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Cosmic Microwave Background

May 30, 2015 1 comment

The Cosmic Microwave Background (CMB) radiation is a very faint but observable form of radiation that is coming to us (and to other places too) from all directions. (By ‘radiation’ here is meant photons of light, or electromagnetic waves, from a wide range of possible frequencies or energies.) In today’s standard model of cosmology, this radiation is believed to emanate from about a time 200,000 to 400,000 years after the Big Bang – a timeframe known as ‘last scattering’ because that was when superheavy collisions between photons of light and other particles (electrons, protons, neutrons, etc) eased off to a degree that photons can ‘escape’ into the expanding space. At the time of last scattering, this radiation was very hot, around {3000}^{\circ}K (in the Kelvin scale). And in time, as space expanded, radiation cooled to its currently observed value of 2.726^\circ K.

One of the amazing facts about this radiation is that it almost perfectly matches Planck’s radiation formula (discovered in 1900) for a black body:

\displaystyle I(\nu) = \frac{8\pi h}{c^3} \cdot \frac{\nu^3}{e^{h\nu/kT} - 1}.

In this formula, \nu is the frequency variable (a positive real number that gives the number of cycles per second of a wave) and I(\nu) is the energy density as a function of frequency \nu.

The other variables are: T is the temperature of the black body which is assumed to be in equilibrium (so the temperature is uniformly constant throughout the body of radiation), c is the speed of light in vacuum, h is Planck’s constant, and k is Boltzmann’s constant from statistical mechanics.

If you plot the graph of this energy density function (against \nu) you get a curve that looks like a skewed ‘normal distribution’. Here are some examples I hijacked from the internet:

Various Planck radiation density graphs depending on temperature T.

Here you see various plots of Planck’s function for different temperatures T. The horizontal axis labels the frequency \nu, and the vertical gives the energy density I(\nu) per frequency. (Please ignore the rising black dotted curve.)

You’ll notice that the graphs have a maximum peak point. And that the lower the temperature, the smaller the frequency where the maximum occurs. Well, that’s what happened as the CMB radiation cooled from a long time ago till today: as the temperature T cooled (decreased) so did the frequency where the peak occurs.

To those of us who know calculus, we can actually compute what frequency \nu gives maximum density and give our answer in terms of the temperature T. All we do is compute the derivative of I(\nu) with respect to \nu and set it to zero and solve the resulting equation for \nu. You will get an equation whose solution isn’t so trivial to solve, so we’ll need some software or a calculator to approximate it. Anyway, I worked it out (and you can check my answer) and obtained the following:

\displaystyle\nu_{\max} = 2.82 \frac{kT}{h}.

(The equal sign here is an approximation!)

The \nu_{\max} is the frequency that gives maximum density and as you can see it is a straight linear function of temperature. The greater the temperature, the proportionately greater the max frequency. The colder the temperature gets the smaller the max-frequency \nu_{\max} gets, and from Planck’s energy equation E_{\max} = h\nu_{\max}, so also does the energy of the radiation drop.

Now plug in the observed value for the temperature of the background radiation, which is T = 2.726 (degrees Kelvin), and working it out we get (approximately)

\displaystyle\nu_{\max} = 160.2 \text{ GHz}

This frequency lies inside the microwave band which is why we call it the microwave radiation! (Even though it does also radiate in other higher and lower frequencies too but at much less intensity!)

Far back in time, when photons were released from their collision `trap’ (and the temperature of the radiation was much hotter) this max frequency was not in the microwave band.

Homework Question: what was the max-frequency \nu_{\max} at the time of last scattering? What frequency band does it belong to? In the visible range? Infrared? Ultraviolet? Higher still? (I’m dying to know! 😉 )

(It isn’t hard as it can be figured from the data above.)

Anyway, I thought working these out was fun.

The CMB radiation was first discovered by Penzias and Wilson in 1965. According to their measurements and calculations (and polite disposal of the pigeons nesting in their antenna!), they measured the temperature as being 3.5^\circ K plus or minus 1 Kelvin. (So the actual value that was confirmed later, namely 2.726, fits within their range.) The frequency of radiation that they detected, however, was not the maximum yielding one but was (as they had it in the title of their paper) 1080 Mc/s — which is ‘mega cycles per second’ and is the same as MHz (megahertz). The wavelength value corresponding to this is 7.35 cm. To do the conversion between wavelength \lambda and frequency \nu for electromagnetic waves we use the simple formula

\displaystyle\lambda\nu = c

where c is the speed of light (in vacuum).

And that’s the end of our little story for today!

Cheers, Sam Postscript.

The sacred physical constants:

Planck’s constant h = 6.6254 \times 10^{-27} \text{ erg sec}

Boltzmann’s constant k = 1.38049 \times 10^{-16} \text{ erg/K}

Speed of light c = 2.9979 \times 10^{10} \text{ cm/sec}.  

