Orders of Magnitude

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: 9.199294508211012 USD (US Dollars). To 2 significant digits: 9.21012 USD. To an order-of-magnitude: 1013

USD.

Number of people in the USA = 301,139,947= 3108.

Number of wage earners = 1.25108, or to an order-of-magnitude: 108

Debt per Wage Earner = 1013 USD 108 =105

USD=$100,000.

Example 3: ChickensHow many chickens are there in the USA?

Average person eats 1 chicken per week or 50 chickens per year. There are roughly 3108 average people in the USA. That’s roughly 50(3108) = 1.51010 chickens consumed per year. 1.51010 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 = 5106

• 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 = (5106/2) (1/20)=125,000• Average tuner does = 2000 hours/2 hours = 1000

tunings• There must be 125 tuners to do 125,000 tunings/year.

QuickTime™ and a decompressor

are needed to see this picture.

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.

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.

Example: Drake Equation• How many civilizations are there in the Universe that

we can communicate with?

QuickTime™ and a decompressor

are needed to see this picture.

N is the number of stars with technological civilizations sending out radio signals.N* is the total number of stars in the galaxy

QuickTime™ and a decompressor

are needed to see this picture.

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

QuickTime™ and a decompressor

are needed to see this picture.

This is an image of galaxy NGC3894

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

An artist’s conception of our galaxy, the milky way

The milky way at night

Drake Equationfp is the fraction of those stars that have planetsne is the average number of planets that can potentially support lifefℓ is the fraction of the above that actually go on to develop life at some pointfi is the fraction of the above that actually go on to develop intelligent lifefc is the fraction of civilizations that develop a technology that releases detectable signs of their existence into spaceL is the length of time such civilizations release detectable signals into space.T is the age of the Galaxy

Drake EquationN* =1011starsfp = 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