Cosmology How big is the Universe.

Cosmology How big is the Universe

Astronomical distances
The SI unit for length, the meter, is a very small unit to measure astronomical distances. These are the units usually used is astronomy: The Astronomical Unit (AU) – this is the average distance between the Earth and the Sun. This unit is more used within the Solar System AU = 1.5x1011 m

Astronomical distances
The light year (ly) – this is the distance travelled by the light in one year. c = 3x108 m/s t = 1 year = x 24 x 60 x 60= 3.16 x 107 s Speed =Distance / Time Distance = Speed x Time = 3x108 x 3.16 x 107 = 9.46 x 1015 m 1 l y = 9.46x1015 m The distance from the Earth to the nearest star (Alpha Centauri A or B) after our Sun is 4.3 ly. The Milky Way Galaxy is about 150,000 light-years across The andromeda galaxy is 2.3 million light-years away. The edge of the observable universe is 46.5 Giga light years away.

Parallax

Parallax/Trigonometric parallax
Parallax is the change of angular position of two observations of a single object relative to each other as seen by an observer, caused by the motion of the observer.

Trigonometric parallax
To measure the distance to relatively close stars, astronomers use a method known as trigonometric parallax. Explanation: As the Earth moves around the Sun, a relatively close star will appear to move across the background of more distant stars. This optical illusion is used to determine the distance of the star. The star itself does not move significantly during the course of the observations.

Trigonometric parallax

To determine the trigonometric parallax the angle to the star is measured and the change in that angle is observed as the position of the Earth changes. We know that in six months the Earth will be exactly on the opposite side of its orbit ,and therefore will be two astronomical units from its location. Using these observations,the angle between the star and the Earth in these two different positions in space,six months apart is measured.As the size of the Earth’s orbit is known, the distance to the star is calculated using trigonometry.

Parallax angle Tan p P (in rads) r (=1 AU) d

Parallax Space Where star/ball appears relative to background
Angle star/ball appears to shift Distance to star/ball “Baseline”

Parallax We know how big the Earth’s orbit is, we measure the shift (parallax), and then we get the distance… Parallax - p (Angle) Distance to Star - d Baseline – R (Earth’s orbit)

Where r=the radius of the Earth’s orbit
Parallax Where r=the radius of the Earth’s orbit This method works well for close stars upto few hundred pc. If any farther, parallax angle is too small to detect.

Minutes of arc & seconds of arc

A parsec(pc) is a measure of distance
A parsec(pc) is a measure of distance. It is an abbreviation of ‘parallax second’ The parsec (pc) – this is the distance at which 1 AU subtends an angle of 1 arc second or it is the distance a star must be from the Sun in order for the angle Earth-star-sun to be 1 arcsecond. d (parsecs) = 1/p(in arcsecs)

1 parsec=1.5x 1011 /tan(1/3600) =3.06 x 1016 m 1 light year=3x108 x365x24x60x60 1 parsec=3.26 light years

1 parsec(pc)=3.086 X 1016 metres Nearest Star 1.3 pc
(206,000 times further than the Earth is from the Sun)

Parallax has its limits
The farther away an object gets, the smaller its shift. Eventually, the shift is too small to see.

Quick Reference 0.5 degree
The width of a full Moon, as viewed from the Earth's surface, is about 0.5 degree. The width of the Sun, as viewed from the Earth's surface, is also about 0.5 degree. 1.5 degrees Hold your hand at arm's length, and extend your pinky finger. The width of your pinky finger is about 1.5 degrees. 5 degrees Hold your hand at arm's length, and extend your middle, ring, and pinky fingers, with the three fingers touching. The width of your three fingers is about 5 degrees. 10 degrees Hold your hand at arm's length, and make a fist with your thumb tucked over (or under) your other fingers. The width of your fist is about 10 degrees. 20 degrees Hold your hand at arm's length, and extend your thumb and pinky finger. The distance between the tip of your thumb and the tip of your pinky finger is about 20 degrees.

Parallax Experiment Using the quick reference angles that I gave you determine how far something is away near your house based on the parallax method. Include a schematic to show the placement of all objects. Your schematic should include relevant distances and calculations.

Flux and luminosity Luminosity - A star produces light – the total amount of energy that a star emits out as light each second is called its Luminosity. Its unit is watt(W).The luminosity of the Sun is 3.90 x W. Radiation flux - If we have a light detector (eye, camera, telescope) we can measure the light produced by the star – the total amount of energy intercepted by the detector divided by the area of the detector is called the flux. Its unit is W/m2 .The radiation flux received from the Sun is about 1000 Wm -2

Flux and luminosity To find the luminosity, we take a shell which completely encloses the star and measure all the light passing through the shell To find the flux, we take our detector at some particular distance from the star and measure the light passing only through the detector. How bright a star looks to us is determined by its flux, not its luminosity. Brightness = Flux.

F and L are linked by inverse square law
Flux decreases as we get farther from the star – like 1/distance2.

