1.5.3 A note on functions and derivatives

This subsection introduces two crucially important mathematical ideas, functions and derivatives, both of which are used throughout physics.


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Learning outcomes

By the end of this unit you should be able to:

  • identify the charactistics of primates and explain the main differences between the two suborders, prosimians and anthropoids;

  • describe the detection of colour and estimation of distance in primates and explain the advantages of stereoscopic trichromatic vision;

  • discuss the various types of communication seen in anthropoids and explain how playback experiments contribute to understanding vocal communica
    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

8.3 Shortage of minerals

You may be familiar with salt licks that are provided for domesticated cattle. In the wild, grass is also often low in minerals (e.g. it has almost no sodium and very little calcium), so grazers may have to go to extraordinary lengths to supplement their diet with additional minerals obtained from the most unlikely places. LoM gives some examples, but the most impressive activity takes place in the caves of Mount Elgon in Kenya [pp. 113–114]. You'll probably recall this spectacular footage
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

5.3 Hindgut fermenters

The odd-toed ungulates (comprising the order Perissodactyla), the horses, tapirs and rhinoceroses, are hindgut fermenters, as are elephants. Update Table 2 with this information. These animals have a relatively simple, small undivided stomach, but this time an even larger caecum and colon where the microbes are housed and whe
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

6.1 Basic isotropy

As we have said, the photons in the 3 K background have been practically free from interaction with anything since about 4 × 105 years after the instant of the big bang. The present angular distribution of the microwave radiation – the way in which it is spread across the sky – is therefore almost the same as it was then. The spectrum we find today depends on the temperatures at that time – for the intensity of the radiation in a particular region of the early Unive
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

5.3 The redshift of the 3 K radiation

The temperature, T, of the radiation is proportional to the most probable photon energy, E, which as we have said is proportional to f, and hence inversely proportional to the wavelength λ. Thus,

According to Equation 1, we have for the redshift, z


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

5.2 The origin of the 3 K radiation

In speaking of the radiation as having a cosmic origin, what do we have in mind? Essentially this:

In the violent conditions of the early evolution of the Universe, a stage was reached where the matter consisted of a plasma of electrons, protons, neutrons, and some light nuclei such as helium. There were no atoms as such for the simple reason that atoms would have been too fragile to withstand the violence of the collisions that were taking place at the temperature that then existed. As
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

5.1 A second major discovery

In the introduction to this unit, we said that there were three pillars of evidence for the big bang. We now turn to the second. It rests on a discovery that ranks in importance with that of Hubble's law. It came about when observations in a new region of the electromagnetic spectrum – the microwave region – became possible. This was due to the invention of new detectors, working at frequencies as high as 30 000 MHz. In 1965, two Bell Telephone scientists, A. Penzias and R. Wilson, were i
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

4.2 Evidence for a big bang

Having interpreted the redshift as indicating a recessional speed proportional to distance, one may extrapolate into the future to predict how the positions of the galaxies will evolve with time. One can also run the sequence backwards, so to speak, to discuss what their positions were in the past. Clearly, at former times the galaxies were closer to each other.

But not only that. Because of the proportional relationship between speed and distance (Equation 6), at a certain time in the
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

4.1 Hubble's discoveries

In this section, we bring together two important features of galaxies – their redshifts and their distances.

This crucial development owes its origins to Edwin Hubble. His pioneering work in 1923 first led to the confirmation that certain of the fuzzy patches in the sky, loosely called ‘nebulae’, were in fact galaxies like our own.

3.3 Extending the distance scale

Having reviewed some of the properties of galaxies, we are now in a position to return to the question of how we are to develop further our methods of measuring distance.

The various steps taken in determining larger distances from known smaller ones are often called ‘rungs in the distance ladder’. The process of constructing a rung has been:

  1. Find a measurable quantity associated with a class of objects.

  2. Observe how the measura
    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

3.2 Some general properties of galaxies

Firstly, we note that galaxies tend to occur in clusters rather than singly. The mutual gravitational attraction of galaxies naturally tends to hold them on paths that remain close to each other. Typically a cluster contains tens or hundreds of galaxies. There are, however, large clusters with thousands of galaxies, and there are some solitary galaxies. Our own Galaxy is a member of a smallish cluster of about 36 galaxies called the Local Group (see Author(s): The Open University

3.1 First steps towards a distance scale

As you will see from Table 2, when it comes to astronomy and cosmology, one is called on to deal with a wide range of distances. (Note that a light-year (ly) is the distance light travels in one year, i.e. 9.46 × 1015 m. The distances are also quoted in a very commonly used astronomical unit of distance: the megapar
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

2 Radiation from the galaxies

Stars occur in great collections called galaxies. The distribution and motion of galaxies provide us with the first important experimental information on which we shall build our understanding of the type of universe we inhabit. So, what do we know about galaxies?

All the stars that can be distinguished by the naked eye – a few thousand in number – belong to one galaxy: our own Milky Way Galaxy. Sometimes it is just written Galaxy, with a capital G, to distinguish it from all
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1 Introducing cosmology

General relativity has a very different conceptual basis from that of Newtonian mechanics. Its success in accounting for the precession of Mercury's orbit, and the bending of light by massive objects like the Sun, gives us confidence that our picture of space and time should be Einstein's rather than Newton's. In this and the following units, we turn our attention to the study of the large-scale structure of spacetime. We see how spacetime as a whole is curved by the gross distribution of mas
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Learning outcomes

By the end of this unit you should be able to:

  • describe the characteristics of light emitted by stars, and hence the information of cosmological interest that can be deduced from it;

  • distinguish between true and false statements relevant to the distribution and motion of stars within galaxies, and of galaxies within clusters and superclusters;

  • outline the methods used for estimating the distances to stars and to galaxies;

  • explain and
    Author(s): The Open University

    License information
    Related content

    Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Introduction

In this unit, we present the three main lines of experimental evidence pointing to the big bang origin of the Universe: (i) the recession of the galaxies; (ii) the microwave remnant of the early fireball; and (iii) the comparison between the calculated primordial nuclear abundances and the present-day composition of matter in the Universe.

A data sheet of useful information is provided as a pdf for your use. You may wish to print out a copy to keep handy as you progress through the unit
Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

Acknowledgements

The material acknowledged below is Proprietary and used under licence (not subject to Creative Commons licence). See Terms and Conditions.

Grateful acknowledgement is made to the following for permission to reproduce:

Figure 1a: Neil Borden/Science Photo Library; Figure: 1b NOAA/Science Photo Library; Figure 1c: Max-Planck-Institute for Radio Astronomy/Science Photo Library; Figure 11: Science Photo Library; Figure 14: Science Museum.


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

6 Appendix: a note on displacement current density

This appendix is optional reading. It is included for the sake of comparison with other texts.

The Ampère–Maxwell law,

is sometimes expressed in the form


Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

5.2 The energy of electromagnetic waves

The energy density of an electric field E is

Although we will not prove it in this unit, a very similar result applies to magnetic fields. The energy density of a magnetic field B is

Author(s): The Open University

License information
Related content

Except for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share