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Ten photons per hour March 23, 2009

Posted by dorigo in astronomy, games, mathematics, personal, physics, science.
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Every working day I walk for about a mile to my physics department in Padova from the train station in the morning. I find it is a healthy habit, but I sometimes fear it also in some sense is a waste of time: if I catched a bus, I could be at work ten minutes earlier. I hate losing time, so I sometimes use the walking time to set physics problems to myself, trying to see whether I can solve them by heart. It is a way to exercise my mind while I exercise my body.

Today I was thinking at the night of stargazing I treated myself with last Saturday. I had gone to Casera Razzo, a secluded place in the Alps, and observed galaxies for four hours in a row with a 16″ dobsonian telescope, in the company of four friends (and three other dobs). One thing we had observed with amazement was a tiny speck of light coming from the halo of an interacting pair of galaxies in Ursa Major, the one pictured below.

The small speck of light shown in the upper left of the picture above, labeled as MGC 10-17-5, is actually a faint galaxy in the field of view of NGC3690. It has a visual magnitude of +15.7: this is a measure of its integrated luminosity as seen from the Earth. It is a really faint object, and barely at the limit of visibility with the instrument I had. The question I arrived at formulating to myself this morning was the following: how many photons did we get to see per second through the eyepiece, from that faint galaxy ?

This is a nice, simple question, but computing its answer by heart took me the best part of my walk. My problem was that I did not have a clue of the relationship between visual magnitude and photon fluxes. So I turned to things I did know.

Some background is needed to those of you who do not know how visual magnitudes are computed, so I will make a small digression here. The scale of visual magnitude is a semi-empirical one, which sets the brightest stars at magnitude zero or so, and defines a decrease of luminosity by a factor 100 per every five magnitudes difference. The faintest stars visible with the naked eye in a moonless night are of magnitude +6, and that means they are about 250 times fainter than the brightest ones. On the other hand, Venus shines at magnitude -4.5 at its brightest -almost 100 times as bright as the brightest stars-, and our Sun shines at a visual magnitude of about -27, more than a billion times brighter than Venus. The magnitude difference between two objects is in a relation with their relative brightness by a power law: L_1/L_2 = 2.5^{-M_1+M_2}; the factor 2.5 is an approximation for the fifth root of 100, and it corresponds to the brigthness ratio of two objects that differ by one unit of visual magnitude.

Ok, so we know how bright is the Sun. Now, if I could get how many photons reach our eye from it every second, I would make some progress. I reasoned that I knew the value of the solar constant: that is the energy radiated by the Sun on an area of 1 square meter on the ground of the Earth. I remembered a value of about 1 kilowatt (it is actually 1.366 kW, as I found out later in wikipedia).

Now, how many photons of visible light arriving per second on that square meter of ground correspond to 1 kilowatt of power ? I reasoned that I did not remember the energy of a single visible photon -I remembered it was in the electron-Volt range but I was not really sure- so I had to compute it.

The energy of a quantum of light is given by the formula E = h \nu, where h is Planck’s constant and \nu is the light frequency. However, all I knew was that visible light has a wavelength of about 500 nanometers (which is 5 \times 10^{-7} m), so I had to use the more involved formula E = hc/\lambda, where now c is the speed of light and \lambda is the wavelength. I remembered that h=6 \times 10^{-34} Js, and that c=3 \times 10^8 m/s, so with some effort I could get E=6 \times 10^{-34} \times 3 \times 10^8 / (5 \times 10^-7) = 4 \times 10^{-19}, more or less.

My brains were a bit strained by the simple calculation above, but I was relieved to get back an energy roughly equal to that I expected -in the eV range (one eV equals 1.6 \times 10^{-19} Joules -that much I do know).

Now, if the Sun radiates 1 kW of power, which is a thousand Joules per second, how many visible photons do we get ? Here there is a subtlety I did not even bother considering in my walk to the physics department: only about half of the power from the Sun is in the form of visible light, so one should divide that power by two. But I was unhindered by this in my order-of-magnitude walk-estimate. Of course, 1 kW divided by 4 \times 10^{-19} makes 2.5 \times 10^{21} visible quanta of light per square meter per second.

Now, visual magnitude is expressed as the amount of light hitting the eye. A human eye has a surface of about 20 square millimeters, which is 20 millionths of a square meter: so the number of photons you get by looking straight at the sun (do not do it) is 1.2 \times 10^{14} per second. That’s a hundred trillions of ‘em photons per second!

I was close to my goal now: the magnitude of the speck of galaxy I saw on Saturday is +15.7, the magnitude of the Sun is -27, so the difference is 43 magnitudes. This corresponds to 2.5^{43}, which you might throw up your hands at, until you realize that every 5 units of the exponent the number increases by 100, so you just do 100^{43/5} which is 100^{8.6} which is 10^{17.2}… Simple, isn’t it ?

