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Let's say I have an picture of the sky or of an landscape and I want to measure an angle in the celestial sphere. Is there any software for that? In real life I would use the empirical hand gestures or a theodolite.

A plate solver matches the stars in an image to a catalogue of star positions. Astrometry.net provides a web service and a software API for this. Given only another user's cell phone photo, Astrometry.net determined that the image scale was 112 arcseconds per pixel. It also estimated an image size of 72 by 54 degrees, apparently using an approximation which is only valid for small angles.

Assuming low distortion (not a fisheye lens), the image is a gnomonic projection of part of the celestial sphere. If $r$ is the distance from the image center in pixels, $p$ is the pixel scale in radians per pixel, and $ heta$ is the angular separation from the center in radians, then

$$rp = an heta$$

For example, I used the measurement tools in GIMP and Stellarium on two bright stars near the center to verify the central image scale of 112 arcsec/pixel or 0.54 mrad/pixel. The edges of the 2320x1740 image are 1160 and 870 pixels away from the center. Solving for $ heta$, the edges are 0.56 and 0.44 radian away from the center, so the image's angular dimensions are 64 by 50 degrees from edge to edge.

Daytime images would of course have the same pixel scale and angular dimensions as astrophotos taken with the same equipment.

## Measuring the Sky

**Learning Goals:** The goal of this lab is for students to familiarize themselves with the celestial sphere, and get experience estimating angles using (literal) rules of thumb, and work on estimating distances using parallax.

**Angular Size**

Whenever you look at an object, you are measuring its angular size - the amount of space it takes up in your field of view in degrees, minutes, and seconds (or radians). You can't directly measure an object's size in centimeters or inches unless you walk up to it and use a ruler. You know that faraway objects look small and nearby objects look big, so your brain puts together an object's angular size with your guess as to its distance to give you an idea of its actual size.

Complex and precise instruments exist and can be constructed for measuring the angular size of objects, but a set of rough measurement tools can be found at the end of most people's arms. Because humans are built to mostly the same proportions, if you hold your arms outstreched with your palms facing forward, your hands will have about the same angular size in your field of vision regardless of whether you are tall, short, big or small. Your fingers and knuckles can be used to make rough measurements of angular sizes and distances on the sky as shown in the diagram to the right. Other useful angular size rulers exist as well. For example, the moon is almost exactly one-half degree in extent as viewed from the surface of the Earth.

## A Direct Application of the Astronomical Triangle: Stellar Parallax

Measuring distances to astrophysical objects is, in general, a tricky business that depends greatly on the scale of the distance you wish to know (See the Cosmic Distance Scale page). For stars that are close by we can use a method known as stellar parallax. Stellar parallax is a method of measuring the distance to nearby sears by using straightforward geometry. If we wish to know the distance to a close star all we need to do (in theory) is observe it two times in one year separated by 6 months (Viewpoint A and Viewpoint B in the diagram below). Since we know the distance from the earth to the sun, our triangle has a base that is two times this distance. The angle &phi is measured by looking at how the star of interest shifts in the sky when compared to stars at a much further distance, which appear to remain fixed. (The reason the figure shows an angle of 2&phi is because our convention is to assign &phi to the angle subtended by the *radius* of the Earth's orbit around the Sun ("R" in the figure) rather than the angle subtended by the *diameter* as indicated.) This technique is powerful because it is a direct measure of distance to a star, but is limited by how well we can detect the apparent shift of a star against the background field of stars. This minimum detectable shift sets the furthest distance that can be measured using this technique, since as objects get farther away the apparent shift, &phi , decreases.

The equation for measuring these distances, d , is quite simple and defines the unit of distance known as the *parsec* as follows:

Note that the angle &phi must be in units of arcseconds, not degrees, if we want to get the distance in *parsecs,* where 1 parsec = 206,265 Astronomical Units (AUs)

1.9×10 13 miles . Thus 1 parsec is the distance that would be obtained by measuring a parallax of 1 arcsecond, using a baseline of the Earth-Sun distance (which in turn defines 1 *Astronomical Unit).* The total angular displacement for such an object over the 6 month span depicted above would be, naturally, 2 arcseconds. Note the even the *nearest* star to us has a parallax of less than 1 arcsecond.