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Children of older fathers

August 23, 2012 Leave a comment

Sometimes when you put separate studies together you could get what seems a confusing picture (or maybe no picture at all?) — particularly seeing how they word their conclusions. Here is an example on studies related to offspring of older fathers.

Study 1. Older fathers have longer telomeres in their chromosomes as they age, so the offspring of older fathers inherit these longer telomeres, enhancing the life expectancy of the offspring.  This was reported by the BBC and is based on a study published in Proceedings of the National Academy of Sciences.

Study 2. Children born to older fathers have a greater likelihood of developing autism or schizophrenia.  Reported by Nature Magazine (here’s Nature’s summary of the actual Nature study) and also quite recently by the BBC.

Study 3. The life expectancy of people with schizophrenia is lower than average, and can be lower by as much as 10 to 15 years. Reported by the BBC and based on a study done by the British Biomedical Research Centre for mental health, and published in the online journal PLoS ONE.

 
These conclusions aren’t necessarily in contradiction to one another (even if verbally they seem to be). It depends on the various rates (such as the smaller fraction of schizophrenics, for example, compared with the many who don’t get mental illness). So if you’re normal and descended from an older father ‘the chances’ are better that your longevity will get a boost (statistically!) if you don’t get a mental illness like schizophrenia. (Barring a Study 4 and Study 5, about which I know nothing and which could wreck my post!)

I’ve often wondered if in some societies around the world these effects and results could be varied depending on the people being studied. There are still numerous factors beyond our control. (Anyway, whatever your condition, I hope this makes you feel better!) 🙂

Earth’s Magnetic Field

August 18, 2012 Leave a comment

 

I know just a little about the Earth’s magnetic field – also called the geomagnetic field. The following are from some notes I wrote a few years ago and came across lately (thought maybe worthwhile sharing in my own words). My notes were based on: Ency. Britannica; Wiki article on geomagnetic field; and ‘chapter 3′ of a physicist’s notes (whose name I’m missing).

1. Magnetic north and Earth’s true north aren’t the same!

2. In fact, it is the magnetic south pole that’s closer to the Earth’s north, by something like 11 degrees. (That is called ‘magnetic declination’, the angle difference from true north.) For precise navigation this 11 degrees could be taken into account.

3. The magnetic field lines (usually written as B in physics) start from magnetic north and end at magnetic south. (At least, that is the convention.) Magnetic fields affect only charged particles (like electrons and protons). These particles move along the field lines by spiraling around them (like a coiled wire).

(They go back and forth. The reason they spiral in doing so is explained by the magnetic force being F = q v x B, where q is the charge on the particle, v is its velocity, and B is the magnetic field. The force is always perpendicular to B and v, which is why they spiral.)

4. The Chinese appear to have been the first to discover the geomagnetic field in their effort to perfect their navigation technology. (About 1100′s AD or so.) Later Sir Edmond Halley (of Halley’s comet) mapped the magnetic field.

5. It was believed 100s of years ago that the geomagnetic field had extra-terrestrial origin. It was the brilliant mathematician Carl Friedrich Gauss (mid 1800s) who showed that the field actually had its origin from the Earth itself, and he gave a mathematical expression for it (using spherical harmonics).

6. The magnetic field of the earth can experience reversal — where magnetic north and south poles are interchanged. But this happens irregularly, from some 700,000 or so years to a few million years. (I think it’s still a mystery as to why this happened in the past.) These reversals are recorded in rocks that register the field’s direction in the past. And in turn, this has been valuable in determining the history and motion of plate tectonics – and discovery of the mid ocean ridges which affect continental motion.

7. The motion of molten iron in the core of the Earth is generally credited for the creation of the geomagnetic field — this is called the geodynamo theory of Sir Bullard (about 1940s-50s). (The Earth’s crust has its contributions too, but they are fairly smaller.) Scientists study the inside of the Earth in part from how the geomagnetic field behaves and changes.

8. There are still some mysteries about geomagnetism. For example, why do the magnetic poles move about 10 km each year?

Some wonders of Natural History

August 5, 2012 Leave a comment

Some wonderful stuff I read about on Natural History magazine, I thought to share in my own simple ways. (Some don’t seem to have link references online, so I have cited them in red.)

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Recent research suggests that some groups of dinosaurs were already on the road to extinction before an asteroid impact — about 150 species of them.
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Did you know that the planet Uranus has its axis of rotation nearly parallel to its orbital plane? At one point it was pointing toward the sun! Secondly, its magnetic pole is some 60 degrees off its rotation axis! (Unlike earth where the magnetic north and rotation axis north are fairly close.)

