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**OPTION E - ASTROPHYSICS E3 Stellar distances Parallax method**

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**Astronomical distances – recap**

The SI unit for length, the metre, is a very small unit to measure astronomical distances. There 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. 1 AU = km or 1 AU = 1.5x1011m

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**Astronomical distances – recap**

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 ly = 9.46x1015 m

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E3 - Stellar distances E.3.1 Define the parsec. The parsec (pc) – this is the distance at which 1 AU subtends an angle of 1 arcsencond. “Parsec” is short for parallax arcsecond 1 pc = 3.086x1016 m or 1 pc = 3.26 ly

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**(206,000 times further than the Earth is from the Sun)**

E3 - Stellar distances 1 parsec = X 1016 metres Nearest Star 1.3 pc (206,000 times further than the Earth is from the Sun)

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E3 - Stellar distances E.3.2 Describe the stellar parallax method of determining the distance to a star. Bjork’s Eyes Space Where star/ball appears relative to background Angle star/ball appears to shift Distance to star/ball “Baseline”

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**E.3.2 Describe the stellar parallax method of determining the distance to a star.**

Parallax, more accurately motion 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. Simply put, it is the apparent shift of an object against the background that is caused by a change in the observer's position over a period of 6 months.

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E3 - Stellar distances E.3.2 Describe the stellar parallax method of determining the distance to a star. 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)

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E3 - Stellar distances E.3.2 Describe the stellar parallax method of determining the distance to a star. For very small angles tan p ≈ p In conventional units it means that

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E3 - Stellar distances E.3.2 Describe the stellar parallax method of determining the distance to a star.

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**Angular sizes 360 degrees (360o) in a circle**

E3 - Stellar distances Angular sizes 360 degrees (360o) in a circle 60 arcminutes (60’) in a degree 60 arcseconds (60”) in an arcminute

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E3 - Stellar distances E.3.3 Explain why the method of stellar parallax is limited to measuring stellar distances less than several hundred parsecs. The farther away an object gets, the smaller its shift. Eventually, the shift is too small to see. Measurements from Earth only allow distances up to 300ly, or roughly 100pc, to be determined with the parallax method. With satellites, distances of around 500 pc can be determined.

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**How Do We Measure the Distance to Stars? - Instant Egghead #46**

https://www.youtube.com/watch?v=vyiauRjJBNQ

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**Go to Option E – Astrophysics SL worksheet**

E.3.4 Solve problems involving stellar parallax. Go to Option E – Astrophysics SL worksheet

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**OPTION E - ASTROPHYSICS E3 Stellar distances**

OPTION E - ASTROPHYSICS E3 Stellar distances Absolute and apparent magnitudes

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**Another thing we can figure out about stars is their colours…**

E3 - Stellar distances E.3.5 Describe the apparent magnitude scale. Another thing we can figure out about stars is their colours… We’ve figured out brightness, but stars don’t put out an equal amount of all light… …some put out more blue light, while others put out more red light!

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E3 - Stellar distances E.3.5 Describe the apparent magnitude scale. Usually, what we know is how bright the star looks to us here on Earth… We call this its Apparent Magnitude “What you see is what you get…”

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**Betelgeuse and Rigel, stars in Orion with apparent magnitudes**

E3 - Stellar distances E.3.5 Describe the apparent magnitude scale. Magnitudes are a way of assigning a number to a star so we know how bright it is 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

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**The historical magnitude scale…**

E3 - Stellar distances 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)

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**Later, astronomers quantified this system.**

E3 - Stellar distances 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)

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**Brighter = Smaller magnitudes Fainter = Bigger magnitudes**

E3 - Stellar distances Brighter = Smaller magnitudes Fainter = Bigger magnitudes Magnitudes can even be negative for really bright objects! 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

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**𝑏 𝑏 0 = 2.512 −𝑚 E.3.5 Describe the apparent magnitude scale.**

E3 - Stellar distances E.3.5 Describe the apparent magnitude scale. Given a star of apparent brightness b, we assign that start and apparent magnitude m defined by: 𝑏 𝑏 0 = 100 −𝑚 5 Where 𝑏 0 =2.52× 10 −8 𝑊 𝑚 −2 is taken as the reference value for apparent brightness Or: 𝑚=− 5 2 𝑙𝑜𝑔 𝑏 𝑏 0 The first equation can also be re-written as: 𝑏 𝑏 0 = −𝑚

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E3 - Stellar distances E.3.6 Define absolute magnitude. 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

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**…we need to know its distance!**

E3 - Stellar distances E.3.6 Define absolute magnitude. In order to get from how bright something looks… to how much energy it’s putting out… …we need to know its distance!

