needed for life, are produced in stars.

Testing the Hypothesis of Nuclear Burning
Science progresses as a result of the clash between theory and experiment, between speculation and measurement. Eddington, in the same lecture in which he first discussed the burning of hydrogen nuclei in stars, remarked:

I suppose that the applied mathematician whose theory has just passed one still more stringent test by observation ought not to feel satisfaction, but rather disappointment --- "Foiled again! This time I had hoped to find a discordance which would throw light on the points where my model could be improved.''

Is there any way to test the theory that the sun shines because very deep in its interior hydrogen is burned into helium? At first thought, it would seem impossible to make a direct test of the nuclear burning hypothesis. Light takes about ten million years to leak out from the center of the sun to the surface and when it finally emerges in the outermost regions, light mainly tells us about the conditions in those outer regions. Nevertheless, there is a way of "seeing'' into the solar interior with neutrinos, exotic particles discovered while trying to understand a different mystery.4

Discovery, Confirmation, and Surprise
A neutrino is a sub-atomic particle that interacts weakly with matter and travels at a speed that is essentially the speed of light. Neutrinos are produced in stars when hydrogen nuclei are burned to helium nuclei; neutrinos are also produced on earth in particle accelerators, in nuclear reactors, and in natural radioactivity. Based upon the work of Hans Bethe and his colleagues, we believe that the process by which stars like the sun generate energy can be symbolized by the relation,

(1)

in which four hydrogen nuclei (1H, protons) are burned into a single helium nucleus (4He, particle) plus two positive electrons () and two neutrinos () plus energy. This process releases energy to the star since, as Aston showed, four hydrogen atoms are heavier than one helium atom. The same set of nuclear reactions that supply the energy of the sun's radiation also produce neutrinos that can be searched for in the laboratory.

This figure is a cross section of the sun. The features that are usually studied by astronomers with normal telescopes that detect light are labeled on the outside, e. g., sunspot and prominences. Neutrinos enable us to look deep inside the sun, into the solar core where nuclear burning occurs.

Because of their weak interactions, neutrinos are difficult to detect. How difficult? A solar neutrino passing through the entire earth has less than one chance in a thousand billion of being stopped by terrestrial matter. According to standard theory, about a hundred billion solar neutrinos pass through your thumbnail every second and you don't notice them. Neutrinos can travel unaffected through iron as far as light can travel in a hundred years through empty space.

In 1964, Raymond Davis Jr. and I proposed that an experiment with 100,000 gallons of cleaning fluid (perchloroethylene, which is mostly composed of chlorine) could provide a critical test of the idea that nuclear fusion reactions are the ultimate source of solar radiation. We argued that, if our understanding of nuclear processes in the interior of the sun was correct, then solar neutrinos would be captured at a rate Davis could measure with a large tank filled with cleaning fluid. When neutrinos interact with chlorine, they occasionally produce a radioactive isotope of argon. Davis had shown previously that he could extract tiny amounts of neutrino-produced argon from large quantities of perchloroethylene. To do the solar neutrino experiment, he had to be spectacularly clever since according to my calculations only, a few atoms would be produced per week in a huge Olympic-sized swimming pool of cleaning fluid.

Our sole motivation for urging this experiment was to use neutrinos to:

enable us to see into the interior of a star and thus verify directly the hypothesis of nuclear energy generation in stars.

As we shall see, Davis and I did not anticipate some of the most interesting aspects of this proposal.

Davis performed the experiment and in 1968 announced the first results. He measured fewer neutrinos than I predicted. As the experiment and the theory were refined, the disagreement appeared more robust. Scientists rejoiced that solar neutrinos were detected but worried why there were fewer neutrinos than predicted.

What was wrong? Was our understanding of how the sun shines incorrect? Had I made an error in calculating the rate at which solar neutrinos would be captured in Davis's tank? Was the experiment wrong? Or, did something happen to the neutrinos after they were created in the sun?

Over the next twenty years, many different possibilities were examined by hundreds, and perhaps thousands, of physicists, chemists, and astronomers5. Both the experiment and the theoretical calculation appeared to be correct.

Once again experiment rescued pure thought. In 1986, Japanese physicists led by Masatoshi Koshiba and Yoji Totsuka, together with their American colleagues, Eugene Beier and Alfred Mann, reinstrumented a huge tank of water designed to measure the stability of matter. The experimentalists increased the sensitivity of their detector so that it could also serve as a large underground observatory of solar neutrinos. Their goal was to explore the reason for the quantitative disagreement between the predicted and the measured rates in the chlorine experiment.

The new experiment (called Kamiokande) in the Japanese Alps also detected solar neutrinos. Moreover, the Kamiokande experiment confirmed that the neutrino rate was less than predicted by standard physics and standard models of the sun and demonstrated that the detected neutrinos came from the sun. Subsequently, experiments in Russia (called SAGE, led by V. Gavrin), in Italy (GALLEX and later GNO led by T. Kirsten and E. Belotti, respectively), and again in Japan (Super-Kamiokande, led by Y. Totsuka and Y. Suzuki), each with different characteristics, all observed neutrinos from the solar interior. In each detector, the number of neutrinos observed was somewhat lower than standard theory predicted.

