Calculating the Astronomical Unit

during a Transit of Venus

using Satellite Data

Dr. Sten Odenwald (NASA/IMAGE)

Dr. Bart DePontieu (Lockheed Martin/TRACE)

Teacher Note: To see the following activity completely worked out for the Transit of Mercury on May 7, 2003, please look at the following page: Calculating the Astronomical Unit During a Transit of Mercury Using Satellite Data.

Introduction to Triangulation:

Suppose you were trying to measure the width of a canyon from your side to the distant canyon wall. You couldn't pace the distance. Could you come up with a way to make this measurement safely?

Teacher Note: Have the class suggest ways that they could measure this distance.

Surveyors and geologists encounter this kind of problem all the time, and over the course of centuries, they have found a simple way to sole this problem. They use a method called 'triangulation'. Here's how it works:

In ordinary land surveying, imagine a distant mountain peak and two observers are located at 'A' and 'B' separated by a few miles (the length 'S'). The base angles at A and B can be measured with an instrument called a theodolite. By knowing the base distance A to B, and the baseline distance S, the distance to the peak can be worked out with a simple scaled drawing or with trigonometry.

Teacher Note: At this point, have the students measure the distance to some remote object in the school yard by using the above triangulation method. Here are some activities to consider.

Introduction to Parallax.

Suppose it was hard for you to measure the two base angles in the triangulation method. This could easily happen if the object were so far away that your instrument could not accurately discern that these angles were different than 90 degrees. For example, if the object is 10 miles away, and your baseline is only 5 feet long, the two base angles would have a measure of 89.9946 degrees. This angle differs from 90 degrees by only 0.0044 degrees which equals 16 seconds of arc (there are 60 minutes or arc/degree x 60 seconds of arc/minute of arc = 3600 seconds of arc per degree!) This would be a very difficult angle to measure even with very expensive modern surveying equipment!

Astronomers run into this problem all the time. To solve this problem, they don't bother measuring the base angles at all. Instead, they measure the vertex angle in the triangle. It turns out that this angle is very easily measured using photographic techniques. The method is called trigonometric parallax or just 'parallax' for short. Here's how it works:

Extend your arm in front of you, hold your thumb up, and alternately open and close your eyes. You will see your thumb's position move against the more distant background in front of you. Astronomers call this the parallax shift angle as the figure below illustrates:

From the above figure, by knowing the distance from your nose to one of your eyes, R, and this Parallax Shift angle in degrees, you can calculate the distance to your thumb (D)! The formula that you use is:

Tan (P/2) = R / D

Teacher Note: It is very important to realize that the Parallax Shift angle is actually twice as large as the angle used in computing the Tangent of R/D. You may want to use this as an opportunity to review basic geometry principles to prove that this is actually the case.

But this same principle applies to measuring distance to objects far away from you too...like the planets. Today, astronomers use photographs of stars taken 6 months apart. During that time, Earth has traveled from one side of its orbit to the other, and the orbit baseline is twice 93 million miles (150 million kilometers). By measuring how far the image of a star has shifted relative to the far more distant stars in the background between, say, January and June, astronomers can accurately measure angles as small as 0.001 seconds of arc or 0.0000003 degrees.

Believe it or not, long before the time when photography had been invented, astronomers were using the parallax technique to measure the distances to the nearby planets. By the time of Kepler in the early 1600s, astronomers knew exactly how far the planets were from the sun in terms of the distance from earth to Sun, but they didn't know exactly how many kilometers this distance equaled. They had a table that looked something like this:

Planetary Data ca 1620 |
||

Planet |
Length of Year |
Distance |

Mercury |
0.24 |
0.38 |

Venus |
0.61 |
0.72 |

Earth |
1.00 |
1.00 |

Mars |
1.88 |
1.52 |

Jupiter |
11.86 |
5.20 |

Saturn |
29.56 |
9.54 |

Teacher Note: Johannes Kepler's detailed work on the orbits of the inner planets resulted in his discovery of 'Kepler's Third Law' which states that the orbital period-squared equals the distance to the sun-cubed. The table entries can be used to show that this is the case since they are all based on the Earth as a 'unit distance and unit time' reference, so that this law is an exact one without a proportionality constant needed. So, by 1639 astronomers knew the relative separations of the planets very accurately, but not their actual distances in miles or kilometers.

