Orders of Magnitude PTYS 206-2 29 January 2008

Example 1: The National Debt How much of the national debt do you owe? The National Debt is $9,199,929,450,821. (as of Sunday night at 9:00 EST). In scientific notation: 9.199294508211012 USD (US Dollars). To 2 significant digits: 9.21012 USD. To an order-of-magnitude: 1013 USD. Number of people in the USA = 301,139,947= 3108. Number of wage earners = 1.25108, or to an order-of-magnitude: 108 Debt per Wage Earner = 1013 USD 108 =105 USD=$100,000.

Example 3: Chickens How many chickens are there in the USA? Average person eats 1 chicken per week or 50 chickens per year. There are roughly 3108 average people in the USA. That’s roughly 50(3108) = 1.51010 chickens consumed per year. 1.51010 is 15 billion chickens. Chickens out number us 50 to 1. Chickens are slaughtered at an age of 6 weeks. The average commercial breeder chicken produces 1000 chicks per year. That means that 15 million breeder chickens are needed to produce the 15 billion chickens that we eat.

Example 4: Piano Tuners in Chicago How many piano tuners are there in Chicago? How many people in Chicago? How many people per household? How many households have pianos? How often are piano’s tuned? How long does it take to tune a piano (include travel)? How many hours does a piano tuner work per year?

Example 4: Piano Tuners in Chicago How many people in Chicago? 5,000,000 = 5106 How many people per household? 2 How many households have pianos? 1/20 How often are piano’s tuned? 1 per year How long does it take to tune a piano (include travel)? 2 hours How many hours does a piano tuner work per year? 2000 # of Tunings/year = (5106/2) (1/20)=125,000 Average tuner does = 2000 hours/2 hours = 1000 tunings There must be 125 tuners to do 125,000 tunings/year.

Enrico Fermi: Born: Rome, Sept 29, 1901 Died: Chicago, Nov 28, 1954 Fermi was one of the leading physicists of the 20th century. He was unusual in that he excelled as a theorist and as an experimentalist. He led the team that achieved the first sustained nuclear reaction, thereby leading the way to nuclear power plants. An ardent foe of fascism he fled his native Italy in 1939 for the US where he had a leading role in the Manhattan project. He later made fundamental contributions to sub-atomic physics. Element 100 on the periodic table (Fermium) is named in his honor. Fermi was known for being able estimate the solutions to complex problems quickly and accurately. The order of magnitude problems we’re discussing are sometimes called Fermi problems.

Example: Drake Equation How many civilizations are there in the Universe that we can communicate with? N is the number of stars with technological civilizations sending out radio signals. N* is the total number of stars in the galaxy

Aside on Stars, Galaxies, and the Universe This image from the Hubble Space Telescope shows the distribution of galaxies in the darkest region of the sky. The objects in the image are galaxies. An average galaxy is made up of billions of stars.

Galaxies An artist’s conception of our galaxy, the milky way This is an image of galaxy NGC3894 The milky way at night

Drake Equation fp is the fraction of those stars that have planets ne is the average number of planets that can potentially support life fℓ is the fraction of the above that actually go on to develop life at some point fi is the fraction of the above that actually go on to develop intelligent life fc is the fraction of civilizations that develop a technology that releases detectable signs of their existence into space L is the length of time such civilizations release detectable signals into space. T is the age of the Galaxy

Drake Equation N* =1011stars fp = 0.5 (half of all stars have planets) Ne = 2 (2 stars/planet will be able to develop life) fℓ=1 (100% of planets that are able will develop life) fi =0.01 (1% of the time the life will evolve to intelligence) fc =0.01 (1% of the time the life will be able to communicate across space) L =104 years (Tech. civilizations last on average 10,000 years) T = 1010 (the galaxy is 10 billion years old) N = 1011  0.5  2 1  0.01  0.01  104 / 1010 =10