What do all of these experimental results mean?

Neutrinos produced in the center of the sun have been detected in five experiments. Their detection shows directly that the source of the energy that the sun radiates is the fusion of hydrogen nuclei in the solar interior. The nineteenth century debate between theoretical physicists, geologists, and biologists has been settled empirically.

From an astrophysical perspective, the agreement between neutrino observations and theory is good. The observed energies of the solar neutrinos match the values predicted by theory. The rates at which neutrinos are detected are less than predicted but not by a large factor. The predicted neutrino arrival rate at the earth depends approximately upon the 25th power of the central temperature of the sun, T x T x...T (25 factors of the temperature T). The agreement that has been achieved (agreement within a factor of three) shows that we have empirically measured the central temperature of the sun to an accuracy of a few percent. Incidentally, if someone had told me in 1964 that the number of neutrinos observed from the sun would be within a factor of three of the predicted value, I would have been astonished and delighted.

In fact, the agreement between normal astronomical observations (using light rather than neutrinos) and theoretical calculations of solar characteristics is much more precise. Studies of the internal structure of the sun using the solar equivalent of terrestrial seismology (i. e., observations of solar vibrations) show that the predictions of the standard solar model for the temperatures in the central regions of the sun are consistent with the observations to an accuracy of at least 0.1%. In this standard model, the current age of the sun is five billion years, which is consistent with the minimum estimate of the sun's age made by nineteenth-century geologists and biologists (a few hundred million years).

Given that the theoretical models of the sun describe astronomical observations accurately, what can explain the disagreement by a factor of two or three between the measured and the predicted solar neutrino rates?

New Physics
Physicists and astronomers were once again forced to reexamine their theories. This time, the discrepancy was not between different estimates of the sun's age, but rather between predictions based upon a widely accepted theory and direct measurements of particles produced by nuclear burning in the sun's interior. This situation was sometimes referred to as the Mystery of the Missing Neutrinos or, in language that sounded more scientific, the Solar Neutrino Problem.

As early as 1969, two scientists working in Russia, Bruno Pontecorvo and Vladimir Gribov, proposed that the discrepancy between standard theory and the first solar neutrino experiment could be due to an inadequacy in the textbook description of particle physics, rather than in the standard solar model. (Incidentally, Pontecorvo was the first person to propose using a chlorine detector to study neutrinos.) Gribov and Pontecorvo suggested that neutrinos suffer from a multiple personality disorder, that they oscillate back and forth between different states or types.

According to the suggestion of Gribov and Pontecorvo, neutrinos are produced in the sun in a mixture of individual states, a sort of split personality. The individual states have different, small masses, rather than the zero masses attributed to them by standard particle theory. As they travel to the earth from the sun, neutrinos oscillate between the easier-to-detect neutrino state and the more difficult-to-detect neutrino state. The chlorine experiment only detects neutrinos in the easier-to-observe state. If many of the neutrinos arrive at earth in the state that is difficult to observe, then they are not counted. It is as if some or many of the neutrinos have vanished, which can explain the apparent mystery of the missing neutrinos.

Building upon this idea, Lincoln Wolfenstein in 1978 and Stanislav Mikheyev and Alexei Smirnov in 1985 showed that the effects of matter on neutrinos moving through the sun might increase the oscillation probability of the neutrinos if Nature has chosen to give them masses in a particular range.

Neutrinos are also produced by the collisions of cosmic ray particles with other particles in the earth's atmosphere. In 1998, the Super-Kamiokande team of experimentalists announced that they had observed oscillations among atmospheric neutrinos. This finding provided indirect support for the theoretical suggestion that solar neutrinos oscillate among different states. Many scientists working in the field of solar neutrinos believe that, in retrospect, we have had evidence for oscillations of solar neutrinos since 1968.

But, we do not yet know what causes the multiple personality disorder of solar neutrinos. The answer to this question may provide a clue to physics beyond the current standard models of sub-atomic particles. Does the change of identity occur while the neutrinos are traveling to the earth from the sun, as originally proposed by Gribov and Pontecorvo? Or does matter cause solar neutrinos to "flip out''? Experiments are underway in Canada, Italy (three experiments), Japan (two experiments), Russia, and the United States that are attempting to determine the cause of the oscillations of solar neutrinos, by finding out how much they weigh and how they transform from one type to another. Non-zero neutrino masses may provide a clue to a still undiscovered realm of physical theory.

Nature: A Wonderful Mystery
Nature has written a wonderful mystery. The plot continually changes and the most important clues come from seemingly unrelated investigations. These sudden and drastic changes of scientific scene appear to be Nature's way of revealing the unity of all fundamental science.