Now, taking the number of photons reaching the eye from the Sun every second, and dividing by the ratio of apparent luminosities of the Sun and the galaxy, I could get N_{\gamma}=10^{14} / 10^{17} = 10^{-3}. One photon every thousand seconds!

Let me stress this: if you watch that patch of sky at night, the number of photons you get from that source alone is a few per hour! With my dobson telescope, which intensifies light by almost 10,000 times, I could get a rate of a few tens of photons per second, and the detail was indeed detectable!

If you are intested in the exact number, which I worked out after reaching my office and the tables of constants in the PDG booklet, I computed a rate of N_{\gamma}=3.4 \times 10^{-3} photons per second with unaided eye, and 22 per second through the eyepiece of the telescope. Without telescope, that galaxy sends to each of us about 10 photons per hour!

UPDATE: this post will remain as one clear example of how dangerous it is to compute by heart! Indeed, somewhere in my order-of-magnitude conversions above I dropped a factor 10^2 -which, mind you, is not horrible in numbers which have 20 digits or so; but when one wants to get back to reasonable estimates for reasonably small numbers, it does count a lot. So, after taking care of some other (more legitimate) approximations, if one computes things correctly, the number of photons from the galaxy seen with the unaided eye is more like two hundred per hour, and in the telescope it is of about 350 per second.

Black holes, the winged seeds of our Universe January 8, 2009

Posted by dorigo in astronomy, cosmology, news, science.
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From Percy Bysshe Shelley’s “Ode to the West Wind” (1819), one of my favourite poems:

[...]O thou,
Who chariotest to their dark wintry bed
The winged seeds, where they lie cold and low,
Each like a corpse within its grave, until
Thine azure sister of the Spring shall blow
Her clarion o’er the dreaming earth, and fill
(Driving sweet buds like flocks to feed in air)
With living hues and odors plain and hill:
Wild Spirit, which art moving everywhere;
Destroyer and preserver; hear, oh, hear!

The winged seeds -of galaxies, and ultimately of everything that there is to see in our Universe- appear today to be black holes: this is what emerges from the studies of Chris Carilli, of the National Radio Astronomy Observatory (NRAO). In a press release of January 6th, Carilli explains that the evidence that black holes are antecedent to galaxy formation is piling up.

In a nutshell, there appears to be a constant ratio between the mass of objects like galaxies and giant globular clusters and the black hole they contain at their center. This has been known for a while -I learned it at a intriguing talk by Al Stebbins at the “Outstanding Questions in Cosmology” conference, in March 2007 at the Imperial College of London. But what has been discovered more recently is that the very oldest objects contain more massive black holes than expected, a sign that black holes started growing earlier than their surroundings.

This is incredibly interesting, and I confess I had always suspected it, when looking at the beautiful spiral galaxies, attracted in a giant vortex by their massive center. I think this realization is a true gate to a deeper understanding of our Universe and its formation. A thought today goes to Louise, who has always held that black holes have a special role in the formation of our Universe.

The Worldwide telescope May 13, 2008

Posted by dorigo in astronomy, computers, cosmology, internet, science.
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Jeff pointed out to me today the remarkable world wide telescope, a site where you can download a software created by Microsoft to browse the heavens as if you were commanding a powerful telescope. The constellations are not maps, but actual pictures, into which you can zoom as much as the images of the digital sky surveys (SDSS and others) allow.

My jaw dropped as I started using the software, which you can download and install on your computer, and which works pretty much like google Earth – downloading the region you are visualizing from the internet. A nice feature is the appearance of a frame of thumbnail pictures around the zoomed area, highlighting the most interesting celestial objects present there. If you click once on each pic the relevant object is highlighted on the map; clicking twice will allow you to download full-resolution image of the object directly from the online databases, including Hubble images.

What I find amazing, however, is the fact that browsing the night sky becomes a thrilling experience at your fingertips in front of the computer. The realism is perfect – these are pictures, in pure google earth style. However, while we never have the need to find a feature on the Earth surface by hovering over it in our real life, that is exactly what we do when we observe the night sky: so the learning experience provided by the program for a user who wants to get better at locating celestial objects is invaluable.

Above you can see a screenshot of part of the WWT window, which I centered on the Deer Lick group of galaxies – NGC7331, a milky way-like galaxy which is the largest member of the group, is on top. Below you can see Stephan’s quintet – a group of five small galaxies of 13th-14th magnitude which is among my favorite targets in deep-sky observing sessions. By zooming in (below), you get to see stars fainter than 18th magnitude, at a resolution comparable to that of  a meter-class instrument. Amazing!

I highly recommend downloading the software. Learning to locate objects will become a wonderful pastime!


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