When you make a scale model of the solar system with its nine planets, how do you know that in this model, the actual distance from Sun to Earth is 93 million miles and not, say, 153 million or 23 million? At the time of Kepler, the best estimates for this distance were as small as 5 million miles! The answer is that you have to come up with a way to actually measure this distance, and just like land surveyors, you can't travel across distance to make this measurement.

Teacher Note: Ask the students to come up with ideas about how to measure the distances to the planets, without using modern technology or spacecraft.

In 1663, Rev. James Gregory who was an astronomer and mathematician considered, suggested that a more accurate measurement of the Earth-Sun distance could be made. He showed how this distance could be measured from observations of the transit of Venus made from various widely separate geographical locations. A few years later, Sir Edmund Halley, made the same suggestion in 1677, and published the details of this technique in 1716. Sir Edmund Halley described in detail how scientists from various nations could observe the upcoming 1761 and 1769 transits of Venus from many parts of the world. From the parallax shifts they would measure, they would be able to measure this distance from the Earth to Venus and the sun with near-perfect accuracy.

Teacher Note: Have students look up James Gregory and Edmund Halley and learn about what other things these two early astronomers were famous for. Note, also, that Gregory is seldom given any credit for his suggestion that the transit of Venus be used this way, while Halley is usually given all the credit. Ask students to explore this historical controversy and explain why it may have happened. Who should really be given the credit?

The astronomical event called the transit of Venus happens because Venus orbits the sun inside the orbit of Earth. This means that every once in a while, Venus will pass across the face of the sun. The following photograph taken in 1882 by the U.S. Naval Observatory shows what this looks like. You can see Venus as a large black spot directly above the center of the sun's disk.

When these rare transit observations were made between 1761 and 1882, hundreds of scientists were involved in dozens of expeditions to the far corners of the world to set up sensitive equipment at a cost of millions of dollars. Some scientists actually died, or were arrested, in obtaining these measurements. Thanks to modern NASA space technology, we can re-do these transit of Venus parallax measurements very simply, so that students can get an idea of how parallax works. What we will do is to use NASA spacecraft data, instead of equipment on the ground.

Teacher Note: To see the following activity completely worked out for the Transit of Mercury on May 7, 2003, please look at the following page: Calculating the Astronomical Unit During a Transit of Mercury Using Satellite Data.

Activity 1: TRACE observes the Transit of Venus

The TRACE satellite orbits Earth and has its sensors trained on the surface of the Sun to search for solar flares. The changing perspective of the location of Venus in its orbit also leads to a parallax shift. Here is a close-up of the consecutive images of Venus as it traveled across the sun on June 8, 2004:

The composite shows the position of Venus roughly every 450 seconds.

Teacher Note: By counting the number of exposures from peak to peak (about 13) students can calculate that the orbit period for this satellite is 5850 seconds = 97.5 minutes, which is typical for satellites orbiting within a few hundred kilometers of earth's surface.

Method 1: Calculation using a Parallax Table

The first method we can use is to examine a table that lists the angular shift of the disk of Venus at various times during the transit. At the moment that the satellite captured each image of Venus in the montage above, it was also able to measure the vertical 'North-South' shift of the center of each image every 450 seconds. The Venus Parallax Table shown below gives the times, and the angular shifts of the centers of each image in the sequence.