The mystery begins in the middle of the nineteenth century with the puzzle: How does the sun shine? Almost immediately, the plot switches to questions about how fast natural selection occurs and at what rate geological formations are created. The best theoretical physics of the nineteenth century gave the wrong answer to all these questions. The first hint of the correct answer came, at the very end of the nineteenth century, from the discovery of radioactivity with accidentally darkened photographic plates.

The right direction in which to search for the detailed solution was revealed by the 1905 discovery of the special theory of relativity, by the 1920 measurement of the nuclear masses of hydrogen and helium, and by the 1928 quantum mechanical explanation of how charged particles get close to each other. These crucial investigations were not directly related to the study of stars.

By the middle of the twentieth century, nuclear physicists and astrophysicists could calculate theoretically the rate of nuclear burning in the interiors of stars like the sun. But, just when we thought we had Nature figured out, experiments showed that fewer solar neutrinos were observed at earth than were predicted by the standard theory of how stars shine and how sub-atomic particles behave.

At the beginning of the twenty-first century, we have learned that solar neutrinos tell us not only about the interior of the sun, but also something about the nature of neutrinos. No one knows what surprises will be revealed by the new solar neutrino experiments that are currently underway or are planned. The richness and the humor with which Nature has written her mystery, in an international language that can be read by curious people of all nations, is beautiful, awesome, and humbling.

Bibliography
l. F.W. Aston, "The Mass-Spectra of Chemical Elements,'' Philosophical Magazine and Journal of Science, 39, 611-625 (1920). In the course of a systematic program to measure the masses of atoms, Aston found that four hydrogen nuclei (protons) are heavier than a helium nucleus (an alpha particle) and two positive electrons [see Eq. (1)]. This fundamental discovery is the experimental basis of our understanding of how stars like the sun shine. The original paper is rarely cited, perhaps because the text is mainly devoted to a description of Aston's new apparatus and to a discussion of the many different masses that he measured. The hydrogen-helium mass difference is only briefly discussed.

2. R.D.E. Atkinson and F.G. Houtermans, "Zur Frage der Aufbaumöglichkeit der Elements in Sternen,''

Z. Physik 54, 656 (1929). An early attempt to calculate the rate of nuclear reactions in stars using the Gamow factor.

3. J.N. Bahcall, "Solar Neutrinos I. Theoretical,'' Phys. Rev. Lett. 12, 300 (1964).

4. H.A. Bethe, "Energy production in Stars,'' Phys. Rev. 55, 436 (1939). If you are a physicist and only have time to read one paper in the subject, this is the paper to read.

5. J.D. Burchfield, Lord Kelvin and The Age of the Earth, (Chicago: University of Chicago Press), 1990. This concise book provides a clear and insightful account of Kelvin's views on the age of the earth and the age of the sun, and on many other topics including natural selection and geological evolution. The author tells an exciting story with historical accuracy.

6. C.L. Cowan Jr., F. Reines, F.B. Harrison, H.W. Kruse, and A.D. McGuire, "Detection of the Free Neutrino: a Confirmation'', Science 124, 103 (1956); F. Reines and C.L. Cowan, "Detection of the Free Neutrino'', Phys. Rev. 92, 830 (1953). These papers describe the first experimental detection of neutrinos.

7. C. Darwin, On the Origin of the Species by Natural Selection, or, The Preservation of Favored Races in the Struggle for Life (London: Murray 1859), p. 285 (Pelican Preprint of first edition, 296--297, 1968).

8. R. Davis Jr., "Solar Neutrinos. II. Experimental,'' Phys. Rev. Lett. 12, 302 (1964).

9. J.N. Bahcall and R. Davis Jr., "An Account of the Development of the Solar Neutrino Problem,'' in Essays in Nuclear Astrophysics, ed. C.A. Barnes, D.D. Clayton, and D. Schramm (Cambridge: Cambridge University Press 1982), p. 243; reprinted in J.N. Bahcall, Neutrino Astrophysics, (Cambridge: Cambridge University Press 1989). For related material, see http://www.sns.ias.edu/~jnb/Papers/Popular/snhistory.html.

10. A.S. Eddington, "The Internal Constitution of the Stars,'' Observatory 43, 353 (1920). This lecture is inspiring.

11. A. Einstein, "Zur Elektrodynamik bewegter Körper,'' Annalen der Physik, 17 (105). English translation in The Principle of Relativity, translated by W. Perrett and G.B. Jeffery with notes by A. Sommerfeld, (Dover Publications: New York), 1923. The logic in this paper is breathtakingly beautiful and incredibly clear.

12. E. Fermi, "Tentativo di una teoria della emissione di raggi ,'' Ric. 4, 491 (1934). Reprinted in Enrico Fermi, Collected Papers: Note e memorie, Vol 1. p. 538 (University of Chicago Press: Chicago) (1962-1965). See also p. 559, 575. Fermi formulated the mathematical theory of neutrino emission in -decay. His first paper on the subject was rejected as "too speculative'' for publication.

13. W.A. Fowler, "Experimental and theoretical nuclear astrophysics: the quest for the origin of the elements,'' Rev. Mod. Phys. 56, 149 (1984).

