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Homework 3

ASTR 130

Due: Sept 27th before class begins

34 possible points Instructions to students: Please type your answers whenever possible. Feel free to edit this document and upload it to Canvas. 1. [6 pts Motivation: understand the origin of chemical elements in the universe.] 1a. [2 pts] Massive stars can synthesize elements up to iron by nucleosynthesis. Why is iron the heaviest element that can be synthesized this way? Iron is the heaviest element that can be synthesized by nucleosynthesis because all the elements from helium up to iron are formed through fusion exothermic reactions that release energy; however, elements heavier than iron require higher temperatures and hence an absorption of energy. 1b. [2 pts] Elements heavier than iron can be found in significant amounts on earth. How were they formed if not by nucleosynthesis in a stellar core? Elements heavier than iron are formed through supernova explosions of massive stars. 1c. [2 pts] Do all these heavier elements get returned to the interstellar medium after the death of a star? If not then where else do they eventually get accumulated? These heavier elements eventually get accumulated into future generation stars, while some formed planets because of the proto-stellar disks. Essentially, when the first stars died, they returned to the interstellar medium with heavier elements. Soon enough there was such an accumulation of heavy elements that allowed for the creation of future generation stars 2. [9 pts motivation: understanding how radioactive decay is used to date the ages of rocks]. You are analyzing Moon rocks that contain small amounts of uranium-238, which decays into Pb (lead) with a half-life of about 4.5 billion years. By measuring the rate of decay, you can measure how old the specimen is with the following formula, where N is how much of the radioactive material you have now, N0 is the amount of radioactive material you started with at the “very beginning”, thalf is the half-life in years, t is the time between formation of the material and the current time (i.e., age in years), and ln is a function called the “natural logarithm”, which you should use a calculator (or Google) to find.

(

−t N =exp 1.44∗t half N0

)

Which implies that, Age ( ¿ t )=−ln

( NN )∗1.44∗t 0

half

Homework 3

ASTR 130

Due: Sept 27th before class begins

2a. [3 pts] In one rock from the lunar highlands, you determine that 57% of the original uranium-238 remains; the other 43% decayed into lead. How old is the rock? N ∗1.44∗t half Age ( ¿ t )=−ln N0 57 Age ( ¿ t )=−ln ∗1.44∗( 4.5 × 109 yrs) 100 9 Age ( ¿ t )=3.642530590 ×10 years

( )

( )

2b. [3 pts] In a rock from the lunar maria, you find that 65% of the original uranium-238 remains; the other 35% decayed into lead. Is this rock older or younger than the highlands rock? By how much? N ∗1.44∗t half Age ( ¿ t )=−ln N0

( )

Age ( ¿ t ) =−ln

65 ( 100 ) ∗1.44∗( 4.5 × 10 yrs) 9

Age ( ¿ t ) =2.791473296 ×10 9 years This rock is younger than the highlands rock. It is younger by a difference of 8.51057294 8 ×10 years. Percentage-wise the highlands rock is roughly 1.3 times the age of maria. 3.642530590 × 109 years−2.791473296 × 109 years =8.51057294 × 108 years 3.642530590 ×10 9 years− 2.791473296 × 109 years +1=1.3 2.791473296 ×109 years 2c. [3 pt] Al-26 has a half-life of 717,000 years. Planetesimals are small rocky bodies (like asteroids) that collide and accrete into planets over timescales of millions of years. If a nascent solar system had a particularly high abundance of Al-26, what might this imply for the water content of the planetesimals (and eventually the planets)? Considering Al-26 has such a short half-life, it must’ve gone through many half-lives over a timescale of millions of years. We know that radioactive decay results in a release of energy in the form of heat. In that sense, if a nascent solar system had a particularly high abundance of Al-26, the amount of heat would multiply to such a magnitude that the water content of the planetesimals (of which would eventually form into planets) would be very little if any at all. 3. [9 pts motivation: understanding measurement uncertainty] When counting the number of instances of a random process, there is an associated uncertainty. For example, if you detect an average of 100 photons per second from a distant star, you are unlikely to detect exactly 100 photons every second. The uncertainty for this type of counting statistics is sqrt(N). In this case, sqrt(N=100) = 10, so you’ll usually detect between 90 and 110 photons. This can be called your uncertainty range – it defines an interval within which a numerical result is expected to lie. (Scientists usually use the term “error” instead of “uncertainty”, which isn’t to say that we did something wrong.)