Department of Physics & Astronomy, 136 Frelinghuysen Rd, Piscataway, NJ 08854

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## Why bother with angular distances?

Since we know how to convert between angular and physical distances, why bother with angular distances at all? Consider what happens, however, if you don't know how far away something is. You see a rock. Is it a huge boulder that's far away, or is it a smaller rock that's closer to you? With a rock, you might be able to walk off the distance. However, with astronomy, it turns out that measuring distances to stars and galaxies can be very challenging. If you don't know the distance, then it's hard to figure out if you're looking at something that's bigger and farther, or if you're looking at something that's smaller and closer.

Angular distance, however, you can measure by staying in one spot (on the Earth, if you're talking astronomy). All you have to do is point from one side to the other side of an object, and measure the angle between the two directions of pointing. Or, if you want to look at the angular distance between two stars, first point at one, then point at the other, and measure the angle between the two directions of pointing.

Because stars can be at different distances away from us, two stars which are a relatively small angular distance apart may be physically farther from each other than two stars that have a large angular distance between them. For example, in the picture below, the *physical* distance between stars A and B is bigger than the *physical* distance between stars A and C, even though to the observer the *angular* distance is smaller.

## Measure an angle in the sky from a photo? - Astronomy

### Scales and Angular Measurement

The apparent sizes of and distances between objects are described with angular measurement. This is important because the objects in the sky are often at greatly differing distances. For example, the Sun is 400 times larger than the moon. It is also 400 times more distant, so it appears to be the same size as the full moon that is, it has the same angular size.

The system of angular measurement used by astronomers is based on divisions of the circle. The circle is divided into 360 degrees. Degrees are divided into 60 minutes of arc, or arc minutes, and each minute is divided into 60 arc seconds.

The Sun and the moon have angular diameters of about half a degree, as would a 4-inch diameter orange at a distance of 38 feet. People with keen eyesight can distinguish objects that are about an arc minute in diameter, equivalent to distinguishing between two objects the size of a penny at a distance of 70 meters (226 feet). Modern telescopes allow astronomers to routinely distinguish objects one arc second in diameter, and less. Chandra can distinguish objects down to about 0.5 arc seconds in diameter, and the Hubble Space Telescope can distinguish objects as small as 0.1 arc seconds. For comparison, 1 arc second is the apparent size of a penny seen at a distance of 4 km (2.5 miles)!

The accompanying illustration shows how you can use your hand to make rough estimates of angular sizes. At arm&rsquos length, your little finger is about 1 degree across, your fist is about 10 degrees across, etc.

The angular diameter is proportional to the actual diameter divided by its distance. If any two of these quantities are known, the third can be determined.

For example if an object is observed to have an apparent diameter of 1 arc second and is known to be at a distance of 5000 light years, it can be determined that the actual diameter is .02 light years.

## Conducting the Activity

### Materials

### Measuring the Earth with Eratosthenes

- The first step is to contact another teacher at your same grade level who lives at least 100 miles directly north or south of you &ndash farther apart is better for this experiment. A direct north-south line between the cities is also important for this, you will need to know as exactly as possible how many miles north or south of you the other school is as opposed to the direct mileage between the cities. Look a map and select a likely city, research their schools on the internet and reach out to someone by email and send them an invitation to join your class in this exciting project. It may take one or two tries, but I bet you can find a partner without too much difficulty!
- When the big day arrives, send an email in the morning to be sure you have sunny weather in both cities. A few minutes before noon, set up the yard sticks in the playground area. One stick should be held vertically, (use a small carpenter&rsquos level for this). Use the compass to lay out the second yardstick flat on the ground so that it points directly north. You have now made a simple sundial! Watch as the shadow moves clockwise when the shadow lies directly along the flat yardstick, measure and record the position where the tip of the shadow falls. Depending on your location and the time of year, the shadow may extend past the end of the flat yardstick &ndash that&rsquos okay, just mark its position with some sidewalk chalk.
- Now that you&rsquove marked the tip of the shadow, stretch a piece of string from the top of the vertical yardstick down to where the tip of the shadow touched the ground. Measure the angle between the vertical stick and the string with a protractor as accurately as you can and record it. Email this information to each other &ndash it will be the
that will be important for this activity!*difference between the angles* - Eratosthenes believed that the Earth was round, and so the angle of the Sun in the sky would be different depending on how far north you were from the equator &ndash and he was right! By setting up a simple ratio and proportion between the difference in the two angles and the distance between the cities, he was able to accurately measure the circumference of the Earth for the first time about 2,300 years ago. Eratosthenes&rsquo calculation for the size of the Earth was accurate to within about 2% of our modern value, how close can your students get? Set up your calculation as shown below!