14. G. Gamow, "Zur Quantentheorie der Atomzertrümmerung,'' Zeit. fur Physik 52, 510 (1928). Derives the Gamow factor using quantum mechanics.

15. S. Hawking, "Gravitationally collapsed objects of very low mass'', Monthly Notices of Royal Astronomical Society, 152, 75 (1971). In this imaginative paper, Hawking speculated that the central region of the sun might contain a black hole and that this could be the reason why the flux of solar neutrinos was less than predicted.

16. H. von Helmholtz, Lecture "On the interaction of natural forces,'' Königsberg, February 7 (1854), in Phil. Mag. 11 [series 4], 489-518 (1856).

17. J.F.W. Herschel, A Treatise on Astronomy (London 1833), p. 211.

18. W.T. Kelvin, "On the Age of the Sun's Heat,'' Macmillan's Magazine, 288--293 (March 5, 1862).

19. J. Marchant, Alfred Russel Wallace, Letters and Reminiscences, I (London: Cassell 1916), p. 242. Letter dated 14 April, 1869.

20. W. Pauli, letter to a physicists' gathering at Tübingen, December 4, 1930. Reprinted in Wolfgang Pauli, Collected Scientific Papers, ed. R. Kronig and V. Weisskopf, Vol. 2, p. 1313 (Interscience, New York) (1964).

21. H.N. Russell, "On the Sources of Stellar Energy,'' Pub. Ast. Soc. Pacific, August (1919). If you like to read mystery stories and to figure out "Who did it'' from limited clues, then you will love this paper. A year before Aston's measurements of the mass of hydrogen and of helium and two decades before Bethe's calculations of nuclear fusion rates, Russell used well-known observations of stars and simple physical reasoning to infer that the rate of the "unknown process'' that supplies stellar energy must increase rapidly with increasing stellar temperature. Incredibly, he also correctly deduced that this dependence of energy production on temperature would lead to stars being stable over very long periods of time. These insights are presented in the text of a closely-reasoned lecture that contains no equations.

22. E. Rutherford, "The Radiation and Emanation of Radium,'' Pt. II, Technics, Aug., 171, (1904) Collected Papers, I: 650.

23. C. Smith and M.N. Wise, Energy and Empire: A biographical study of Lord Kelvin, (Cambridge: Cambridge University Press), 1989. This book is a stimulating and authoritative account of Kelvin, his science, and his life. Chapters 15-17 deal with the age of the sun, the cooling of the earth, and the age of the earth.

24. C. F. von Weizsäcker, "Über Elementumwandlungen in Innern der Sterne. II,'' Physikalische Zeitschrift, 39, 633 (1938). The CNO cycle is described in the last paragraph of Section 7.

--------------------------------------------------------------------------------

1 von Helmholtz and Mayer were two of the codiscoverers of the law of conservation of energy. This law states that energy can be transformed from one form to another but the total amount is always conserved. Conservation of energy is a basic principle of modern physics that is used in analyzing the very smallest (sub-atomic) domains and the largest known structure (the universe), and just about everything in between. We shall see later that Einstein's generalization of the law of conservation of energy was a key ingredient in understanding the origin of solar radiation. The application of conservation of energy to radioactivity revealed the existence of neutrinos.

2 According to the modern theory of stellar evolution, the sun is heated to the enormous temperatures at which nuclear fusion can occur by gravitational energy released as the solar mass contracts from an initially large gas cloud. Thus, Kelvin and other nineteenth-century physicists were partially right; the release of gravitational energy ignited nuclear energy generation in the sun.

3 The sensitive dependence of the Gamow factor upon the relative energy of the two charged particles is, we now understand, what provides the temperature "thermostat'' for stars.

4 The existence of neutrinos was first proposed by Wolfgang Pauli in a 1930 letter to his physics colleagues as a "desperate way out" of the apparent non-conservation of energy in certain radioactive decays (called -decays) in which electrons were emitted. According to Pauli's hypothesis, which he put forward very hesitantly, neutrinos are elusive particles which escape with the missing energy in -decays. The mathematical theory of -decay was formulated by Enrico Fermi in 1934 in a paper which was rejected by the journal Nature because "it contained speculations too remote from reality to be of interest to the reader.'' Neutrinos from a nuclear reactor were first detected by Clyde Cowan and Fred Reines in 1956.

5 Perhaps the most imaginative proposal was made by Stephen Hawking, who suggested that the central region of the sun might contain a small black hole and that this could be the reason why the number of neutrinos observed is less than the predicted number.

## Thursday, 5 June 2008

### Space Math III Educator Guide

Audience: Educators

These activities comprise a series of 36 practical mathematics applications in space science. This collection of activities is based on a weekly series of space science problems distributed to teachers during the 2006-2007 school year. The problems in this booklet investigate science and mathematics concepts such as radiation effects on humans and technology, solar science, algebra, trigonometry, and calculus. The problems are authentic glimpses of modern engineering issues that arise in designing satellites to work in space. Each word problem has background information providing insight into the basic phenomena of the sun-Earth system, specifically space weather. The one-page assignments are accompanied by one-page teacher's sheets with answer keys.

