# How Far the Stars?

 Last: 8. The Sun as a Star Next: 10. The Lives of Stars

Since ancient times, many people have suggested that the stars are like the sun, but seen at such great distances that they appear faint instead of dazzlingly bright. One test of this idea is to (1) measure distances to stars, (2) calculate how bright the Sun would appear if seen at such distances and (3) compare the result with actual stars.

## Topics

• How Far the Sun?
• Parallax: Reaching for the Stars
• The Inverse-Square Law
• The Stellar Distance Scale

## How Far to the Sun?

At this point, we have some unfinished business to complete:

1. Ancient astronomers developed a chain of reasoning giving a fairly accurate value for the distance to the Moon in terrestrial units, but could only guess at a distance to the Sun.
2. Copernicus, Kepler, and later astronomers developed an accurate scale model for the solar system, in which all distances were expressed in terms of AU, where 1 AU is the Earth's distance from the Sun.
3. However, they could not express the distances between planets in terrestrial units such as miles or kilometers.

To express distances between planets in terrestrial units, we need to determine the value of 1 AU.

### A First Distance Scale Revisited

Ancient astronomers constructed the first astronomical distance scale:

• The Earth's radius was shown to be 3800 mi = 6120 km.

• The Moon's radius was shown to be about 25% of the Earth's radius, or 950 mi = 1530 km.

• Once the Moon's actual size was known, it's apparent size of about 0.5°, together with the small-angle formula, established that the Moon's distance is about 60 times the Earth's radius, or roughly 216000 mi = 348000 km.

• The Sun was known to be farther away than the Moon since the Moon can hide it during a solar eclipse. The exact distance was not known, but was clearly much greater than the Moon's distance.

### The Keplerian Model

Kepler assigned each planet, including the Earth, an elliptical orbit around the Sun with a definite semi-major axis (in AU) and period.

 Planet a(AU) P(yr) Mercury 0.387 0.241 Venus 0.723 0.615 Earth 1.000 1.000 Mars 1.524 1.881 Jupiter 5.203 11.857 Saturn 9.537 29.424

Measuring the distance (in terrestrial units) to any planet would provide enough information to find the value of 1 AU.

### The Parallax Method: Theory

To measure the distance from the Earth (E) to a planet (P), observations are made at two different places. Suppose that one observation is made with P exactly overhead and aligned with a distant star. The other observation is made with P on the horizon; the angle between P and the distant star is θ.

The distance D from E to P can be calculated in terms of the Earth's radius, R:

 D = 1 2 π R&oplus 360° θ

You may recognize this equation -- or something like it -- from an earlier lecture. Eratosthenes used it to calculate the size of the Earth. Now the same relationship is being used to calculate distances to planets!

### Parallax: Distance to Mars

On October 1, 1672, Mars passed in front of a moderately bright star. Realizing that this presented an opportunity to measure the distance to Mars, Cassini organized simultaneous observations from Paris (Europe) and from Cayenne (South America).

Giovanni Cassini [Wikipedia]

According to Kepler's model, Mars was 0.43 AU at this time. Cassini used the observed parallax and the known distance between Paris and Cayenne to to determine the distance to Mars, and find a value for 1 AU (which was 8% too small):

 DMars = 6.0×107 km = 0.43 AU 1 AU = 1.38×108 km

### Parallax: Transits of Venus

 A transit of Venus happens when Venus passes between us and the Sun. Such passages take several hours, and during that time the planet can be easily observed as a small black dot moving across the face of the Sun. Parallax [Wikipedia]

In 1761, 1769, 1874, and 1882, teams of astronomers scattered across the globe to observe transits of Venus. They hoped to use the parallax of Venus (relative to the Sun) to determine its distance, and thus set the scale of the solar system. However, the results were disappointing -- atmospheric blurring made it difficult to determine the planet's position against the Sun.

### Modern Measurements of the AU

In 1877, observations of Mars using the rotation of the Earth to carry a single observer along a West-East baseline finally gave an accurate value:

 1 AU = 1.49×108 km

For this purpose, asteroids were even better targets than Mars; their star-like appearances make parallaxes easier to measure. In 1930, observations of the near-Earth asteroid Eros gave

 1 AU = 1.4960×108 km

Since 1958, radar has been used to measure distances directly by timing a radio signal bounced off another planet. The currently accepted value is

 1 AU = 1.49597870×108 km

## Parallax: Reaching for the Stars

To measure stellar distances, a baseline much larger than the Earth is required. The Earth's orbit provides a suitable baseline for measuring distances to nearby stars; observations made 6 months apart are separated by a baseline of

AU = 2.99×108 km .

Over the course of a year, a nearby star appears to trace a small oval in the sky when compared to more distant ones. By measuring the parallax angle p (defined as half the long axis of this oval), the distance D to the star can be found:

 D = 1 2 π (1 AU) 360° p

By now this equation should seem familiar...

### Question 9.1

 Suppose we could measure stellar parallaxes from Mars instead of from the Earth. Over the course of one Mars year, nearby stars would move in ovals which are the same size as seen from Earth, because the stars are just as far away from Mars. larger than seen from Earth, because Mars has a larger orbit. smaller than seen from Earth, because Mars is a smaller planet.

