Estimating the radius of the moon with a digital camera By Joe Coady.

Estimating the radius of the moon with a digital

cameraBy Joe Coady

Aim

• Using only a digital camera and the gravitation constant G, calculate the radius of the moon.

• Targeted at first year physics students in university and possibly high school students as a way to teach newtons laws of gravitation though experiment.

• In general most students should have a digital camera and therefore should be able to do this practical at home, as well as working well as a group activity.

Newton’s law of universal gravitation

• Newton 1687 Sir Isaac Newton formulated his law of universal gravitation

¿𝐺𝑚1𝑚2

𝑟2𝐹¿𝑚1𝑎1

Where: F = Force felt by one object in the direction of the other object

= Mass of the first object

= Mass of the second object

r = Distance between the two objects

G = Gravitational constant,

= Acceleration of the first object.

* S. Johnson, “The Gravitational Force,” in PHYSICS, John Wiley & Sons, inc, 2007, pp. 96-97

*

Newton’s law of universal gravitation

• Newton 1687 Sir Isaac Newton formulated his law of universal gravitation

¿𝐺𝑚1𝑚2

𝑟2𝐹¿𝑚1𝑎1

Where: F = Force felt by one object in the direction of the other object

= Mass of the first object

= Mass of the second object

r = Distance between the two objects

G = Gravitational constant,

= Acceleration of the first object.

* S. Johnson, “The Gravitational Force,” in PHYSICS, John Wiley & Sons, inc, 2007, pp. 96-97

*

The camera

= Radius of the moon in pixels

= Width of the photo in pixels

= Focal length of the camera lens

= Physical width of the CCD

= Distance to the Moon

𝑟=𝑥𝑓𝑛 ′𝑛𝑑

Working backwards

To calculate the distance to the moon, rearrange newtons law of gravitation

𝑎𝑚=𝐺𝑚𝑒

𝑟 2

Mass of EarthGravitational acceleration of the moon

Length of a sidereal day

𝑡=2𝜋𝜃𝑇

= Rotation of the first photo to match the second (degrees).

= The time interval between photos (seconds).

= The length of a sidereal day (seconds).

Orbital period of the moon

P = (radians/sidereal day)

Centripetal acceleration

𝑑=(𝐺𝑚𝑒

𝑃2 )13

Therefore the distance of the moon is related to the orbital period and earths mass

Radius of the Earth

t = Time between disappearances (seconds)T = Length of a sidereal day (seconds) = Angle which earth rotates in t seconds (radians) = Radius of the earth (m)h = Displacement of the camera from its ground position (m)

Mass of the Earth

The acceleration due to gravity can be considered constant for objects near the earth’s surface as any variance in height (Δh) should be negligible relative to the radius of the earth (r >> Δh)

When = 0

Therefore the acceleration due to gravity on the earths surface can be calculated as the difference in position over the time squared.

Mass of the Earth

Using Newtons law once again, it can now be rearranged to solve for the mass of earth given the acceleration of objects near its surface and its radius.

Finally, combining the distance to the moon equation with the camera equation we can estimate the radius of the moon based on the mass of the earth and the moons orbital period

Results – Sidereal Day / Moon orbital period

Photo Time Taken Relative Time (s)

Relative Angle (°)

Sidereal DayEstimate

1 18:50:42 0 0 02 19:20:18 1776 ± 1 7.43 ± 0.01 23h 54m ± 2m3 19:45:00 3258 ± 1 13.63 ± 0.01 23h 54m ± 1m

Day Moon Position Moon RadiusOne (375 ± 2, 299 ± 2) 22 ± 2Two (1071 ± 2, 1055 ± 2) 22 ± 2

Orbital Period: 1 revolution every 28 ± 4 days

Results – Radius of the Earth

Frame 1 (Tripod ) reference frame 7225Frame 2 (Ground) 7508 ± 3Frame Difference (Frame 2 – Frame 1) 283 ± 3Time Difference 11.3 ± 0.1 sHeight difference between frames 2.22 ± 0.05 mFrame Rate 25 fpsCalculated Radius of the Earth 6500 ± 100 km

Results – Acceleration of objects

Frame 1 (t0) 5436 ± 2Frame 2 (s(t)) 5516Frame Difference (Frame 2 – Frame 1) 80 ± 2Time Difference 0.333 ± 0.008 sPixel to inch ratio 0.0846 ± 0.0007 (inches / pixel)Pixels Displacement 255 ± 4Real displacement 21.6 ± 0.4 inchesHeight difference between frames 2.22 ± 0.05 mFrame Rate 240 fpsGravitational Acceleration 9.9 ± 0.3

Results – Radius of the moon

Moon Diameter (pixels) 747 ± 2Moon Radius (pixels) 374 ± 1Moon Radius (km) km

Final ResultsCalculation Experimental Value Expected Value % Error

Earth Radius 6500 ± 100 km 6378.1 km* 1.9 %Earth Mass kg 5.9726 kg* 5.5 %Gravitational Acceleration 9.9 ± 0.3 9.8 * 1.0 %

Distance to the Moon 378 * 12 %

Radius of the Moon km 1736.0 km* 0.81 %

At this point in time, the regular method of calculating the gravity, size and mass of objects is done though satellites, so this method doesn’t really have any impact of in the future of science.

It does however, stand to be a good teaching mechanism for undergraduate students as is the original intention of the method.

* D. D. R. Williams, “Moon Fact Sheet,” NASA, 13 August 2015. [Online]. Available: http://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html. [Accessed 20 10 2015]