Space Math III  [12MB PDF file]

Individual sections:
Introductory Pages
Sample Problem, An Introduction to Space Radiation
Problem 1, Unit Conversion Exercises
Problem 2, Radiation Background and Lifestyles
Problem 3, A Perspective on Radiation Dosages
Problem 4, Having a Hot Time on Mars!
Problem 5, Calculating Total Radiation Dosages at Mars
Problem 6, Single Event Upsets in Aircraft Avionics
Problem 7, The Deadly Van Allen Belts?
Problem 8, Systems of Equations in Space Science
Problem 9, Monster Functions in Space Science
Problem 10, Parametric Functions and Substitution
Problem 11, Radon Gas in the Basement
Problem 13, Moving Magnetic Filaments Near Sunspots
Problem 14, Correcting Bad Data Using Parity Bits
Problem 15, Data Corruption by High-Energy Particles
Problem 16, The Pressure of a Solar Storm
Problem 17, Are U Nuts?
Problem 18, Lunar Meteorite Impact Risk
Problem 19, Beyond the Blue Horizon
Problem 20, Measuring the Speed of a Solar Tsunami
Problem 21, Do Fast CMEs Produce Intense SPEs?
Problem 22, Atmospheric Shielding from Radiation Part I
Problem 23, Atmospheric Shielding from Radiation Part II
Problem 24, Atmospheric Shielding from Radiation Part III
Problem 25, Introduction to Radiation Shielding
Problem 26, Astronomy as a Career
Problem 27, Solar Storms; Odds, Fractions and Percentages …
Problem 28, A Study of Astronaut Radiation Dosages
Problem 29, Hinode Satellite Power
Problem 30, Hinode -- A Close-up of a Sunspot
Problem 31, Compound Interest
Problem 32, Solar Flare Reconstruction
Problem 33, A Lunar Transit of the Sun from Space
Problem 34, The Hinode Satellite Views the Sun
Problem 35, The Sunspot Cycle -- Endings and Beginnings
Problem 36, Super-fast Solar Flares

Source: NASA

## Sunday, 1 June 2008

### How White Dwarfs Get Their 'Kicks'

Arip Nurahman

Department of Physics
Faculty of Sciences and Mathematics, Indonesia University of Education

and Follower Open Course Ware at MIT-Harvard University, M.A., U.S.A.

A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored thermal energy. White dwarfs comprise roughly 6% of all known stars in the solar neighborhood. The unusual faintness of white dwarfs was first recognized in 1910 by Henry Norris Russell, Edward Charles Pickering, and Williamina Fleming; p. 1 the name white dwarf was coined by Willem Luyten in 1922.

(Wikipedia)

NASA's Hubble Space Telescope is providing strong evidence that white dwarfs, the burned-out relics of stars, are given a "kick" when they form.

The sharp vision of Hubble's Advanced Camera for Surveys uncovered the speedy white dwarfs in the ancient globular star cluster NGC 6397, a dense swarm of hundreds of thousands of stars.

Before the stars burned out as white dwarfs, they were among the most massive stars in NGC 6397. Because massive stars are thought to gather at a globular cluster's core, astronomers assumed that most newly minted white dwarfs dwelled near the center.

Hubble, however, discovered young white dwarfs residing at the edge of NGC 6397, which is about 11.5 billion years old.

"The distribution of young white dwarfs is the exact opposite of what we expected," said astronomer Harvey Richer of the University of British Columbia in Vancouver. "Our idea is that as aging stars evolve into white dwarfs, they are given a kick of 7,000 to 11,000 miles an hour (3 to 5 kilometers a second), which rockets them to the outer reaches of the cluster."

Richer suggested that white dwarfs propel themselves by ejecting mass, like rockets do. Before stars evolve into white dwarfs, they swell up and become red giants. Red giant stars lose about half their mass by shedding it into space. If more of this mass is ejected in one direction, it could propel the emerging white dwarf through space, just as exhaust from a rocket engine thrusts the rocket from the launch pad, Richer proposed.

Observations of some planetary nebulae display similarly directed outflows. (Planetary nebulae are the glowing material ejected by red giant stars.) The jets in those planetary nebulae are shown to flow in opposite directions. If they are not perfectly balanced, Richer reasoned, the stronger jet could accelerate the white dwarf in the opposite direction.

The idea that young white dwarfs are born with a kick was suggested 30 years ago to explain why there were so few of them in open star clusters. In 2003 Michael Fellhauer of the University of California at Santa Cruz and colleagues calculated that if white dwarfs were given a small boost, they could be expelled from open clusters. It is easier, however, for white dwarfs to escape the weak gravitational clutches of open clusters than to rocket out of globular clusters, which are as much as 100 times more massive than open clusters.