Standard candle: The problem in using the flux equation to measure how far away a star is how to determine the star’s full output power-its luminosity L. The luminosity of some stars can be determined separately from other measurements. These stars are known as standard candles. In astrophysics, a standard candle is a distant star of known luminosity. If we see one star in a galaxy, we can then calculate the distance to the star, and therefore the distance to the galaxy. This method is suitable upto 29 Mpc. Two of the most important std candles are Cepheid variables and type 1 A supernovae.

At distances greater than 29Mpc, trigonometric parallax cannot be relied upon to measure the distance to a star. When we observe another galaxy, all of the stars in that galaxy are approximately the same distance away from the earth. What we really need is a light source of known luminosity in the galaxy. If we had this then we could make comparisons with the other stars and judge their luminosities. In other words we need a ‘standard candle’ –that is a star of known luminosity. The outer layers of Cepheid variable stars undergo periodic expansion and contraction, producing a periodic variation in its luminosity.

Cepheid variables: to be calculated.

Cepheid variable stars are useful to astronomers because of the period of their variation in luminosity turns out to be related to the absolute magnitude(the actual brightness of a star)of the Cepheid. Thus the luminosity of the Cepheid can be calculated by observing the variation in brightness.

The process of estimating the distance to a galaxy (in which the individual stars can be imagined) might be as follows: Locate a Cepheid variable in the galaxy Measure the variation in brightness over a given period of time. Use the luminosity-period relationship for Cepheids to estimate the star’s luminosity. Use the luminosity, the brightness and the inverse square law to estimate the distance to the star.

Cepheid calculation - Example

From the left-hand graph we can see that the period of the cepheid is 5.4 days. From the second graph we can see that this corresponds to a luminosity of about 103 suns (3.9 x 1029 W).

Now using the relationship between apparent brightness, luminosity and distance
D = (L/(4πF))½ D = (3.9 x 1029/(4 x π x 9.15 x 10-10))½ D= 5.8 x 1018 m = 615 ly = 189 pc

Supernova explosions

Supernova explosions type Ia
Supernova explosions (type Ia)can be used as standard candles to find the distance to the more distant galaxies, as it is believed that the maximum luminosity of these exploding stars is same all over the universe. Type Ia is used to measure the rate of expansion of the universe.

Brightness Some stars are so bright that you can see them even in a lighted city, while others are so dim that you can only see them through a telescope. A star’s brightness depends on three things: how big the star is (size) how hot the star is (temperature) how far away from Earth the star is (distance)

Brightness cont. The bigger and hotter the star, the more light it gives off. However, brightness also depends on distance. A star that is nearer to Earth looks brighter than one that gives off the same amount of light, but is farther away.

We call this its Apparent Magnitude
Usually, what we know is how bright the star looks to us here on Earth… We call this its Apparent Magnitude

Betelgeuse and Rigel, stars in Orion with apparent magnitudes
The Magnitude Scale Magnitudes are a way of assigning a number to a star so we know how bright it is. Logarithmic scale is used to tell how bright a star looks. Similar to how the Richter scale assigns a number to the strength of an earthquake Betelgeuse and Rigel, stars in Orion with apparent magnitudes 0.3 and 0.9 This is the “8.9” earthquake off of Sumatra

Or how bright it would appear if it was at a distance of
Magnitude Absolute magnitude is the actual amount of light that a star gives off. Apparent magnitude is the amount of a star’s light that is observed on Earth. If two stars are the same distance from Earth, the one with the greatest absolute magnitude will be the brightest. If one star is farther away than the other, the one that is closer will appear the brightest even if it puts off less light or has less absolute magnitude. Or how bright it would appear if it was at a distance of 10 pc away from Earth.

The historical magnitude scale…
Greeks ordered the stars in the sky from brightest to faintest… …so brighter stars have smaller magnitudes. Magnitude Description 1st The 20 brightest stars 2nd stars less bright than the 20 brightest 3rd and so on... 4th getting dimmer each time 5th and more in each group, until 6th the dimmest stars (depending on your eyesight)

Later, astronomers quantified this system.
Because stars have such a wide range in brightness, magnitudes are on a “log scale” Every one magnitude corresponds to a factor of 2.5 change in brightness Every 5 magnitudes is a factor of 100 change in brightness (because (2.5)5 = 2.5 x 2.5 x 2.5 x 2.5 x 2.5 = 100)

Example: Magnitude Description Dimmer 21 Can’t see with naked eye 7
Limit of what naked eye can see 2 Saturn Vega -1 Sirius(brighter star) Brighter -4 -26 Venus Sun

Brighter = Smaller magnitudes Fainter = Bigger magnitudes
Magnitudes can even be negative for really bright stuff! Object Apparent Magnitude The Sun -26.8 Full Moon -12.6 Venus (at brightest) -4.4 Sirius (brightest star) -1.5 Faintest naked eye stars 6 to 7 Faintest star visible from Earth telescopes ~25

However: knowing how bright a star looks doesn’t really tell us anything about the star itself!
We’d really like to know things that are intrinsic properties of the star like: Luminosity (energy output) and Temperature

…we need to know its distance!
In order to get from how bright something looks… to how much energy it’s putting out… …we need to know its distance!