5. The actual circumference of the Earth is 24,900 miles. The example above was done by my own students several years ago and shows a value within 4% of the true size of the Earth &ndash pretty good for kids using some string and a protractor! How close will your students get!

- Eratosthenes obviously didn&rsquot have a telephone or the internet, how do you think he managed to do this activity in ancient Egypt? (Egypt was then part of the Greek/Macedonian empire.)
**Answer:**Eratosthenes did not take both measurements on the same day! The astronomer took a measure of the solar angle in the town of Syene in southern Egypt on the summer solstice. He then walked to the town of Alexandria in northern Egypt and carefully measured the distance along the way and measured the solar angle again on the summer solstice in the following year.

- We sometimes think of ancient peoples as &lsquoprimitive&rsquo or even &lsquoignorant&rsquo. What do you think of the ancient Greek culture of Eratosthenes now that you know that people in this era were able to measure the size of the Earth and Moon, and even measure the distance between them accurately?
**Answer:**The ancient cultures were not all ignorant or primitive! Many cultures have had &lsquodark ages&rsquo where learning was not advanced, but ancient cultures were in many ways remarkably advanced!

## Part 1: The Small Angle Formula

In astronomy, the sizes of objects in the sky are often given in terms of their angular size as seen from Earth, rather than their actual sizes. For a given observer, the distance to the object D, the size of the object (or separation) d, and angle θ in radians (as portrayed in the picture above) form a right triangle with the trigonometric relationship:

Since these angular diameters are often small, we can use the small angle approximation which will give us:

So we can rewrite our small angle approximation as:

When dealing with astronomically distant objects, where angle sizes are extremely small, it is often more practical to present our angles in terms of arcseconds, which is 1/3600th of one degree. Since one radian equals 3600⋄(180/π) ≈ 206265 arcseconds, we can then rewrite this as the Small Angle Formula:

where θ is now measured in arcseconds, d is the physical size or separation, and D is the distance to the object.

Since it is easy to measure the angular size of astronomical objects, we often use this to solve for other unknowns, such as the distance or the diameter of a celestial body. If two objects are roughly the same distance from the observer, you can also use the formula to find the distance between the two objects. As you are aware from the background section of this lab (and hopefully your own experience), an object's angular size depends on its physical size (in feet, meters, etc.) and its distance.

**Lab Exercise**

Using trigonometry, if you know the length of two sides of a right triangle, you can find the angles. Suppose the building in the picture to the left is 50 feet tall and 300 feet away from you (A=50ft, B=300 ft), and that you can use the small angle approximation.

**1. What is the angular size of the building in degrees?**

**2. Now find the angular size of something in the lab. Be sure to record how large it is and how far away it is from you.**

**3. Does the angular size depend on where you stand in relation to the object?**

## Measuring Double Stars with a Micrometer

The observation and measurement of visual double stars is an important area of study in astronomy and astrophysics. This type of observing is an area of research well suited for amateur participation. The amateur can still carry out important scientific work, and make a valuable contribution to astronomy. While accurate measurements of visual double stars does not necessarily require the use of an expensive micrometer the process is never the less made easier and highly accurate using such instruments.

To find the apparent separation of a pair of close stars, or double stars as they are called, with a telescope one must use a measuring device such as a micrometer. Micrometers are usually made with adjustable webs, needlepoints, or an eyepiece reticle with graduated lines ruled in the glass. The webs, needlepoints, or reticle lines are positioned at the focal plane, in focus with the image, and magnified. An image can be aligned between the micrometer webs, points, or the ruled lines of the reticle and the separation noted on a dial or scribed on the reticle in either fractions of an inch or millimeters (See Figures 1 through 3).

Figure 2. The Eyepiece Reticle Micrometer. A cross-section view is presented in view (a) and the image of the reticle and Mars is shown in ( .