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**Absolute Magnitude (M):**

E3 - Stellar distances E.3.6 Define absolute magnitude. 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 (M): The magnitude an object would have if we put it 10 parsecs away from Earth

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**Absolute Magnitude (M)**

E3 - Stellar distances E.3.6 Define absolute magnitude. 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”

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**Absolute Magnitude (M)**

E3 - Stellar distances Absolute Magnitude (M) Knowing the apparent magnitude (m) and the distance in pc (d) of a star its absolute magnitude (M) can be found using the following equation: Example: Find the absolute magnitude of the Sun. The apparent magnitude is -26.7 The distance of the Sun from the Earth is 1 AU = 4.9x10-6 pc Therefore, M= – log (4.9x10-6) + 5 = = +4.8

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**So we have three ways of talking about brightness:**

E3 - Stellar distances 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

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**E.3.7 Solve problems involving apparent magnitude, absolute magnitude and distance.**

E.3.8 Solve problems involving apparent brightness and apparent magnitude.

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**OPTION E - ASTROPHYSICS E3 Stellar distances Spectroscopic parallax**

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**E.3.9 State that the luminosity of a star may be estimated from its spectrum.**

Spectroscopic parallax is an astronomical method for measuring the distances to stars. Despite its name, it does not rely on the apparent change in the position of the star. This technique can be applied to any main sequence star for which a spectrum can be recorded.

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**E.3.10 Explain how stellar distance may be determined using apparent brightness and luminosity.**

The Luminosity of a star can be found using an absorption spectrum. Using its spectrum a star can be placed in a spectral class. Also the star’s surface temperature can determined from its spectrum (Wien’s law) Using the H-R diagram and knowing both temperature and spectral class of the star, its luminosity can be found.

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E.3.11 State that the method of spectroscopic parallax is limited to measuring stellar distances less than about 10 Mpc. E.3.12 Solve problems involving stellar distances, apparent brightness and luminosity.

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**Distance measurement by parallax apparent brightness spectrum**

Distance measured by parallax: Distance measurement by parallax apparent brightness spectrum Chemical composition of corona Wien’s Law (surface temperature T) Luminosity L = 4πd2 b d = 1 / p L = 4πR2 σT4 Stefan-Boltzmann Radius

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**OPTION E - ASTROPHYSICS E3 Stellar distances Cepheid variables**

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**E.3.13 Outline the nature of a Cepheid variable.**

Cepheid variables Cepheid variables are stars of variable luminosity. The luminosity increases sharply and falls of gently with a well-defined period. The period is related to the absolute luminosity of the star and so can be used to estimate the distance to the star. A Cepheid is usually a giant yellow star, pulsing regularly by expanding and contracting, resulting in a regular oscillation of its luminosity. The luminosity of Cepheid stars range from 103 to 104 times that of the Sun.

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**E.3.14 State the relationship between period and absolute magnitude for Cepheid variables.**

The relationship between a Cepheid variable's luminosity and variability period is quite precise, and has been used as a standard candle (astronomical object that has a known luminosity) for almost a century. This connection was discovered in 1912 by Henrietta Swan Leavitt. She measured the brightness of hundreds of Cepheid variables and discovered a distinct period-luminosity relationship.

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**E.3.14 State the relationship between period and absolute magnitude for Cepheid variables.**

A three-day period Cepheid has a luminosity of about 800 times that of the Sun. A thirty-day period Cepheid is 10,000 times as bright as the Sun. The scale has been calibrated using nearby Cepheid stars, for which the distance was already known. This high luminosity, and the precision with which their distance can be estimated, makes Cepheid stars the ideal standard candle to measure the distance of clusters and external galaxies.

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**E.3.14 State the relationship between period and absolute magnitude for Cepheid variables.**

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**E.3.15 Explain how Cepheid variables may be used as “standard candles”.**

The luminosity of a Cepheid variable can be determined from its period. The brightness of the Cepheid (b) can be determined from its apparent magnitude. Then, from the relationship 𝒃= 𝑳 𝟒𝝅 𝒅 𝟐 the distance to the Cepheid can be determined. If a Cepheid variable is located in a particular galaxy, then the distance to the galaxy may be determined. The Cepheids method can be used to find distances up to a few Mpc.

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**Surface temperature (T)**

E.3.16 Determine the distance to a Cepheid variable using the luminosity–period relationship. Distance measured by spectroscopic parallax / Cepheid variables: Apparent brightness Luminosity class spectrum Chemical composition Cepheid variable Spectral type H-R diagram Surface temperature (T) Wien’s Law Period Luminosity (L) Stefan-Boltzmann L = 4πR2 σT4 b = L / 4πd2 Distance (d) Radius

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