The whole point of knowing the distance using the parallax method is to figure out luminosity…
Once we have both brightness and distance, we can do that! It is often helpful to put luminosity on the magnitude scale… Absolute Magnitude: The magnitude an object would have if we put it 10 parsecs away from Earth

Absolute Magnitude (M)
removes the effect of distance and puts stars on a common scale The Sun is in apparent magnitude, but would be 4.4 if we moved it far away Aldebaran is farther than 10pc, so it’s absolute magnitude is brighter than its apparent magnitude Remember magnitude scale is “backwards”

So we have three ways of talking about brightness:
Apparent Magnitude - How bright a star looks from Earth Luminosity - How much energy a star puts out per second Absolute Magnitude - How bright a star would look if it was 10 parsecs away

Star colour Star colour is dependent on surface temperature.
Hot stars=Blue or white(about 30,000 K) Cool stars=Red or Orange(about 3000 K) The Sun=Orange or Yellow(about 5,500 K)

Wein’s law:

T = 12000K. From an HR diagram we can see this corresponds to a brightness of about 100x that of the sun (= 100 x 3.9 x 1026 = 3.9 x 1028 W)

Stefan-Boltzmann law: The mathematical relationship between energy radiated per second and temperature is given by the Stefan-Boltzmann law (also known as Stefan’s Law): The luminosity of a star can therefore be related to its temperature as described in the Stefan-Boltzmann law. Since stars are approximately spherical, we know that A =4πr2, where r is the radius of the star.

Exam Hint: Questions on this topic will typically ask you to:
Sketch a second blackbody curve on the graph you are given, for a body of different temperature Use Wien’s law to estimate temperature or peak wavelength Use the Stefan-Boltzmann law to find radius, luminosity or temperature Use any of the above in conjunction with the Hertzsprung-Russell diagram

H-R diagram: The H-R diagram is a plot of stellar luminosity against surface temperature. It is a diagram not a graph. The vertical axis is luminosity, scaled in “Sun-powers”,i.e multiples of Sun’s luminosity .It has no units. The scale is logarithmic. The horizontal axis is the surface temperature T of the star in kelvin. This scale is also logarithmic and goes from high temperatures from the left to low temperatures on the right.

The H-R diagram

Doppler effect(Red Shift / Blue Shift:
Wave source approaching (blue shift) wave fronts are squished together wavelength is shorter, frequency is higher so it appears bluer Wave source moving away (red shift) wave fronts are spread apart wavelength is longer, frequency is lower so it appears redder Q: What happens to the light the left eye sees? Red Shift Blue Shift No Shift Q: What happens to the light the right eye see? Red Shift Blue Shift No Shift

The Sun & The Stars The sun is a star.
The sun and the stars are all made out of hydrogen. Light from hydrogen leaves a distinctive pattern of black lines (an absorption spectrum).

What We See Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us. This is the absorption spectrum (pattern of black lines) from a close star like our sun.

What We See Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us. If the nearby galaxy wasn’t moving, the absorption spectrum (the pattern of black lines) would be in the same place.

Is the pattern red shifted or blue shifted?
What We See Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us. Is the pattern red shifted or blue shifted?

What We See The pattern is red shifted
Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us. The pattern is red shifted since it is shifted towards the red side of the spectrum.

What We See The pattern is red shifted. — So —
Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us. The pattern is red shifted. — So — Is the nearby galaxy moving towards us or away from us?

The bigger the shift, the faster the light source is moving.
What We See The bigger the shift, the faster the light source is moving. Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us.

Which is moving away faster: the nearby galaxy or the faraway galaxy?
Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us.

What We See Which is moving away faster:
Close Star Nearby Galaxy Faraway Galaxy Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us. Which is moving away faster: the nearby galaxy or the faraway galaxy?

What We See Close Star Nearby Galaxy Faraway Galaxy
Really Faraway Galaxy The pattern of black lines come from hydrogen. As we look at objects further from earth, is the pattern shifted? Is it a red shift or a blue shift?  Red shift, so the faraway galaxies, are going away from us.

Since the launch of the Hubble telescope, the uncertainty in measurements, and hence the uncertainty in the Hubble constant ,has reduced significantly. The best modern value considered accurate to within 5⁒, is Hₒ = 71kms⁻1Mpc⁻1.Within an accurate value for Hₒ,astronomers can now use Hubble’s law to determine distances to newly observed objects.

According to Hubble’s law,it implies that the universe as a whole is expanding. If we imagine time running backwards from the present,then the Universe would contract until a moment where everything was in the same place.This would be the time of the Big Bang, when everything first exploded outwards from that single point. Thus,if we can find the Hubble constant,it will tell us how quickly the Universe is expanding.From this we can work out when it all started.

Age of the Universe:

4.33 X 10¹⁷ s.

Star Formation & Explosion Life Cycle Of A Star (Documentary)
Life cycle of a star(IOP)

The life cycle of Stars

Fate of the universe

Cosmology How big is the Universe.

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