What Happens When Two Black Holes Collide?

March 10, 2013

A student asked this question about black holes during a discussion, and I didn’t have a good answer. Now there’s this:

A study last year found unusually high levels of the isotope carbon-14 in ancient rings of Japanese cedar trees and a corresponding spike in beryllium-10 in Antarctic ice.

The readings were traced back to a point in AD 774 or 775, suggesting that during that period the Earth was hit by an intense burst of radiation, but researchers were initially unable to determine its cause.

Now a separate team of astronomers have suggested it could have been due to the collision of two compact stellar remnants such as black holes, neutron stars or white dwarfs.

— via The Weather Channel (2013): Black Hole Collision May Have Irradiated Earth in 8th Century.

From the original article:

While long [Gamma Ray Bursts (GRBs)] are caused by the core collapse of a very massive star, short GRBs are explained by the merger of two compact objects … [such as] a neutron star with either a black hole becoming a more massive black hole, or with another neutron star becoming either a relatively massive stable neutron star or otherwise a black hole.

— Hambaryan and Neuhäuser (2013): A Galactic short gamma-ray burst as cause for the 14C peak in AD 774/5 in

More info via The Telegraph, and the original article discussing the spike in carbon-14 in tree rings is here.

Citing this post: Urbano, L., 2013. What Happens When Two Black Holes Collide?, Retrieved May 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Albedo and Absorption

March 6, 2013

Ice melts around an embedded leaf, taking the pattern of the leaf.

Darker colored objects absorb more light than lighter colored objects. Darker objects reflect less light; they have a lower albedo. So a deep brown leaf embedded in the ice will absorb more heat than the clear ice around it, warming up the leaf and melting the ice in contact with it. The result, is melting ice with shape and pattern of the leaf. It’s rather neat.

Citing this post: Urbano, L., 2013. Albedo and Absorption, Retrieved May 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Living next to Chernobyl

November 10, 2012

Hanna Zavorotnya, 78, lives in Chernobyl’s dead zone. Image from RENA EFFENDI/ INSITUTE via The Telegraph.

We were talking about environmental disasters, specifically nuclear radiation, and looking at pictures of Chernobyl, when a student asked if anyone still lived there. The city and surrounding region was evacuated, however some 1,200 people returned to their homes. Holly Morris has an interesting article on how “The women living in Chernobyl’s toxic wasteland” survive. Curiously, 80% of the remaining survivors are female.

Citing this post: Urbano, L., 2012. Living next to Chernobyl, Retrieved May 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

A Fatal Dose of Bananas

October 17, 2011

Banana: 1 μS.

In my last physics exam, I asked how many bananas would it take to deliver a fatal dose of radiation. This question came up when we were discussing different types of radiation and looking at this graph. One banana gives you about 0.1 microSieverts, while the usually fatal dosage is about 4 Sieverts. That means 4 million bananas. Michael Blastland uses the instantly fatal dosage of 8 Sieverts to make his estimate of eight million.

Usually Fatal Dose: 4 S.

My students were insistent, “Would eating four million bananas really kill you with radiation?”

My answer was, “Yes. But other problems might arise if you try to eat four million bananas.”

Citing this post: Urbano, L., 2011. A Fatal Dose of Bananas, Retrieved May 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Global Temperature Model: An Application of Conservation of Energy

August 19, 2011

Energy cannot be either created or destroyed, just changed from one form to another. That is one of the fundamental insights into the way the universe works. In physics it’s referred to as the Law of Conservation of Energy, and is the basic starting point for solving a lot of physical problems. One great example is calculating the average temperature of the Earth, based on the balance between the amount of energy it receives from the Sun, versus the amount of energy it radiates into space.

The Temperature of Radiation

Anything with a temperature that’s not at absolute zero is giving off energy. You right now are radiating heat. Since temperature is a way of measuring the amount of energy in an object (it’s part of its internal energy), when you give off heat energy it lowers your body temperature. The equation that links the amount of radiation to the temperature is called the Stefan-Boltzman Law:

 E_R = s T^4

where:
ER = energy radiated (W/m-2)
T = temperature (in Kelvin)
s = constant (5.67 x 10-8 W m-2 K-4)

Now if we know the surface area of the Earth (and assume the entire area is radiating energy), we can calculate how much energy is given off if we know the average global temperature (the radius of the Earth = 6371 km ). But the temperature is what we’re trying to find, so instead we’re going to have to figure out the amount of energy the Earth radiates. And for this, fortunately, we have the conservation of energy law.

Energy Balance for the Earth

Simply put, the amount of energy the Earth radiates has to be equal to the amount of energy gets from the Sun. If the Earth got more energy than it radiated the temperature would go up, if it got less the temperature would go down. Seen from space, the average temperature of the Earth from year to year stays about the same; global warming is actually a different issue.

So the energy radiated (ER) must be equal to the energy absorbed (EA) by the Earth.

 E_R = E_A

Now we just have to figure out the amount of solar energy that’s absorbed.

Incoming Solar Radiation

The Sun delivers 1367 Watts of energy for every square meter it hits directly on the Earth (1367 W/m-2). Not all of it is absorbed though, but since the energy in solar radiation can’t just disappear, we can account for it simply:

  • Some if the light energy just bounces off back into space. On average, the Earth reflects about 30% of the light. The term for the fraction reflected is albedo.
  • What’s not reflected is absorbed.