The problem however is explaining the auroras near the magnetic poles of Uranus. It was believed that they could be explained by huge coronal mass ejections from the sun, but this seems in question. That’s an open question for anyone who wishes to research it.
Aurora Uranus, by Harvey Leifert, Natural History, May 2012, page 6.
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It may sound like a myth to read stories about people using pigeons to send messages in ancient times. But in fact pigeons have flight patterns that are based on their GPS sensing the earth’s magnetic field. There’s a part of the pigeon brain closely connected to their inner ears that interacts with the magnetic field, giving them this GPS ‘6th’ sense to navigate. Neuroscientists are still studying this interesting phenomenon.
“GPS: Global Pigeon System,” by Adam Hadhazy, Natural History, May 2012, page 8.
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The amazing ants are back! An ant leaves its colony and goes on an unknown journey. It gets infected with a fungus that is contagious. It returns to its colony. What do the ants in the colony do when it returns? Do they kill it? Ostracize it? Contain it? No. They welcome it back home and they come to its rescue, licking it clean of the infection; they share the infection; but because each gets a little of the infection they don’t get as affected as much and eventually get over it. All is well. They heal their friend and as a bonus they develop their immune system to make it stronger. No wonder I loved ants when I was a boy!! 🙂
“Ant-i-fungal,” by Judy Rice, Natural History, May 2012, page 8.

Comparison between UN resolutions: Israel and Turkey

http://t1.gstatic.com/images?q=tbn:ANd9GcSfz0GdPOBLzJmRw0_icoWQoxTEqR9N_mYQ-KlBN5JhXlXbFyObWwAs I was browsing through some of these UN resolutions related to Israel and comparing them with those on Turkey’s invasion and occupation of Cyprus, I noted a distinct difference in tone in the way the two states are addressed and treated.

For example, in UN Security Council Resolution 497 — which is related to Israel’s annexation of the Golan Heights from Syria following the 1967 Six Day War — it derides Israel for imposing “its laws, jurisdiction and administration in the occupied Syrian Golan Heights” and says that they are “null and void and without international legal effect.”  It demands that “Israel, the occupying Power, should rescind forthwith its decision.”

In a UN General Assembly resolution on the Golan, it patronizes Israel with harsh language such as: “illegal settlements”; “Israel has failed”; “Israel law null and void and has no validity”; referring to Golan as “occupied by Israel”; “Demands once more that Israel withdraw”; etc.

By stark contrast, when you look at the resolutions regarding Turkey’s invasion and occupation of another legal state, Cyprus — and which is still ongoing since 1974 — you don’t read that kind of patronizing language.  For example, in UN Security Council resolution 367 (1975), it instead “calls upon all States to respect sovereignty … of the Republic of Cyprus” — with no mention of Turkey by name. In paragraph 2 of this resolution it says that it “Regrets unilateral decision” declaring a part of Cyprus would be a “Federated Turkish State” (by Turkey, which is again not mentioned explicitly).

Turkey is hardly criticized in its occupation of Cyprus compared with how Israel has been prejudicially treated by the United Nations.  In fact, Turkey is only mentioned once in Resolution 367, even obliquely – and with the generous verb: ‘regret’. You don’t see the kind of mean language used against Israel in the Turkey resolution on Cyprus.

Clearly, there is distinct discrimination (by the so-called “international law”) against Israel. Nothing new, of course, since that has been known for a long time, but it is interesting to see it where it may not be obvious to look.

Memorial Day

https://i0.wp.com/i.i.com.com/cnwk.1d/i/tim/2012/05/28/MDay_AP120528026731_370x278.jpgIn Honor of all American men & women in uniform on this blessed Memorial Day. From those who fought in the Revolutionary War days that created the United States of America, to those who fought and are still fighting today, we owe them our lives and our freedom. Thank you and God bless them.

The Refugee Problem in the Muslim World

December 31, 2011 Leave a comment

There is an obsession in the media about the Palestinian refugee problem at the expense of several other refugee problems in that part of the world that are only given casual cursory interest.  Let’s briefly cite some of these other ones.

  • Saddam’s Al-Anfal Campaign in the 1980s led to a million Kurdish refugees. A clear case of ethnic cleansing.
  • Human Rights Watch says that “By the mid-1990s, more than 3,000 villages had been virtually wiped from the map, and, according to official figures, 378,335 Kurdish villagers had been displaced and left homeless.
  • Assyrian Christians have constantly been leaving Iraq over the last few 100 years (unlike Muslims who have grown in number). Now their number is only a small fraction.
  • The Lebanese civil war caused around 900,000 Lebanese to be displaced.
  • The 1974 Cyprus Crisis led to around 200,000 Greek Cypriot refugees and 65,000 Turkish Cypriot refugees (UN figures). These are near half the populations of both groups.
  • Nearly 800,000 Jews have been made refugees by Muslim states that they lived in for generations, thru persecution and intimidation. Most of them were sheltered and given a home in Israel; others fled to Western nations.
  • There are also many refugees from among the minorities living in Muslim states, such as the Bahais in Iran, who sought refuge in Western nations, in Israel, Turkey and others.
  • 100,000s Algerians have fled and taken refuge in Europe during the Algerian War of Independence.
  • The current Syrian civil war has led to the creation of 19,000 Syrian refugees near Turkey’s border; some 8,000 refugees fled to Lebanon, 1500 to Jordan, and 6,000 to Libya.
  • The Armenian refugees under the Turks, as well as the genocide against the Armenians.
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