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.

Jean's Criterion

M_c > M_J

Molecular Cloud in Orion

• T = 10 K;
• n = 1E4 particles/ccm;
• ρ₀ = 3E-14 g/cm
• M_J = 10 Mʘ

Problem 2

Describe a planetary nebula.

The expanding shell of gas around a white dwarf progenitor is called a planetary nebula. These beautiful glowing clouds of gas were given this name in the 19th century because, when viewed through a small telescope, they look somewhat like giant gaseous planets.

Describe the central star of a planetary nebula.

A planetary nebular owes its appearance to the UV light emitted by the hot condensed central star. The UV photons are absorbed by the gas in the nebula, causing the atoms to become excited or ionized. When the electrons cascade back down to lower energy levels, photons are emitted, whose wavelengths are in the visible portion of the electromagnetic spectrum. As a result, the cloud appears to glow in visible light.

What kind of star forms a planetary nebula?

AGB stars

What kind of star will the central star of a planetary nebula soon become?

White Dwarfs

Sketch the spectrum of a PN, and label the main characteristics.

Estimate the lifetime of a PN.

Combined with characteristic length scales of around 0.3pc, their estimated ages are on the order of 10,000 years. After only about 50,000 years, a planetary nebula will dissipate into the ISM. Compared with the entire lifetime of a star, the phase of planetary nebula ejection is fleeting indeed.

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.

The stars just leaving the main sequence have masses near 0.9 Mʘ.

Why are these stars leaving the main sequence?

The stars have consumed all of the hydrogen in their cores and therefore must evolve.

How old are these stars and, therefore, how old is a typical globular cluster?

Their age is from 11 × 10⁹ years to 13 × 10⁹ years

Use the HR diagram below to determine the age of this cluster (Write down the steps!)

When a cluster of stars is formed at about the same time, the life span of these stars will depend on their individual masses.

The most massive stars will leave the main sequence first, followed steadily in sequence by stars of ever lower masses.

Thus the stars will evolve in order of their position on the main sequence, proceeding from the most massive at the left toward the right of the HR diagram.

The current position where stars in this cluster are leaving the main sequence is known as the turn-off point. By knowing the main sequence lifespan of stars at this point, it becomes possible to estimate the age of the cluster

The age of this cluster is “Much Older Cluster,” at approximately ~ 13 × 10⁹ years

Problem 5

The mass of a black hole is 15 M⊙. What is its radius?

R × 1 Mʘ = 3 km

R × 15 Mʘ = 45km

Does a black hole have a surface at that radius?

There is no solid surface at the boundary of a black hole.

What, then, is the significance of a black hole’s radius?

The “radius” of a black hole is actually its event horizon.

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?

The gravity of a black hole of 1 Mʘ is the same as the gravity of the sun

What are the maximum masses of white dwarfs, neutron stars, and black holes?

The maximum mass of a white dwarf is 1.4 M⊙. The maximum mass of both neutron stars and black holes is 10⁹ Mʘ .