First it would be wise to find the orientation of the micrometer in the telescope field relative to the sky. As we know the celestial sky is marked off in the astronomer’s mind and we refer to the direction west and east as Right Ascension (RA). RA is graduated in hours, minutes and seconds, and equated with the longitude of the celestial sky. The north-south direction or Declination (DEC) equates to the latitude of the celestial sky. Usually, double stars are not exactly the same visual magnitude so we consider the primary as the brightest of the two and the companion as the dimmer.

The position angle of a double star pair or polar coordinates are relative to the celestial north, so we must rotate the micrometer so that the two stars drift exactly east to west along the **centerline (T)** web or etched **centerline** in the reticle eyepiece field. For this drift check we will use the micrometer with the telescope drive off and select any star at or very near the celestial equator. If the primary, or brightest, star is centered in the field and the dimmer star is on the west side of the primary then the micrometer is correctly oriented to the north (See Figure 4).

To translate the separation to some usable angle or linear dimension the image scale of the telescope must be calculated. Image scale is usually expressed in degrees, minutes, or seconds of arc per inch or millimeter. To find the image scale in seconds of arc (arcsec) per millimeter, divide 206,265 (seconds of arc in 360 degrees) by the focal length (F.L.). For example, the image scale for a 16-inch (406.4mm) f/7 aperture telescope with a F.L. of 2844.8 mm is:

image scale = 205,625 / 2844.8 = 72.5 arcsec per mm

Since many of the objects subtend very small angles, usually in the seconds of arc, we must increase the effective focal length (**EFL**) of our telescope to allow the image to be large enough to be separated by several increments. A large image also results in a higher resolution in the micrometer readings. The Barlow lens is a good way to accomplish this. If a 5x Barlow is used on the above telescope then the EFL will be:

EFL = 2844.8mm x 5 = 14,224mm

With the increase in effective focal length the image scale then becomes:

image scale = 206,265 / 14,224mm = 14.5 arcsec / mm

Let's use a micrometer to measure a double star from ** The Aitken Double Star catalogue**, ADS-111 (RA00:09:21, DEC -27:59.3), 1992 [kappa-1 Scl]. This 6-magnitude pair is separated by 1.4 seconds of arc and position angle 261 degrees. Since this pair is nearly at the same visual magnitude then find the star that appears a little brighter and position it in the center of the field on the

**centerline (T)**web and the

**fixed (F)**.

After the image is positioned at the focal plane between the webs or hash markes of the reticle, the telescope drives are adjusted so one star is centered in the movable (M) and centerline (T) webs, and the fixed (F) and centerline (T) webs. The separation is read from the micrometer thimble and spindle, or lines on the reticle and noted (See Figure 5).

You can find the separation by subtracting the micrometer "**zero**," that is, the dial reading where the webs or points are centered on each other. To determine the micrometer zero one positions the movable web exactly over the fixed web, then reading the micometer. The Darbainian B-Filar Micometer this author uses has a micrometer zero at **10.6772**. If we measure the separation of the double stars using a web type Bi-filar micrometer with a read of 10.7737 mm -- the separation of the double stars: separation = 10.7737 - 10.6772 = 0.0965 mm

Given the above image scale of 14.5 arcsec/mm then the size of the image is:

separation = 0.0965 x 14.5 = 1.4 arcsec

Of course, this was just a check to see if this particular double is still situated as catalogued and it appears that it is still separated by 1.4 seconds of arc and at 261 degrees position angle.

In order to join in the fun with other double star observers you may wish to write to Ronald C. Tanguay at 306 Reynolds Drive, Saugus, MA 01906-1533 and ask about the "Double Star Observer" newsletter.

Couteau, Paul, ** Observing Visual Double Stars**, ISBN 0-262-03077-2, The MIT Press, Cambridge, MA.

** Webb Society Observers Handbook**, Double Stars

Gerald North, ** Advanced Amateur Astronomy** (Edinburgh University Press)

An on-line catalogue of speckle interferometry maintained the Georgia State University is at:

http://www.chara.gsu.edu/CHARA/DoubleStars/Speckle/intro.html

**Suppliers of micrometers:**

Lyot-Carriichel microrneter (SAF) - write to Edgar Sou1ie’ "Les Dryades," 19 avenue Salengro, 92290 Chatenay-Maiabry, FRANCE

## Measuring Tip: You Are a Ruler!

Gareth Branwyn is a freelance writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor to Boing Boing, Wink Books, and Wink Fun. And he has a new best-of writing collection and “lazy man’s memoir,” called Borg Like Me.

