Astronomy 1144: Introduction to Stars, Galaxies, and Cosmology

Todd Thompson
Department of Astronomy
The Ohio State University

Lecture 4: Matter & Gravity

Key Ideas

Structure of Atoms
Nuclei of protons & neutrons orbited by electrons
Elements, Isotopes, & Radioactivity

Four Fundamental Forces
Strong & Weak Nuclear Forces
Electromagnetic & Gravitational Forces

Nature of Gravity
Inverse Square Law
Orbits & Circular Speed
Escape Speed & Gravitational Binding Energy

Key Equations

FGrav=GM1M2/d2

P2=4 π2a3/[G(M1+M2)]

Vcirc=(GM/r)1/2

Vescape=(2GM/r)1/2

UGrav= 2GM2/R

Matter

Ordinary matter encompasses a wide range of phenomena:
  • Fundamental particles (quarks & leptons)
  • Subatomic particles (protons & neutrons) composed of fundamental particles
  • Single atoms composed of protons, neutrons, & electrons (Hydrogen, Helium, Gold, etc.)
  • Simple molecules composed of 2 or more atoms bound together
  • Macromolecules (DNA, complex polymers)
  • Macroscopic objects (rocks, people, planets ...)
Most ordinary matter in the universe is in the form of atoms.

Atoms

Atoms consist of two basic parts:

Nucleus of heavy subatomic particles:

  • proton: positively charged
  • neutron: uncharged (neutral)

Cloud of Electrons orbiting the Nucleus:

  • negatively charged particles (leptons)
  • masses are 1/1836th the mass of a proton

Atoms are mostly empty space:

  • Only 1 part in 1015 of the volume is occupied.
  • Volume is threaded by electromagnetic fields.
A simple, yet familiar cartoon:
Simple Atoms (Schematic)
(Graphic by R. Pogge)

Real atoms don't look like this, but it can help you visualize the locations and numbers of the different components (nucleus and electrons).

Chemical Elements

We distinguish atoms into Elements by the total number of protons in their nucleus.

Atomic Number:

  • 1 proton = Hydrogen
  • 2 protons = Helium
  • 3 protons = Lithium ... and so on

Number of electrons = Number of protons

All elements are Chemically Distinct

Top Ten Most Abundant Elements

10) Sulfur
9) Magnesium
8) Iron
7) Silicon
6) Nitrogen
5) Neon
4) Carbon
3) Oxygen
2) Helium
1) Hydrogen
In general, the ordinary matter in the Universe is mostly Hydrogen and Helium (roughly 75% and 24%, respectively), and 1% every other element.

Known Elements

There are 117 elements currently known from laboratory studies:
  • 87 are metals
  • 11 are gasses
  • 2 occur as liquids (Bromine & Mercury)
  • 26 are radioactive
  • 24 are made only in the lab

In addition, each element can have a number of isotopes.

Isotopes

A given element can have many different Isotopes
  • Same number of protons.
  • Different number of neutrons.

Example: Carbon Isotopes:

12C has 6 protons and 6 neutrons
13C has 6 protons and 7 neutrons
14C has 6 protons and 8 neutrons

All isotopes of a given element are chemically identical, but they have different masses.

Radioactivity

If a nucleus has too many or too few neutrons, it becomes unstable to radioactive decay

Examples: 

    3H (1p+2n) -> 3He (2p+1n) + e- + neutrino

    14C (6p+8n) -> 14N (7p+7n) + e- + neutrino 
                      (basis of radioactive carbon dating)

Free neutrons are also unstable: 

    n -> p + e- + neutrino

All of the above are examples of "beta decay", in which one of the neutrons turns into a proton, shedding its excess mass by spitting out an electron and a neutrino.

These reactions all involve the weak nuclear force.