### The Parsec

In practice, even distances to neary stars are hard to measure, because the angles involved are so tiny. The nearest star has a parallax angles p = 0.76'', where 1'' = (1° ⁄ 3600) is a second of arc. Since stellar parallaxes are usually measured in seconds of arc, it's convenient to simplify the parallax equation by defining a new unit of distance, the parsec:

 1 pc = 360 × 3600 2 π AU  = 206264.8 AU = 3.086×1013 km

With this definition, the parallax equation becomes:

 D = 1'' p pc

(It turns out that 1 pc = 3.26 ly.)

### Question 9.2

Suppose star B is twice as far as star A. Which star has a larger parallax angle?

1. Star B has four times the parallax angle of star A.
2. Star B has two times the parallax angle of star A.
3. Star B has the same parallax angle as star A.
4. Star B has half the parallax angle of star A.
5. Star B has one quarter the parallax angle of star A.

### Stellar Distances: Results

The nearest known stars are α Centauri, a pair of sun-like stars at a distance of 1.32 pc, and their faint companion Proxima Centauri, which is marginally closer. A handfull of bright stars, including Sirius, Vega, Procyon, and Altar, lie within a sphere 10 pc in radius centered on the Sun. The same sphere also contains dozens of faint stars, most too dim to be seen without a telescope!

 Parallax measurements made from Earth are limited by the effects of our atmosphere; consequently, distances greater than about 20 pc are hard to measure. More accurate measurements can be made from space; the Hipparcos satellite has measured distances to over 100000 stars within 100 pc. This animation shows the 150 nearest stars, with the Sun at the center.

### A Stellar Distance Scale

By building step by step on earlier measurements, we have now constructed a distance scale out to nearby stars:

• Improved versions of Eratosthenes' method yield a value for the Earth's radius:
R = 6378 km .

• Parallax measurements made from two (or more) locations on Earth yield distances to other planets in terms of the Earth's radius. (These have now been checked by radar measurements.)

• Distances to nearby planets set the scale for our model of the solar system and thereby determine the Earth's average distance to the Sun:
AU = 1.4960×108 km.

• Parallax measurements made from two (or more) places around the Earth's orbit yield distances to nearby stars in terms of our distance to the Sun. (A check on these distances will be discussed next time.)

### Question 9.3

Suppose we discovered an error in our calculation of the AU, and the correct value was 10% larger than we had thought. What impact would this have on our values for stellar distances?

1. Stellar distances would be unchanged.

2. Stellar distances would increase by 10%.

3. Stellar distances would decrease by 10%.

## The Inverse-Square Law

Light from a central source spreads out in all directions. Define an object's brightness as the total amount of light a fixed area receives. Then the relationship between brightness B, luminosity L, and distance D is

 B = L 4 π D 2

This is called the Inverse-Square law because the brightness is proportional to the inverse (that is, one over) the square of the distance.

### How Bright is the Sun? I

Assuming a given value for the Sun's total energy output or luminosity, we can work out how bright it appears at a distance of 1 AU, which is where we observe it.

LSun = 3.8×1026 watt

D = 1 AU = 1.50×108 km = 1.50×1011 m

 B = LSun 4 π D 2 = 3.8×1026 watt 4 π (1.50×1011 m) 2 = 1350 watt ⁄ m2

Our result is that at our distance from the Sun, each square meter receives 1350 watt.

### How Bright is the Sun? II

How bright is the Sun from other planets? We can plug in their distances and work through the calculation, but it's easier to form ratios: if the same luminosity L is seen at two distances D1 and D2:

B1 : B2 = D22 : D12

For example, Venus is 0.723 AU from the Sun. Compared to sunlight on Earth, sunlight on Venus is

BV : BE = DE2 : DV2 = (1 AU)2 : (0.723 AU)2 = 1 : 0.523 = 1.91 : 1

In other words, sunlight on Venus is 91% brighter than on Earth.

### How Bright is the Sun? III

Now we can ask how bright the Sun would appear if seen from the same distance as a nearby star. Let's pick the distance of αCen.

D&alpha = 1.32 pc = 2.7×105 AU

Bα : BE = DE2 : Dα2 = (1 AU)2 : (2.7×10 AU)2 = 1 : 7.29×1010 = 1.37×10-11 : 1

In other words, the Sun would appear 1.37×10-11 times fainter at the distance of αCen.

At their actual distances, αCen appears 2.09×10-11 times fainter than the Sun. So if they were viewed at the same distance αCen would appear 2.09 ⁄ 1.37 = 1.53 times brighter then the Sun, or Lα = 1.53 LSun.

To summarize, αCen and the Sun are not exactly the same, but they are similar in terms of brightness. This is evidence that the Sun is a star.

### Neighbors of the Sun

We can play the same game with other stars near the Sun. It turns out that some are much brighter than the Sun, some are like the Sun, and most are much fainter than the Sun. For a sample of the 150 stars nearest the Sun, this table shows how many we find in various ranges of luminosity (measured in units of LSun).

 From(LSun) To(LSun) Numberof Stars 10.0 100.0 4 Vega: L=50LSun 1.0 10.0 14 αCen: L=1.5LSun 0.1 1.0 25 τCet: L=0.46LSun 0.01 0.1 21 0.001 0.01 57 0.0001 0.001 29

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