Richer and his team, therefore, decided to test the acceleration theory in a globular cluster. The astronomers chose NGC 6397 because, at 8,500 light-years away, it is one of the closest globular star clusters to Earth. About 150 globular clusters exist in the Milky Way, each containing up to a million stars.

The team studied 22 young white dwarfs less than 800 million years old and 62 older white dwarfs between 1.4 and 3.5 billion years old. The astronomers distinguished the younger from the older white dwarfs based on their color and brightness. The younger white dwarfs are hotter and therefore bluer and brighter than the older ones.

Globular clusters sort out stars according to their mass, governed by a gravitational pinball game between stars. Heavier stars slow down and sink to the cluster's core, while lighter stars pick up speed and move across the cluster to its outskirts. Richer's team found that the older white dwarfs were behaving as expected: They were scattered throughout the cluster according to weight.

The young white dwarfs, however, were found unexpectedly at the edge of the cluster, puzzling Richer and his team.

Their expected neighborhood is near the center because their progenitor stars were the heaviest stars present in the cluster. These fledgling white dwarfs are so young that they have not had enough encounters with other stars to spread them across the cluster, suggesting that some other mechanism (a kick) is at work.

"The first time we plotted up the distribution and found a difference, we thought, 'My goodness, what is happening?'" said team member Saul Davis, a graduate student at the University of British Columbia in Vancouver. "For a long time, we thought we had made a mistake. But no matter what we did, it didn't go away."

The team considered other explanations for the young white dwarfs' location. They could have been part of binary systems and gotten kicked out by their partners. Or perhaps they were given a boost after encountering heavier stars. The team, however, ruled out those explanations through computer simulations.

Richer hopes to study other globular clusters for runaway white dwarfs. The results will appear in the January 2008 issue of the Monthly Notices of Royal Astronomical Society Letters.

Other members of the team are I.R. King (University of Washington, Seattle); J. Anderson (Space Telescope Science Institute, Baltimore, Md.); J. Coffey (University of British Columbia, Vancouver); G.G. Fahlman (NRC Herzberg Institute of Astrophysics, National Research Council Canada, Saanich, British Columbia); J. Hurley (Swinburne University of Technology, Hawthorn, Australia); and J.S. Kalirai (University of California at Santa Cruz.)

## Composition and structure

Although white dwarfs are known with estimated masses as low as 0.17[27] and as high as 1.33[28] solar masses, the mass distribution is strongly peaked at 0.6 solar mass, and the majority lie between 0.5 to 0.7 solar mass.[28] The estimated radii of observed white dwarfs, however, are typically between 0.008 and 0.02 times the radius of the Sun;[29] this is comparable to the Earth's radius of approximately 0.009 solar radius. A white dwarf, then, packs mass comparable to the Sun's into a volume that is typically a million times smaller than the Sun's; the average density of matter in a white dwarf must therefore be, very roughly, 1,000,000 times greater than the average density of the Sun, or approximately 106 grams (1 tonne) per cubic centimeter.[1] White dwarfs are composed of one of the densest forms of matter known, surpassed only by other compact stars such as neutron stars, black holes and, hypothetically, quark stars.[30]

White dwarfs were found to be extremely dense soon after their discovery. If a star is in a binary system, as is the case for Sirius B and 40 Eridani B, it is possible to estimate its mass from observations of the binary orbit. This was done for Sirius B by 1910,[31] yielding a mass estimate of 0.94 solar mass. (A more modern estimate is 1.00 solar mass.)[32] Since hotter bodies radiate more than colder ones, a star's surface brightness can be estimated from its effective surface temperature, and hence from its spectrum. If the star's distance is known, its overall luminosity can also be estimated. Comparison of the two figures yields the star's radius. Reasoning of this sort led to the realization, puzzling to astronomers at the time, that Sirius B and 40 Eridani B must be very dense. For example, when Ernst Öpik estimated the density of a number of visual binary stars in 1916, he found that 40 Eridani B had a density of over 25,000 times the Sun's, which was so high that he called it "impossible".[33] As Arthur Stanley Eddington put it later in 1927:[34], p. 50
We learn about the stars by receiving and interpreting the messages which their light brings to us. The message of the Companion of Sirius when it was decoded ran: "I am composed of material 3,000 times denser than anything you have ever come across; a ton of my material would be a little nugget that you could put in a matchbox." What reply can one make to such a message? The reply which most of us made in 1914 was—"Shut up. Don't talk nonsense."
As Eddington pointed out in 1924, densities of this order implied that, according to the theory of general relativity, the light from Sirius B should be gravitationally redshifted.[23] This was confirmed when Adams measured this redshift in 1925.[35]