The Fundamental Forces

All interactions among matter are governed by four fundamental forces:
Strong & Weak Nuclear Forces
Bind protons to neutrons inside nuclei
Mediate nuclear reactions and radioactivity

Electromagnetic Force
Binds electrons to nuclei inside atoms, and atoms to atoms inside molecules
Mediates chemical reactions

Gravitational Force
Binds massive objects together on large scales
Mediates orbital motions

Gravitation

Gravitation binds matter to matter over long distances:
  • Long-range attractive force
  • Weakest force of nature.
  • Obeys the Inverse Square Law of distance:
    F=GM1M2/d^2
(Graphic by R. Pogge)

Additional reading on gravitation here.

Gravitational Force

The Force of Gravity between any two objects depends only upon:

The masses of the two objects:

  • More massive objects exert a stronger gravitational force.

The distance between them:

  • The force gets stronger as the two objects move closer together.

  • The force gets weaker as the two objects move farther apart.

It does not depend on the shapes, colors, or compositions of the objects.

Kepler's Third Law

Kepler's 3rd (or "Harmonic") Law of orbital motions
P2 = a3
only applies to planets orbiting our Sun, and P must be measured in years and a in AU.

Newton, however, derived a more general version

Generalized Kepler third law
(Graphic by R. Pogge)

This is a universal form of Kepler's 3rd law applicable to any two massive objects orbiting each other anywhere in the Universe.

The important quantities in this form are:
P = period of the orbit
a = semi-major axis of the orbit
M1+M2 = combined masses of the two bodies
This gives us a way of measuring the masses of objects from observation of their orbital period and semi-major axis.

Additional information on Kepler's laws here.

Circular Speed

In the simple case of a tiny mass in a circular orbit around a more massive body we can simplify the formula above to define the smaller object's Circular Speed:
Circular Speed
Speed needed to sustain a circular orbit at a given distance from a massive object:
Circular Speed
(Graphic by R. Pogge)

The circular speed depends on two things:

  • The mass of the larger parent body (M)
  • The radius of the orbit (r)
This gives us another way to measure masses for objects in circular orbits around larger bodies, now measuring orbital speeds (Doppler Effect) and orbital radii.

In general, orbital motions are the most important way we have to estimate the masses of astronomical objects.

Escape Speed

Another consequence of gravitating bodies is that there is a minimum speed an object must be given in order for it to escape from the gravity of its parent body.

This is called the Escape Speed:

escape speed
(Graphic by R. Pogge)

The escape speed depends on the Mass of the body (M) and its radius (R). Basically, the stronger the gravity at the surface of a body, the faster you have to be going to escape from its gravitational clutches.

Examples:

Earth's Surface
Vesc = 11.2 km/sec (40,320 km/hr)
Sun's Surface
Vesc = 615 km/sec (~2.2 Million km/hr)

Gravitational Binding Energy

Imagine you wanted to take apart a gravitating body. You might do it piece-by-piece, taking a rock and throwing it with the escape speed. As the body got steadily smaller, the escape speed would be less.

If you added up all the energy you would need for each piece, you would derive the Gravitational Binding Energy for the body:

gravitational binding energy
(Graphic by R. Pogge)

This depends on two quantities: the Mass (M) and the Radius (R) of the body.

The formula above is a "proportionality", it tells us how the binding energy scales with the mass and radius of the object. The constant out front we need depends on details of how matter is distributed in the object. For example, a sphere of uniform density would have a constant of 3/5.

For the Earth, the Gravitational Binding Energy is about 2x1032 Joules, or about 12 days of the Sun's total energy output!

Implications

For objects of the same Radius but with different masses, the more massive object will have

  • Faster orbital and escape speeds
  • Greater gravitational binding energy ("more tightly bound")

For objects of the same Mass but different Radii, the larger object (bigger radius) will have:

  • Slower circular and escape speeds
  • Less gravitational binding energy ("less tightly bound")

Gravity: The Universal Glue

Gravity is the force that rules in the domain of astrophysics:
  • Holds planets and stars together
  • Controls the orbits of moons around planets
  • Controls the orbits of planets around stars
  • Binds stars into galaxies
  • Binds galaxies into groups and clusters
  • Binds galaxy clusters into superclusters
  • Binds the Universe together...
Throughout this course we will be exploring the consequences of gravitation.

Updated 9/2/2013 Todd Thompson
Original version by Rick Pogge