# 1 Modelling the Universe .pdf

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1

MODELING THE UNIVERSE

It is almost impossible to conduct direct experiments in astronomy. The timescales,

size scales and energy scales rule out virtually any attempt to create astronomical conditions in the laboratory. Thus astronomy is par excellence the science that requires

modeling at every stage of the process, from simulating the performance of telescopes

in orbit, to analysing data, to performing detailed simulations of the evolution of stars

and galaxies on cosmic timescales. Indeed, modeling provides the foundation for the

astonishing increase in our knowledge of the Universe over the last half century.

In my lecture I will illustrate how modeling is used in astronomy using a wide

range of examples. How did modeling permit the flawless repair of the Hubble Space

Telescope in orbit? What do astronomers mean when they say that they ‘listen’ to the

stars? What is the ‘field of streams’ and how does it give us insight into the ‘Dark

Matter’ that pervades our Galaxy? I will draw on modeling examples using the work

of some of my PhD students. How do you make a star cluster in a computer? How can

you detect a planet that will never be ‘seen’ directly because it is still embedded in the

disc of dust and gas from which it formed?

To prepare for this lecture, have a think about the following questions.

1)To google:

What is an emission line? What is an absorption line? What determines whether

you see lines in absorption or emission (...and where and how was helium first discovered?).

2) To calculate:

A computer code tracks the passage of a comet around the Sun. The comet is on

a highly elliptical orbit and travels between the ‘Oort Cloud’ (at a large distance RO

from the Sun) to a radius Rmin << RO where it nearly grazes the Sun. At radius R0

the comet has gravitational energy WO (which is less than zero) and positive kinetic

energy KO , with a total energy T = KO + WO (< 0). As it orbits around the Sun, the

individual values of kinetic energy and potential energy (K and W ) vary but - by the

principle of energy conseration - the total energy T = K + W is conserved.

Show that if W at a distance r from the Sun is proportional to 1/r and that if

KO << T then |K/T | ∼ RO /r − 1.

Now consider the case that the computer code introduces an error of 1% in the

calculation of K. Is this error going to be more serious (in terms of giving a wrong

value for the total energy T ) when the comet is near Rmin or near RO ? What will

happen to the comet in the computer simulation if the result of this error is to make

T > 0? (We will discuss in the lecture how computer codes try and get round this

problem and how this is important for a whole class of simulations of galaxies, star

clusters and planetary systems called ‘Nbody codes’)

3) To think about

Another important class of modeling codes involves gas (‘hydrodynamical codes’)

and in this case the gas motion responds to gradients in pressure in the gas. In order to

calculate this, it is important to be able to accurately determine the local gas density

in the code. One type of hydrodynamical code (called a Lagrangian code) models the

2

gas as lots of particles and the most obvious way to calculate the gas density in this

case is to simply place a sampling volume around each particle and calculate the density

by multiplying the number of particles in the volume by the mass of each particle and

then dividing by the volume. Have a think about what problems this might cause.

How should you pick the size of your sampling volume? Think about the effect on

the forces on a particle as neighbouring particles enter and leave the sampling volume.

To see how these problems are dealt with in practice try googling ‘Smoothed Particle

Hydrodynamics’ (SPH). (I shall be showing a number of animations of star and cluster

formation using SPH.)

4) And finally:

The first bit of this question requires a knowledge of circular motion and Newton’s

law of gravitation. If you haven’t covered this then you can simply assume the answer

and move on to (*). Show that the speed of a satellite in circular motion at radius r

about the centre of the Earth is given by v 2 = GM/r (where M is the mass of the Earth

and G is constant). Why does atmospheric friction cause a satellite to speed up?

(*) A disc of dusty gas orbits a young star with its axis of rotation being in the z

direction. The gas at each radius r is in ciruclar motion with the speed given above.

Sketch the disc in the x-y plane and draw on it contours of equal projected velocity along

the line of sight as seen by an observer situated at some point well outside the disc on

the y axis.

The Doppler effect shifts the frequency of an emission line (see question 1) from

its ‘rest frequency’ to a different ‘observed’ frequency with the shift being simply proportional to the velocity of the emitting material projected along the observer’s line of

sight. If the emitting material approaches the observer, the line is blue shifted and if

it is receding then the line is redshifted. Consider emission lines being produced from

the gas in the disc sketched above and assume that the amount of light that is shifted

by a given amount depends on the area of the disc that is emitting at each projected

velocity. If the line emission is at a single ‘rest frequency’ have a think about how much

light is received at each frequency by the observer. This doesn’t require a detailed calculation - you should try and sketch the ‘line shape’ (i.e. how much light is received by

the observer as a function of frequency). (We will be discussing how line shapes can be

used to detect discs around young stars and how can even show the effects of embedded

planets in such discs.)

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