Venus Parallax Data |
|||

Time |
Displacement |
Time |
Displacement |

1 |
0.0228 |
11 |
0.0160 |

2 |
0.0247 |
12 |
0.0180 |

3 |
0.0247 |
13 |
0.0200 |

4 |
0.0233 |
14 |
0.0288 |

5 |
0.0213 |
15 |
0.0233 |

6 |
0.0193 |
16 |
0.0213 |

7 |
0.0180 |
17 |
0.0193 |

8 |
0.0160 |
18 |
0.0180 |

9 |
0.0153 |
19 |
0.0167 |

10 |
0.0153 |
20 |
0.0147 |

Procedure:

__Step 1__ Have the
students identify the largest (0.0247 degrees) and smallest (0.0147 degrees) shift of
the images.

__Step 2__ Compute the difference between the largest
and smallest displacements to obtain the Parallax Shift.
The answer should be 0.0247 - 0.0147 = 0.010 degrees.

__Step 3__ Determine the diameter of the TRACE orbit in kilometers as follows:

From the TRACE satellite orbit data at the time of the transit on June 8, 2004 the satellite was in a polar orbit (the red line in the figure below) with an altitude above Earth's surface of 580 kilometers.

Adding the radius of the Earth (6378 kilometers) to this number (580 kilometers), we get an orbit radius R = 6958 kilometers. We now know the orbit baseline for the TRACE Venus parallax measurement.

Teacher Note: The TRACE spacecraft has an orbit that changes its position throughout the year so that the plane of the orbit is always perpendicular to the line connecting the sun and earth. This allows the spacecraft solar panels to face the sun 24 hours a day and 365 days a year. It also lets the spacecraft keep a round-the-clock watch on the sun.

__Step 4:__
Now that we have determined the Parallax Shift angle, P, from Step 2
and the orbit radius, R, from Step 3, we can use the formula:

Tan (P/2) = R / D

to determine the distance from Earth to Venus (D). First, solve this equation for D:

D = R/Tan (P/2)

Teacher Note: Remind the students that the Parallax Shift angle, P, we measured in Step 2 is actually twice the angle we need to form the arctangent relationship R/D

D = 6,958 km/ Tan(0.0100/2)

D = 6,958 km/ 0.000087

D = 79,900,000 kilometers

At the time of the transit on June 8, 2004, the distance between the Earth and Venus was predicted to be 0.29 Astronomical Units. To find our estimate for the Astronomical Unit, we use the proportion: 1.0/L = 0.29/ D. Solving for L, this means that

L = 1.0 x D / 0.29

L = 275 million kilometers.

The actual value for the Astronomical Unit is 149.5 million kilometers! How could we be so far off?

The TRACE satellite obtained the Venus images by first shifting the solar disk image to the center of the picture frame, then taking the 'picture'. Since the solar image was re-centered each time, this solar parallax was subtracted from the full parallax angle we need to measure. To recover this 'instrumental effect' in the data, you have to add back the 0.0084 degrees for the spacecraft repointing, to the parallax angle you measured on the image. The full Venus parallax angle P is then 0.010 + 0.0084 = 0.0184 degrees.

Teacher note: Explain to students that scientists always have to examine the details of how data is often taken in order to determine how their instruments alter the data. This process is called 'Calibration', and is very similar to what we have to do with our bathroom weight scales every once in a while. We have to make a slight change in the scale so that it registers 'zero pounds' when no one is standing on it. Ask students to identify other examples of the calibration process in other instruments they use to measure with.

The distance between the Earth and Venus which produces this measured parallax shift P = 0.0184 degrees can again be computed from:

Tan (P/2) = R / D

From the example where the parallax angle P= 0.0184 degrees, we get a distance from Earth to Venus, of

D = 6958 km / Tan (0.0092)

D = 6958 km / (0.00016)

D = 43.5 million kilometers.

__Step 10:__ At
the time of the transit on June 8, 2004, the distance between
the Earth and Venus was predicted to be 0.29 AU. To find our
estimate for the Astronomical Unit, we use the proportion:

1.0/L = 0.29/43.5 million km.

This means that L = 1.0 x (43.5 million / 0.29) so that L = 150 million kilometers.

Teacher note: The actual value for the Astronomical Unit is 149.5 million kilometers!

Method 2: Parallax calculation using image analysis

We can also use the TRACE Venus Transit image to determine the distance to Venus. To do this, we first have to determine the scale of the print so we can measure the parallax angle q.