### @garethb2

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They don’t call feet, feet, for nothin.’ Measurement started with counting segments of the human body (forearm, hand, finger, foot). So, if you find yourself without a ruler, make sure you know how to count it old-school.

It’s good to know things like the width of the open and closed spans of your hand, the length of a finger and its joints, the actual length of your foot. Memorize them. Write them on your body for a day to remember them.

When I posted the question of anything to add to this ancient of tips on a private maker group on Facebook, several members chimed in:

**Matt Friedrichs** Similar to knowing body part length: When you don’t have a tape measure with you, find a body part that matches the length, and then measure that part when you get back to the shop. I use this all the time when I’m working on a shed or something and have left the tape around the corner. Or, when I’m trying to match bolts, I compare to fingers.

**Ben Daigneau** Some fields (geology here) calibrate the length of their stride so they can walk off distances. Not as precise as a foot or a finger. Also, not exactly a body part, but a lot of people wear belts most of the time. I guess a person could have a few known marks on there.

### Measuring Sky Angles

You can also use your hands to measure degrees of the sky. There is a method common in astronomy for measuring sky angles. Here’s how they describe it on One Minute Astronomer:

- Stretch your thumb and little finger as far from each other as you can. The span from tip to tip is about 25 degrees
- Do the same with your index finger and little finger. The span is 15 degrees
- Clench your fist at arms length, and hold it with the back of your hand facing you. The width is 10 degrees
- Hold your three middle fingers together they span about 5 degrees
- The width of your little finger at arm’s length is 1 degree.

### Measuring Distance with Your Thumb

I leave you with one more body part measuring trick, using your thumb to measure distance. This is taken from the old Your Body Ruler – A User’s Manual, a doc that has been online since the web Jurassic:

I hold out my arm, look at my thumb, and see a distant car half as high. Cars are about 5 feet (1.5 meters) high. So my thumb appears 10 feet (3 meters) wide. And since I know (see below) my thumb is x30 times as far as it seems tall… I know the car is something like 300 feet (90 meters) away!

Do you have any body ruler tips to share? Please post them in the comments below.

### By Gareth Branwyn

### Gareth Branwyn

Gareth Branwyn is a freelance writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor to Boing Boing, Wink Books, and Wink Fun. And he has a new best-of writing collection and “lazy man’s memoir,” called Borg Like Me.

## Handy Measuring Tool

Many of the numbers we use in science have never been measured directly we only know them from indirect measurements. How far is it to the sun? What is the diameter of Saturn’s rings? Here’s a way to use nonstandard measurements and simple ratios for estimating sizes or distances.

### Video Demonstration

### Tools and Materials

- Your hand
- Meter stick or measuring tape
- Partner
- String
- Tape

### Assembly

### To Do and Notice

*Measuring with your extended hand*

Open your hand and fully extend your fingers. You will be using the distance between the tip of your thumb and the tip of your little finger as a measuring tool for finding height. (In some Spanish-speaking cultures, this is called a *cuarta*.) The height of your extended hand, from the thumb to the little finger, is 1 unit.

Measure the length of your arm using extended hands (or *cuartas*) as your measuring unit. What is the ratio of the height of your extended hand to the length of your arm measured in hands? Round it to a simple fraction. Most people are about 1 to 3, or 1:3, meaning that their arms are about 3 *cuartas* long.

Close one eye and look down your straightened arm to locate an object in the distance that you can just barely span the height of with your extended hand. The top of the object should just line up with the top of your thumb, and the bottom of the object should just line up with the tip of your little finger. At this distance, the ratio of the height of the object you’ve obscured to the distance between the object and your eye is the same as the hand-to-arm length ratio you determined earlier. Therefore, whatever you can obscure with your extended hand *is three times farther away from you than it is tall*.

Mark your location on the floor with a piece of tape. Use the tape as a reference, and measure your distance to the object in centimeters.

