The Agency for Toxic Substances and Disease Registry which is an agency of the Center for Disease Control (CDC), recommends an annual dose of no more than 500 mrem/year (including the 300 mrem or so from background).
Calculating the Radiation from Cassini -- Outline of Calculation
Here's an outline of how we'll estimate our radiation exposure from a
- 1. We'll assume ALL the plutonium aboard Cassini gets vaporized.
- 2. Then we'll calculate the resulting concentration of plutonium in the
- 3. We'll use that concentration to estimate how much each person is likely
- 4. Finally, we'll figure how much energy gets absorbed in the body due to the alpha radiation emitted by that plutonium and express that energy (called the dose) in mrem and see how it compares to the level of background radiation described above.
Ready to start?
NOTATION: To write numbers and equations in ASCII text, I'll use the
I'll also use this notation to describe units like meter^2 for square meters, etc...
Concentration of Plutonium in the Atmosphere
Cassini carries about 33 kilograms (72 pounds) of PuO2 ceramic. That contains about 29 kilograms of plutonium. Let's assume the unlikely "worst-case" accident occurs and Cassini burns up entering the Earths atmosphere at 43,000 mph. The RTGs might rupture and release some of the plutonium. The Apollo 13 Lunar Module burned up returning from the Moon at 25,000 mph but the RTG stayed intact and didn't release any plutonium, so it might seem reasonable that the Cassini RTG would stay at least partially intact and release only a fraction of its plutonium, but we want to consider the worst possible accident so we'll assume ALL of the plutonium gets vaporized.
Burning up high in the atmosphere (say 50 km), the bits of plutonium will be distributed all over the Earth by high altitude winds and gradually settle to Earth over a period of years, just like bits of dust from meteors and volcanic eruptions. The volume of the Earth's atmosphere up to a height, h, is just V = 4 pi R^2 h where R is the radius of the Earth.
R is 6380 km (which is 6.38 x 10^6 meters) and h is 50 km, so
N^2 = N squared = N x N
N^3 = N cubed = N x N x N
2.0 x 10^3 = 2000, etc...
V = 4 (3.14) (6.38 x 10^6 meters)^2 (5.0 x 10^4 meters)
V = 2.6 x 10^19 meters^3.
Dividing this volume into the mass of the plutonium shows us the
concentration in the atmosphere is 1.1 x 10^-18 kilograms per cubic meter
(kg/m^3). I'll call this number "C" (for "concentration") in future equations.
How Much Gets Inhaled?
Remember, this plutonium is only dangerous if it gets inhaled and stays in the body. Only particles of plutonium close to the Earth's surface (no higher than your nose) can be inhaled. A small fraction of the plutonium in the atmosphere will settle down toward the surface over several years.
To overestimate the risk again, we'll assume that ALL the plutonium over the land area of the Earth is inhaled. Since 3/4 of the Earths surface is covered by the oceans, we'll assume that means that 1/4 of the plutonium contained in the concentration described above gets into your lungs (remember PuO2 is not soluble, so what lands in the oceans settles to the bottom of the ocean and doesn't enter the food chain).
We could ignore the plutonium that settles into uninhabited areas like Antartica, but once again we'll OVERestimate the risk and include those areas.
In reality, only a small fraction of the plutonium particles will be small enough to be inhaled. Anything larger than about 10 microns will be filtered y the nose or the cillia in the lungs and exhaled. To be safe however, we will once again OVERestimate the danger and assume ALL the particles get inhaled (Are you keeping track of how much we do this?).
A human being inhales about 0.1 liters (0.1 liter = 0.0001 meter^3) of air every second (depending upon the size of the person). Multiply by the number of seconds in one year and we find we inhale 3150 meter^3 of air per year I'll call this volume "Vyr".
Multiplying this volume by the atmospheric concentration, C, tells us how much plutonium each person breates in a year. If every cubic meter of that air contains the concentration of plutonium calculated above, every person on the surface of the Earth will inhale an amount of plutonium we'll call m, where
m = 1/4 C Vyr
m = 1/4 (1.1 x 10^-18 kilogram/meter^3) (3150 meter^3)
m = 8.7 x 10^-16 kilogram.
That's how much plutonium we would each inhale in one year. If we recall our high school chemistry, we can multiply that mass by ogadro's Number (6.02 x 10^23) and divide by the gram atomic weight of plutonium (0.238 kilogram) to see that we will each inhale about 2.2 x 10^9 plutonium atoms.
Now lets ask how much radiation we get from that plutonium.
Energy Absorbed From the Plutonium -- The Number of Alpha Particles
Remember that each atom of plutonium will emit it's alpha particle at a random time. The half-life, we'll call it "T," measures how long it takes for half the atoms in any sample to emit their alpha particles. The half-life for Pu-238 is 88 years. The mathematics of probability tells us that if we
have N atoms, the number that will emit alpha particles in one year is equal
to N ln(2)/T, where ln is the natural logarithm (the ln button on your calculator) and T is the half-life.
This means that the amount of radiation emitted decreases with time as the number of Pu-238 atoms decreases, but once again we'll overestimate the risk and assume N stays constant at the value N = 2.2 x 10^9 which we determined above. So the number of alpha particles emitted by the plutonium inside the body is:
Nalpha = N ln(2)/T
Nalpha = (2.2 x 10^9) ln(2) /88years
Nalpha = 1.7 x 10^7 alpha particles per year.
The Energy of Alpha Particles
An atom of plutonium emits an alpha particle with an energy of 5 MeV. The MeV is a unit of energy used in nuclear physics. For our purposes, I'll express that energy in the more common metric unit of joules (5 MeV = 8.0 x 10^-13 joule). Remember from the discussion above that a rad is defined as 1 joule absorbed in a body mass of 100 kilograms. Since each alpha particle packs an energy of 8.0 x 10^-13 joule, that means each alpha particle contributes 8.0 x 10^-13 rad to a 100 kilogram person.
If we multiply that by the number of alpha particles emitted per year, we find that the radiation dose from the Cassini plutonium is:
dose = Nalpha x 8.0 x 10^-13 rad
dose = (1.7 x 10^7) (8.0 x 10^-13) rad/year
dose = 1.4 x 10^-5 rad/year
To convert this to units of rem which measure the biological effects, we multiply by 10. So:
dose = 1.4 x 10^-4 rem/year
which is only 0.14 mrem/year! Remember that we get about 300 rem/year from natural background, so this additional 0.14 mrem from a hypothetical Cassini accident is INSIGNIFICANT! You get a lot more radiation from x-rays, airline flights, and just random changes in the background.
Let me repeat that result:
And remember this was an OVERestimate of the "worst-case" accident! If the radiation stayed constant at this rate for 50 years, it would add up to 7 mrem over that time. Of course, it actually decreases with time, so this rough estimate compares very favorably with the much more careful analysis done by NASA which predicts an exposure of about 1 mrem over 50 years.
So much for the obviously false statements of the Cassini opponents that an accident like this would "kill millions"!
Compliments of Proposition One Committee
Cassini dose = 0.14 mrem/year
Background dose = 300 mrem/year