Procedure:

__Step 1:__ Print the TRACE image above on a separate piece of paper.

__Step 2:__ Cut the image out, and carefully tape it by its outer edge to a large sheet of paper. To give yourself room for Step 2, place the image above the center of the page, half way to its top edge.

__Step 3:__ Through trial and error using a string,
a tack and a pencil, draw a circle whose perimeter
passes exactly across the edges of the solar disk
shown in the picture. Measure the diameter of this
solar circle in millimeters. A value near 287 millimeters is reasonable if you are working from the normal print version of the above image.

__Step 4:__ At the time of the transit, the angular
diameter of the sun is known by astronomers to
be 0.53
degrees (1891 arcseconds) .
Calculate the scale of the solar circle in degrees per millimeter
by dividing 0.53 degrees by the measured diameter of the
sun circle (287 millimeters) . Call this number 'A'. A value near 0.0018 degrees per millimeter is appropriate.

Skip to Step 6 below

Alternate Method:

A simpler, though less accurate, method is to use the astronomical fact that the diameter of Venus in this image is 0.016 degrees (57.8 arcseconds) , and use this to calculate the scale of this image.

__Step 1:__ Print the TRACE image on a piece of paper. A typical size
of the image would be about 34mm x 203 mm.

__Step 2:__ Measure the diameter of five of the
Venus disks with a millimeter ruler. For the above
image size, a typical measurement series would be 8.5, 8.8, 9.0, 8.5, 9.0

Teacher Note: The edges of the Venus disk vary in sharpness and quality. Have the students make note of their experiences measuring Venus's disk diameter. This is a significant source of error in determining the scale of the print. You may want to compare the difficulty of this method with the first method where you measured the diameter of the sun. It is far easier to measure accurately a distance of 287 millimeters for the solar diameter, than it is to measure 9 millimeters for the diameter of Venus. An uncertainty of reading the ruler to 0.2 millimeters in the solar diameter case is only an uncertainty of (0.2/287) x 100% = 0.07%, while for the Venus diameter it is a whopping (0.2/9) x 100% = 2%.

__Step 3:__
Average these numbers to get the average diameter of Venus
on the print, 'D'. For the measurements listed in Step 2 yields 8.8 millimeters.

__Step 4:__ Calculate the scale of the photograph
by dividing the angular diameter of Venus (0.016 degrees) by
its measured diameter (8.8 millimeters). For the example
above, A = 0.016/8.8 = 0.0018 degrees per millimeter.

__Step 5:__ For
this exercise, Method 1 and 2 yielded the same image scale of 0.0018 degrees/mm so we will use this answer. Students may need to average their results from the two methods.

__Step 6:__ Notice that the track of Venus's images across the sun is a wavy curve. Draw a line through the centers of the highest disks of the Venus image trace. Draw a second line through the centers of the lowest disks in the Venus track.

__Step 7:__ Use a millimeter ruler to measure
the separation between these two lines. Call this number 'S'.
Make several measurements along the two parallel lines
just in case they are tilted. In the above example, a value
near 6.0 millimeters was obtained. Values from 5.5 to 6.5 millimeters
are also reasonable, but students should try to
measure the separations to better then 0.5 millimeters and
average the data using at least 9 points.

Teacher note: Measuring 9 points along the two 'parallel' lines will reduce the random measurement error in the final averaged number to about 0.5/3 = 0.15 millimeters.

__Step 8:__
Calculate the parallax angle, P, in degrees by using
the formula P = A x S. In the example, P = (6.0 mm) x (0.0018 deg/mm)
= 0.011 degrees. This is similar to the result from Method 1 using the data table.

__Step 9:__ The TRACE
satellite obtained the Venus images by first
shifting the solar disk image to the center of
the picture frame, then taking the 'picture'.
Since the solar image was re-centered each time,
this solar parallax was subtracted from the full
parallax angle we need to measure. To recover
this 'instrumental effect' in the data, you have
to add back the 0.0084 degrees for the spacecraft repointing,
to the parallax angle you measured on the image. The
full Mercury parallax angle
P
is
then 0.011+ 0.0084 = 0.0194 degrees.

