Homeworks 8

Homework 8 due on 11/02/09

Problem 1

A globule in a molecular cloud (75 % H, 25 % He) has a mean density of 10⁹ particles/ccm. Use the Jean’s criterion to find the temperature T at which a star of 2 solar masses can form. Do stars with lower or higher mass form under the same condition? Explain the meaning of the Jean’s lengths R, and calculate it for the example above. Compare the result with the typical size of globules in the Orion nebula.

Problem 2

Describe a planetary nebula. Describe the central star of a planetary nebula. What kind of star forms a planetary nebula? What kind of star will the central star of a planetary nebula soon become? Sketch the spectrum of a PN, and label the main characteristics. Estimate the lifetime of a PN.

Problem 3

Why do the orbits of the planets in the solar system all lie in a plane, and why does that plane lie on an extension of the sun’s equator into space? What happened to the material in the protoplanetary disk that did not become part of the planets?

Present models of the process suggest that a cloud of gas, which was called the solar nebula and was about 100 times the Earth-Sun distance across and 2-3 times the mass of the Sun, was jolted into action by a nearby exploding star (a supernova). This shockwave squeezed the nebula and caused it to begin to collapse under its own gravity. As it did so conservation of angular momentum resulted in a flat disc of spinning material, called a protoplanetary disc, that surrounded a developing "proto-star", the future Sun, at its center.

The disc was effectively a proto-planetary soup of material, which slowly coaslesced to form initially planetessimals (baby planets) and then the larger individual planets we see today. The inner part of the disc, closest to the proto-star, were too hot for volatile and gaseous materials to condense, so the inner planets (Mercury, Venus, Earth, Mars) are all metal and silicate-rich "rocky" bodies.

Farther out, where the disc was cooler, lighter elements such as hydrogen could be captured and the gas giants Jupiter and Saturn formed. The asteroid field between Jupiter and Mars was the result of Jupiter's gravity. The massive planet prevented the debris in this part of the solar system from merging together to form an additional planet so it remains as an asteroid belt.

Since all of the planets formed from a disc of material they all lie on the same plane. They also all spin because the material that formed them was itself spinning. As the planets formed they also underwent a kind of cosmic gravitational billiards where resonanaces caused by their gravitational fields nudged everything around until it arrived in its present position. A space scientist called Adrian Brunini recently published a paper in Nature in which he modelled the early solar system and found that these resonances could account for the positions and eccentric orbits of some planets, and why Uranus is spinning on its side - it's been gravitationally tipped over during its development.

The material in the protoplanetary disk that did not become part of the planets became asteroids instead.

Problem 4

Draw the HR diagram of the stars in a typical globular cluster, labeling the various parts of the diagram. What is the mass of the stars at the main-sequence turn off of a typical globular cluster. Why are these stars leaving the main sequence? How old are these stars and, therefore, how old is a typical globular cluster? Use the HR diagram below to determine the age of this cluster (Write down the steps!)

Problem 5

The mass of a black hole is 15 M⊙. What is its radius? Does a black hole have a surface at that radius? What, then, is the significance of a black hole’s radius? Suppose you measure the gravity from a black hole with a mass of 1 M⊙ at a distance of 1 AU from its center. How does the gravity compare to the gravity from the sun at 1 AU? What are the maximum masses of white dwarfs, neutron stars, and black holes?