Homework 3

ASTR 130

Due: Sept 27th before class begins

4.6 Billion years ago Sun forms as the pre-solar nebula collapses

Big Bang!

4.5 Billion years ago Planet and Moon Formation: A giant impactor hit the Earth and the debris blown into space caused the formation of the moon

3a. [2 pts] You obtain a piece a coal with 20,000 atoms of carbon-14, which has a half-life of 5730 years. You decide to call your cable company’s customer service, and they put you on hold for 5730 years. How many carbon-14 atoms do you expect to remain? I expect 10,000 atoms of carbon-14 to remain after the 5730 years. This is because after 5730 years, one half-life would have passed for this coal which would reduce the number of atoms of carbon-14 by a factor of 2. 20,000 atoms of carbon-14 reduced by a factor of 2 equals 10,000 atoms of carbon-14. 3b. [3 pts] What’s the uncertainty range for the number of carbon-14 atoms? Sqrt(10,000)=100; The uncertainty range for the number of carbon-14 atoms is 9900-10100 carbon-14 atoms. 3c. [2 pts] How many carbon-14 atoms do you expect after 11,460 years? I would expect 5,000 carbon-14 atoms after 11,460 years. If there are 10,000 carbon-14 atoms after 5730 years, another 5730 years would mean two half-lifes would have passed resulting in the initial 20,000 carbon-14 atoms to reduce by a factor of 4. 20,000 atoms of carbon-14 reduced by a factor of 4 equals 5,000 atoms of carbon-14. 3d. [2 pt] In real life, scientists must measure the ratio of the original product (in this case, C-14) to the daughter product (in this case, C-12) in order to infer the original amount of the original product (C-14). What is one major reason that the inferred age of the substance (rock, bone, etc.) might be wrong and why? (Hint: The half-lives of radioactive elements are known very precisely, so this is not a major source of error.) The inferred age of the substance might be wrong due to error in measurement accuracy. Determining the age of a substance depends on how much parent isotope the sample started with and how much is left. To estimate the initial amount of parent isotope the daughter sample must be used. But a major issue with this is assuming that both isotopes survive in the sample without any sort of contamination. This assumption is almost erratic to hold as it is very possible for the daughter isotope to escape from the sample, adding to either the daughter or parent isot s the uncertainty in mple. nditions on the ear 4. [10 pts. Motivation: u 4a. [2 pts] Sketch a time y solar system that rmation of the Sun and planets, formation of th e heavy bombardm Early Solar System (from the formation of the Sun and planets to late heavy bombardment (LHB))

4.1 Billion years ago to 3.8 Billion Late Heavy Bombardment (LHB) Earth was struck by large amoun Provides evidence for the existe

ASTR 130

ago riod where the meteorites. lunar rocks.

Homework 3

27th before class begins

Present Day

4b. [2 pts] What evidence do we have for the late heavy bombardment? The evidence we have for the late heavy bombardments is the craters on the Moon. Unfortunately, old rocks or craters have not been found on Earth as a result of plate tectonics and heavy erosion. 4c. [1 pt] What effect should collisions during the LHB have on the surface temperature of the Earth (qualitatively and quantitatively)? Collisions during the LHB quantitatively would have increased the surface temperature to 2000 K and vaporized all the ocean water. Qualitatively, the collisions during the LHB would leave a lasting impact due to the increased surface temperature in the sense that the Earth would look like a big ball of magma. One can even go as far as saying that it looks like hell, hence the name of the “Hadean” (for Hades—Greek god of the underworld) era. 4d. [2 pts] Explain why zircon crystals are unusually good for determining the date of formation. Zircon crystals are unusually good for determining the date of formation because their lattice structure admits uranium but no the daughter product (lead). Hence, the uranium/lead ratio can be employed in determining the date of formation. 4e. [3 pts] Explain why the zircon crystals from Jack Hills in Australia indicate that the Earth had cool temperatures and liquid water at the time of the LHB. The zircon crystals from Jack Hills in Australia indicates that the Earth had cool temperatures and liquid water at the time of the LHB due to isotopic oxygen ratios. The high ratios suggest that the material, from which the crystals were formed, consisted of recycled rocks and some form of liquid water....