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Using a video camera to measure the radius of the Earth

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2013 Phys. Educ. 48 731

(http://iopscience.iop.org/0031-9120/48/6/731)

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Using a video camera to measurethe radius of the EarthJoshua Carroll and Stephen Hughes

Department of Chemistry, Physics and Mechanical Engineering, Science and EngineeringFaculty, Queensland University of Technology, Brisbane, Queensland, Australia

E-mail: joshua.carroll@qut.edu.au

AbstractA simple but accurate method for measuring the Earth’s radius using a videocamera is described. A video camera was used to capture a shadow rising upthe wall of a tall building at sunset. A free program called ImageJ was usedto measure the time it took the shadow to rise a known distance up thebuilding. The time, distance and length of the sidereal day were used tocalculate the radius of the Earth. The radius was measured as6394.3 ± 118 km, which is within 1.8% of the accepted average value of6371 km and well within the experimental error. The experiment is suitableas a high school or university project and should produce a value for Earth’sradius within a few per cent at latitudes towards the equator, where at sometimes of the year the ecliptic is approximately normal to the horizon.

Introduction

The first person to measure the radius of theEarth is thought to be the Greek mathematicianEratosthenes (276–195 BCE) [1]. Eratosthenesused the distance between the Egyptian cities ofAlexandria on the western side of the Nile Deltaand Syene near Aswan in the south of Egypt.He knew the angle of the Sun’s elevation ateach of those locations exactly one sidereal dayapart. From these measurements he estimated theEarth’s circumference, from which he determinedEarth’s radius to be around 6317 km (figure 1),which is just 1% less than the accepted value of6371 km [1, 2].

Many other simple ways already exist forcalculation of the Earth’s radius [3–8]. All thesemethods use the fact that the Earth takes onesidereal day to complete a full rotation, whilst therays of the Sun are used as a reference. The ideafor this experiment was to reduce the errors and be

Figure 1. Eratosthenes’ method of using the differencein the angle of the Sun’s incoming rays at two locationsa known distance apart.

0031-9120/13/060731+05$33.00 c© 2013 IOP Publishing Ltd P H Y S I C S E D U C A T I O N 48 (6) 731

J Carroll and S Hughes

Figure 2. The Parliamentary Annexe building of theQueensland State Government used to perform therising shadow experiment.

able to collect the data in one location over a shortperiod of time. The method presented in this paperis essentially similar to recording the time of thesunset at different levels up a tall building [8].

While the Earth’s radius and methods tocalculate it have been well known for some time,this new method serves as an easy and accurateapproach that can be repeated by anyone withthe required equipment. The calculations rely onsimple geometry and the result is close to theaccepted value of the radius of the Earth.

MethodologyA tall building was selected with a clear viewbetween the horizon where the Sun sets andthe side of the building facing that direction.The building chosen for this experiment was theParliamentary Annexe building of the QueenslandState Government, next to the Gardens Pointcampus of QUT (figure 2).

Two easily identifiable points were selectedon the building and a Leica Viva TS11 ElectronicDistance Measurement (EDM) unit used tomeasure the height difference between these twopoints (figure 3). The Leica Viva TS11 hasinternal software that uses geometry to calculatethe height of a selected point.

Figure 3. Schematic diagram of the Leica Viva TS11used to measure the height between two points.

An iPhone 4S camera was placed on a steadytripod and set to video mode, with the auto-focusand auto-exposure lock feature set to prevent anyunwanted exposure or focus changes, and used torecord the vertical progress of the shadow of thehorizon up the face of the building. No cloudsobscured the setting Sun. The experiment wasperformed on the 23 August 2012.

The sequence of digital video images wasanalysed using a program called ImageJ [9],which was used to find the difference in thegreyscale luminance between two points selectedon the building, over the time of the video. Usingthis difference it was possible to see when thestart of the rising shadow reached the selectedpoint at the bottom of the building, and later whenit reached the selected point at the top of thebuilding. Accuracy was improved by analysingthe video frames as the shadow passed these twopoints.

Data analysisThe Leica Viva TS11 measures distance usingthe time of flight of a laser beam bounced off adistant surface. Then, using the angle at whichthe device is from the ground level, the internalsoftware of the TS11 uses Pythagoras’ theorem tocalculate the height of the point above the groundwith an error in the range of ±0.01 m. Findingthe height of two easily distinguished points, one

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Using a video camera to measure the radius of the Earth

Figure 4. The results of the difference in luminanceover time between the top and bottom sections of thebuilding. It is clear that shortly before 200 s into thevideo the rising shadow reaches the selected bottompoint. Then the shadow reaches the selected top pointof the building shortly after 250 s of video. Under closeinspection this time difference is 52.5 s ± 0.5 s.

near the top and the other near the bottom ofthe building and then subtracting the two givesthe height difference between them. This heightdifference was measured as 46.86 ± 0.01 m.