Teacher note: Explain to students that scientists always have to examine the details of how data is often taken in order to determine how their instruments alter the data. This process is called 'Calibration', and is very similar to what we have to do with our bathroom weight scales every once in a while. We have to make a slight change in the scale so that it registers 'zero pounds' when no one is standing on it. Ask students to identify other examples of the calibration process in other instruments they use to measure with.

__Step 9:__ The distance between the Earth and Venus which produces this measured parallax shift can be computed from:

Tan (q) = R / D

From the example where the parallax angle q = 0.0194 degrees, we get a distance from Earth to Venus, of

D = 6958 km / Tan (0.0194/2)

D = 6958 km / (0.000169)

D = 41.2 million kilometers.

__Step 10:__ At
the time of the transit on June 8, 2004, the distance between
the Earth and Venus was predicted to be 0.29 AU. To find our
estimate for the Astronomical Unit, we use the proportion:

1.0/L = 0.29/41.2 million km.

This means that L = 1.0 x (41.2 million / 0.29) so that L = 142 million kilometers.

Teacher note: The actual value for the Astronomical Unit is 149.5 million kilometers! Our answer is only 'off' by 5% which is excellent. You may also note that if we had altered measuring the shift in the Venus disks on the image by 0.5 millimeters, our answer would have ranged from as little as 136.8 million kilometers for a shift of 5.5 millimeters, to as much as 150 million kilometers for a shift of 6.5 millimeters. As it turns out, the actual value of 150 million kilometers for the Earth-Sun distance is within our 'error bars' (luckily!!).

Here is a table of the historical measurements of the distance from the Earth to the Sun since the time of Ptolemy. How well did your student's estimate stand up?

Table of Earth-Sun Distance Estimates using Parallax Method |
|||

Name |
Date (A.D) |
Value (miles) |
Method |

Hipparchos |
130 |
5 million |
Earth shadow on Moon |

Cassini |
1672 |
98 million |
Mars parallax |

Various |
1761 |
109 million |
Transit of Venus |

Various |
1769 |
88 million |
Transit of Venus |

Foucault |
1862 |
92 million |
Speed of Light |

Hall |
1862 |
92 million |
Parallax of Mars |

Stone |
1862 |
92 million |
Parallax of Mars |

Various |
1874 |
91.7 million |
Transit of Venus |

Houzeau |
1882 |
92.7 million |
Transit of Venus |

Harkness |
1889 |
92,797,000 |
Transits of Venus 1761-1882 |

Jones |
1941 |
91,500,000 |
Parallax of asteroid Eros |

Various |
1960 |
92,670,000 |
Motion of Pioneer 5 satellite |

NASA/JPL |
1961 |
92,955,820 |
Radar echo from Venus |

__Instrument Calibration Factor__: As viewed from a satellite orbiting Earth, the sun will appear to shift by the parallax effect by an amount we can now calculate.

Tan (P/2) = (6958 km/150 million km)

P/2 = 0.0026 degrees

P = 0.0052 degrees

This angle amounts to 18.7 arcseconds. This means that the solar image has to be re-centered during the TRACE orbit to eliminate this parallactic motion. The instrumental factor is 0.0084 degrees so this indicates that 0.0084 - 0.0052 = 0.0032 degrees is due to some other instrumental effect, possibly related to the re-centering process. This small angle amounts to 11 arcseconds, which is about one part in 170 of the diamater of the solar disk.

__Extension Assignment__: Have the students discuss in their journals all of the various steps that have taken to get their answer, and to estimate how accurately each measurement could be made. What were the largest sources of uncertainty in this procedure? Have the students construct a histogram of their final estimates and calculate the mean, median and mode.

Acknowledgment:

The Transition Region and Coronal Explorer, TRACE, is a mission of the Stanford-Lockheed Institute for Space Research, and part of the NASA Small Explorer program.