Such densities are possible because white dwarf material is not composed of atoms bound by chemical bonds, but rather consists of a plasma of unbound nuclei and electrons. There is therefore no obstacle to placing nuclei closer to each other than electron orbitals—the regions occupied by electrons bound to an atom—would normally allow.[23] Eddington, however, wondered what would happen when this plasma cooled and the energy which kept the atoms ionized was no longer present.[36] This paradox was resolved by R. H. Fowler in 1926 by an application of the newly devised quantum mechanics. Since electrons obey the Pauli exclusion principle, no two electrons can occupy the same state, and they must obey Fermi-Dirac statistics, also introduced in 1926 to determine the statistical distribution of particles which satisfy the Pauli exclusion principle.[37] At zero temperature, therefore, electrons could not all occupy the lowest-energy, or ground, state; some of them had to occupy higher-energy states, forming a band of lowest-available energy states, the Fermi sea. This state of the electrons, called degenerate, meant that a white dwarf could cool to zero temperature and still possess high energy. Another way of deriving this result is by use of the uncertainty principle: the high density of electrons in a white dwarf means that their positions are relatively localized, creating a corresponding uncertainty in their momenta. This means that some electrons must have high momentum and hence high kinetic energy.[36][38]

Compression of a white dwarf will increase the number of electrons in a given volume. Applying either the Pauli exclusion principle or the uncertainty principle, we can see that this will increase the kinetic energy of the electrons, causing pressure.[36][39] This electron degeneracy pressure is what supports a white dwarf against gravitational collapse. It depends only on density and not on temperature. Degenerate matter is relatively compressible; this means that the density of a high-mass white dwarf is so much greater than that of a low-mass white dwarf that the radius of a white dwarf decreases as its mass increases.[1]

The existence of a limiting mass that no white dwarf can exceed is another consequence of being supported by electron degeneracy pressure. These masses were first published in 1929 by Wilhelm Anderson[40] and in 1930 by Edmund C. Stoner.[41] The modern value of the limit was first published in 1931 by Subrahmanyan Chandrasekhar in his paper "The Maximum Mass of Ideal White Dwarfs".[42] For a nonrotating white dwarf, it is equal to approximately 5.7/μe2 solar masses, where μe is the average molecular weight per electron of the star.[43], eq. (63) As the carbon-12 and oxygen-16 which predominantly compose a carbon-oxygen white dwarf both have atomic number equal to half their atomic weight, one should take μe equal to 2 for such a star,[38] leading to the commonly quoted value of 1.4 solar masses. (Near the beginning of the 20th century, there was reason to believe that stars were composed chiefly of heavy elements,[41], p. 955 so, in his 1931 paper, Chandrasekhar set the average molecular weight per electron, μe, equal to 2.5, giving a limit of 0.91 solar mass.) Together with William Alfred Fowler, Chandrasekhar received the Nobel prize for this and other work in 1983.[44] The limiting mass is now called the Chandrasekhar limit.

If a white dwarf were to exceed the Chandrasekhar limit, and nuclear reactions did not take place, the pressure exerted by electrons would no longer be able to balance the force of gravity, and it would collapse into a denser object such as a neutron star.[45] However, carbon-oxygen white dwarfs accreting mass from a neighboring star undergo a runaway nuclear fusion reaction, which leads to a Type Ia supernova explosion in which the white dwarf is destroyed, just before reaching the limiting mass.[46]

White dwarfs have low luminosity and therefore occupy a strip at the bottom of the Hertzsprung-Russell diagram, a graph of stellar luminosity versus color (or temperature). They should not be confused with low-luminosity objects at the low-mass end of the main sequence, such as the hydrogen-fusing red dwarfs, whose cores are supported in part by thermal pressure,[47] or the even lower-temperature brown dwarfs.[48]

### Mass-radius relationship and mass limit

It is simple to derive a rough relationship between the mass and radii of white dwarfs using an energy minimization argument. The energy of the white dwarf can be approximated by taking it to be the sum of its gravitational potential energy and kinetic energy. The gravitational potential energy of a unit mass piece of white dwarf, Eg, will be on the order of −GM/R, where G is the gravitational constant, M is the mass of the white dwarf, and R is its radius. The kinetic energy of the unit mass, Ek, will primarily come from the motion of electrons, so it will be approximately N p2/2m, where p is the average electron momentum, m is the electron mass, and N is the number of electrons per unit mass. Since the electrons are degenerate, we can estimate p to be on the order of the uncertainty in momentum, Δp, given by the uncertainty principle, which says that Δp Δx is on the order of the reduced Planck constant, ħ. Δx will be on the order of the average distance between electrons, which will be approximately n−1/3, i.e., the reciprocal of the cube root of the number density, n, of electrons per unit volume. Since there are N M electrons in the white dwarf and its volume is on the order of R3, n will be on the order of N M / R3.[38]
Solving for the kinetic energy per unit mass, Ek, we find that
$E_k \approx \frac{N (\Delta p)^2}{2m} \approx \frac{N \hbar^2 n^{2/3}}{2m} \approx \frac{M^{2/3} N^{5/3} \hbar^2}{2m R^2}.$
The white dwarf will be at equilibrium when its total energy, Eg + Ek, is minimized. At this point, the kinetic and gravitational potential energies should be comparable, so we may derive a rough mass-radius relationship by equating their magnitudes:
$|E_g|\approx\frac{GM}{R} = E_k\approx\frac{M^{2/3} N^{5/3} \hbar^2}{2m R^2}.$
Solving this for the radius, R, gives[38]
$R \approx \frac{N^{5/3} \hbar^2}{2m GM^{1/3}}.$
Dropping N, which depends only on the composition of the white dwarf, and the universal constants leaves us with a relationship between mass and radius:
$R \sim \frac{1}{M^{1/3}}, \,$
i.e., the radius of a white dwarf is inversely proportional to the cube root of its mass.
Since this analysis uses the non-relativistic formula p2/2m for the kinetic energy, it is non-relativistic. If we wish to analyze the situation where the electron velocity in a white dwarf is close to the speed of light, c, we should replace p2/2m by the extreme relativistic approximation p c for the kinetic energy. With this substitution, we find
$E_{k\ {\rm relativistic}} \approx \frac{M^{1/3} N^{4/3} \hbar c}{R}.$
If we equate this to the magnitude of Eg, we find that R drops out and the mass, M, is forced to be[38]
$M_{\rm limit} \approx N^2 \left(\frac{\hbar c}{G}\right)^{3/2}.$