Use the known ratio and the distance to the object to find the height of the object without measuring it. This is called an *indirect measurement*.

Now measure the height of the object directly with your measuring tape. How well did the Handy Measuring Tool method work?

Calculate your error: Find the difference between the height you found indirectly and the height you measured directly. Divide this difference by the actual measured height of the object. Multiply this quotient by 100 to get percent error.

What angle is swept out by your hand when your fingers are extended? (Hint: When both of your arms are outstretched at shoulder level, they sweep out a 180˚ angle.)

*Measuring with Your Closed Fist*

Using the same procedure, find the ratio for your fist-to-arm length ratio. Will it be a larger or smaller ratio than the extended hand-to-arm length ratio?

Measure the length of your arm length in fists (do not include your thumb) when your arm is extended at shoulder height. Be careful not to roll your fist along your arm.

What is the ratio of height of your fist to length of your arm? Round it to a simple fraction. Most people have a 1:7 ratio.

Close one eye and look down your straightened arm to locate an object that you can *just cover* with your fist. The top of the object should just line up with the top of your fist, and the bottom of the object should just line up with bottom of your fist. At this distance, the ratio of the height of the object you’ve obscured to the distance between the object and your eye is the same as the fist-to-arm length ratio you determined (for most people, 1:7). Make sure you are facing the object and keeping your elbow straight. The ratio changes if you bend your arm.

Mark your location on the floor with a piece of tape. Use the tape as a reference, and measure your distance to the object in centimeters.

Use the known ratio and the distance to the object to find the height of the object without measuring it.

Measure the height of the object directly to check your work.

Calculate your error: Find the difference between the height you found indirectly and the height you actually measured. Divide this difference by the actual measured height of the object. Multiply this quotient by 100% to get percent error.

What angle is swept out by your fist? (Hint: Compare your fist to something known, like your extended hand, or to a known angle.)

### What’s Going On?

When you use your extended hand to just obscure the height of a distant object, that object sweeps out, or subtends, the same angle as your extended hand does on your eye’s retina. Two similar triangles are created one is embedded within the other. The embedded triangle’s base is the length of your arm, and its altitude is the height of your hand with the fingers extended. The larger triangle’s base is the distance to the faraway object, and its altitude is the height of the distant object. These heights and distances are proportional:

A geometric proof of why this works can be shown using similar triangles (click to enlarge the diagram below).

You can model these triangles with a long piece of string. Ask a partner to extend her hand and find an object that she can just obscure with her hand when her arm is extended. Tape one end of the string to the top of the object. Bring the string toward your partner’s eye, noticing how the top of her thumb and the top of the object she obscured with her hand line up. Ask her to hold the string next to her eye then continue with the rest of the string back to the bottom of the object. Pull the string taut, and you’ll see a large triangle, with two sides formed by the string and one side by the object. A smaller triangle is embedded in this large triangle. It has two sides formed by the string, and its third side formed by her hand. These two triangles are similar.

You can find the angle subtended by your hand by comparing it to an easy “known” angle. When both of your arms are outstretched at shoulder level, they sweep out a 180˚ angle. How many extended hands fit in this 180˚ angle? By carefully stacking your extended hand unit from horizon to horizon, you should find that you can fit about 9 of them in the 180˚ angle. 180˚/9 = 20˚ therefore, your extended hands measures a 20˚ angle.

### Going Further

This is a tool that is used by astronomers to measure objects in the sky.

When your hand is extended in front of you, and your elbow is straight, your hand obscures (or subtends) about 20˚ (assuming that your eye is the vertex of the angle) of your vision. If your extended hand subtends about 20˚ when your elbow is straight, how many degrees do you think your fist subtends when your arm is straight?

You can measure the sizes of different parts of your hand to determine the ratios and angles in the table below. For example, about two fists can fit in one extended hand. If the extended hand sweeps out, or subtends, 20˚, then your fist sweeps out about 10˚.

Here are common simple ratios for other Handy Measuring Tools:

With various combinations of these angular tools, you can measure many objects in the night sky. Can you determine the angular diameter of the Big Dipper or the full moon?

### Related Snacks

##### Size and Distance

##### Oil Spot Photometer

Compare the brightness of two light sources with an oil spot on a white card.