So now, if we know how many square meters of sunlight hit the Earth, we can calculate the total energy absorbed by the Earth.

The solar energy absorbed (incoming minus reflected) equals the outgoing radiation.

With this information, some algebra, a little geometry (area of a circle and surface area of a sphere) and the ability to convert units (km to m and celcius to kelvin), a student in high-school physics should be able to calculate the Earth’s average temperature. Students who grow up in non-metric societies might want to convert their final answer into Fahrenheit so they and their peers can get a better feel for the numbers.

What they should find is that their result is much lower than that actual average surface temperature of the globe of 15 deg. Celcius. That’s because of how the atmosphere traps heat near the surface because of the greenhouse effect. However, if you look at the average global temperature at the top of the atmosphere, it should be very close to your result.

They also should be able to point out a lot of the flaws in the model above, but these all (hopefully) come from the assumptions we make to simplify the problem to make it tractable. Simplifications are what scientists do. This energy balance model is very basic, but it’s the place to start. In fact, these basic principles are at the core of energy balance models of the Earth’s climate system (Budyko, 1969 is an early example). The evolution of today’s more complex models come from the systematic refinement of each of our simplifications.

Advanced Work

If students do all the algebra for this project first, and then plug in the numbers they should end up with an equation relating temperature to a number of things. This is essentially a model of the temperature of the Earth and what scientists would do with a model like this is change the parameters a bit to see what would happen in different scenarios.

Feedback

Global climate change might result in less snow in the polar latitudes, which would decrease the albedo of the earth by a few percent. How would that change the average global temperature?

Alternatively, there could be more snow due to increased evaporation from the oceans, which would mean an increase in albedo …

This would be a good chance to talk about systems and feedback since these two scenarios would result in different types of feedback, one positive and one negative (I’m not saying which is which).

Technology / Programming

Setting up an Excel spreadsheet with all the numbers in it would give practice with Excel, make it easier for the student to see the result of small changes, and even to graph changes. They could try varying albedo or the solar constant by 1% through 5% to see if changes are linear or not (though they should be able to tell this from the equation).

A small program could be written to simulate time. This is a steady-state model, but you could assume a certain percent change per year and see how that unfolds. This would probably be easier as an Excel spreadsheet, but the programming would be useful practice.

Of course this could also be the jumping off point for a lot of research into climate change, but that would be a much bigger project.

References

Yochanan Kushnir has a page/lecture that treats this type of zero-dimesional, energy balance model in a little more detail.

Citing this post: Urbano, L., 2011. Global Temperature Model: An Application of Conservation of Energy, Retrieved May 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Radiation dosages

March 22, 2011

Radiaton dosages from different sources. Graph by http://xkcd.com/radiation/.

xkcd has published an excellent graph showing where different dosages of radiation come from and how they affect health. It’s a complex figure, but it’s worth taking the time to look through. I find it easiest to interpret going backward from the bottom right corner that show the dosages that are clearly fatal.

One Seivert (1 Sv).

One red square of 100 red blocks is equal to one seivert, which is the radiation dosage that will kill you if you receive it all at once. Note:

  • If you were next to the reactor core during the Chernobyl nuclear accident, you would have gotten blasted by 50 Sv.
  • 8 Sv will kill you, even with treatment.
  • Getting 0.1 Sv over a year is clearly linked to cancer.
  • One hour on the grounds of the Chernobyl nuclear plant (in 2010) would give you 0.006 Sv.
  • Your normal, yearly dose is about 0.004 Sv, just about how much was measured over a day at two sites near Fukushima.
  • Eating a banana will give you 0.000001 Sv.

While I did not find equivalent exposure levels, the nuclear bombs dropped on Hiroshima and Nagasaki lead to many deaths and sickness from radiation created by the explosions. Within four months, there were 140,000 fatalities in Hiroshima, and 70,000 in Nagasaki (Nave, 2010). The Manhattan Engineer District, 1946 report describes the radiation effects over the first month:

Radiation effects for the month following the dropping of the nuclear bombs on Nagasaki and Hiroshima. (Table from The Manhattan Engineer District (1946) via atomicarchive.com).

The effects were not limited to the explosion itself, though. There is one estimate, that 260,000 people were indirectly affected:

Radiation dose in a zone 2 kilometers from the hypocenter of the atomic bomb was the largest. Also, those who entered the city of Hiroshima or Nagasaki soon after the atomic bomb detonation and people in the black rain areas were exposed to radiation. … some people were exposed to radiation from black rain containing nuclear fission products (“ashes of death”), and others to radiation induced by neutrons absorbed by the soil upon entering these cities soon after the atomic bomb detonation.

— Hiroshima International Council for Health Care for the Radiation Exposed (HICARE): Global Radiation Exposures.

HICARE also has a good summary of what happened at Chernobyl, where 31 people died at the time of the accident, about 400,000 were evacuated, and anywhere between 1.6 and 9 million people were exposed to radiation. Modern pictures of the desolation of Chernobyl are here. The Wikipedia article has before and after pictures of Hiroshima and Nagasaki.

Citing this post: Urbano, L., 2011. Radiation dosages, Retrieved May 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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