Radius-mass relations for a model white dwarf.
To interpret this result, observe that as we add mass to a white dwarf, its radius will decrease, so, by the uncertainty principle, the momentum, and hence the velocity, of its electrons will increase. As this velocity approaches c, the extreme relativistic analysis becomes more exact, meaning that the mass M of the white dwarf must approach Mlimit. Therefore, no white dwarf can be heavier than the limiting mass Mlimit.
For a more accurate computation of the mass-radius relationship and limiting mass of a white dwarf, one must compute the equation of state which describes the relationship between density and pressure in the white dwarf material. If the density and pressure are both set equal to functions of the radius from the center of the star, the system of equations consisting of the hydrostatic equation together with the equation of state can then be solved to find the structure of the white dwarf at equilibrium. In the non-relativistic case, we will still find that the radius is inversely proportional to the cube root of the mass.[43], eq. (80) Relativistic corrections will alter the result so that the radius becomes zero at a finite value of the mass. This is the limiting value of the mass—called the Chandrasekhar limit—at which the white dwarf can no longer be supported by electron degeneracy pressure. The graph on the right shows the result of such a computation. It shows how radius varies with mass for non-relativistic (blue curve) and relativistic (green curve) models of a white dwarf. Both models treat the white dwarf as a cold Fermi gas in hydrostatic equilibrium. The average molecular weight per electron, μe, has been set equal to 2. Radius is measured in standard solar radii and mass in standard solar masses.[43][49]
These computations all assume that the white dwarf is nonrotating. If the white dwarf is rotating, the equation of hydrostatic equilibrium must be modified to take into account the centrifugal pseudo-force arising from working in a rotating frame.[50] For a uniformly rotating white dwarf, the limiting mass increases only slightly. However, if the star is allowed to rotate nonuniformly, and viscosity is neglected, then, as was pointed out by Fred Hoyle in 1947,[51] there is no limit to the mass for which it is possible for a model white dwarf to be in static equilibrium. Not all of these model stars, however, will be dynamically stable.[52]

## References

1. ^ a b c d e f g h Extreme Stars: White Dwarfs & Neutron Stars, Jennifer Johnson, lecture notes, Astronomy 162, Ohio State University. Accessed on line May 3, 2007.
2. ^ The One Hundred Nearest Star Systems, Todd J. Henry, RECONS, April 11, 2007. Accessed on line May 4, 2007.
3. ^ a b c d White Dwarfs, E. Schatzman, Amsterdam: North-Holland, 1958.
4. ^ a b c d How Degenerate Stars Came to be Known as White Dwarfs, J. B. Holberg, Bulletin of the American Astronomical Society 37 (December 2005), p. 1503.
5. ^ a b c d The Potential of White Dwarf Cosmochronology, G. Fontaine, P. Brassard, and P. Bergeron, Publications of the Astronomical Society of the Pacific 113, #782 (April 2001), pp. 409–435.
6. ^ a b c d e Late stages of evolution for low-mass stars, Michael Richmond, lecture notes, Physics 230, Rochester Institute of Technology. Accessed on line May 3, 2007.
7. ^ a b On Possible Oxygen/Neon White Dwarfs: H1504+65 and the White Dwarf Donors in Ultracompact X-ray Binaries, K. Werner, N. J. Hammer, T. Nagel, T. Rauch, and S. Dreizler, pp. 165 ff. in 14th European Workshop on White Dwarfs; Proceedings of a meeting held at Kiel, July 19–23, 2004, edited by D. Koester and S. Moehler, San Francisco: Astronomical Society of the Pacific, 2005.
8. ^ a b A Helium White Dwarf of Extremely Low Mass, James Liebert, P. Bergeron, Daniel Eisenstein, H.C. Harris, S.J. Kleinman, Atsuko Nitta, and Jurek Krzesinski, The Astrophysical Journal 606, #2 (May 2004), pp. L147–L149. Accessed on line March 5, 2007.

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