High-quality bitmap images were exportedfrom the video every 5 s. Using ImageJ, theluminance for each image was recorded at theselected points at the top and bottom of thebuilding. This was done using the ‘Rectangular’and ‘Straight’ selection tools, with the maximumand mean grey values of the pixels at both placesrecorded, rounded, and then exported for eachimage.

All these values were entered into aspreadsheet and for each image the differencebetween the top and bottom section calculated.The result was a difference in luminance every5 s which, when plotted, clearly showed the risetime for the shadow (figure 4). From seeing theapproximate times these two events occurred,frames were then exported for every 1 s oneither side of these two events. The same processwas repeated for 0.5 s intervals—although 0.5 sappeared to be the limit with this method, assmaller time intervals gave too large an error.

The time measured for the shadow to travelbetween the two selected points with a heightdifference of 46.859± 0.01 m was 52.5 s± 0.5 s.This value can be used with the known value of

Figure 5. At time t = 0, the whole building hasdirect sunlight completely covering one face. As theEarth rotates, the light from the Sun no longer hitsthe whole building due to the horizon blocking thesunlight. Later, at a time t = 1t, sunlight can nolonger directly reach the building, and the building iscompletely covered by the horizon’s shadow. While theSun’s light is not exactly parallel due to refraction fromthe atmosphere [10], over this small time period thedifference in the two points is so minimal that any errordue to refraction is negligible. The size and positions ofthe building are exaggerated for clarity.

the sidereal day to calculate the angle the Earthrotated in this time, as shown below.

θ

2π=

1t

1 Sidereal day

∴ θ =1t

1 Sidereal day2π

Pythagoras’ theorem can be used to calculate theradius of the Earth, as shown in figure 5.

From the value obtained for θ we can usePythagoras’ theorem to come up with the solutionto R, the radius of the Earth.

cos θ =R

R+ h

R =h cos θ

(1− cos θ).

Since θ will be very close to zero, and cos 0 = 1,the above equation can be simplified to

R =h

(1− cos θ)

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θ is calculated from θ = 2π 1tT , where T is

the sidereal day, which is 23 h, 56 min and 4.1 s(86 164.1 s) and1t the time for the shadow to riseup the building of height h.

This gives

θ = 6.2831×52.5

86 164.1= 0.003 828 36 rad.

(Note: the full precision of cos θ must be used inthe calculation to prevent rounding errors.)

Inserting this angle into the equation abovefor the radius of the Earth using a building heighth = 46.86 m gives:

R = 6.3943× 106± 0.118× 106 m.

The error was calculated by repeating the aboveradius calculation with a time increased by 0.5 s,i.e. 53 s and a building height increased by 1 cm,i.e. 46.87 m. This gives the error as 118 km.

This differs from the known average value ofthe Earth’s radius by:

R

Raverage=

6394 km± 118 km6371 km

= 1.8%.

It should be borne in mind that the radius of theEarth varies between 6353 and 6382 m, a rangeof 31 km, which is likely to be much smaller thanthe experimental error.

DiscussionFrom the results it is clear that by using avideo camera and applying digital imaging on theresulting video, a value for the Earth’s radius canbe obtained that is fairly close to the acceptedvalue of 6371 km [2].

While a laser was used to measure the heightof the building, it is possible to use anothermethod, further simplifying this experiment. Byhaving an object of a known length, such asa metre rule, and placing it a known distanceaway, an image of this object can be used tofind its dimension in pixels. Using ImageJ thepixel length of the object can be used to calibratethe image, enabling the height of the building inpixels to be obtained and converted to metres [10].

The main limitation in this experiment is theerror caused from the timings at both events ofthe shadow reaching the two chosen points on thebuilding. While other digital imaging programsexist that do basically the same thing as ImageJ,none could be found that accepted more than one

image at a time, or a video. If there was a programwhere a section of building could be selected,such that a table of the luminance over time foreach frame on that section could be exported,then the error would be significantly smaller. As astandard video is 30 frames s−1, the time betweenframes would be on the order of 1

30 s. This meansthe moment the shadow reaches the selected spotwould be much easier to pinpoint, with a timeerror of ± 1

30 s rather than ±0.5 s.If the timing error was improved, eventually

the error due to the difference between the Sun’sapparent and actual elevation would becomesignificant. This difference arises due to therefractive index of the atmosphere being slightlyhigher than the vacuum of space, meaning theSun’s rays are not exactly parallel with theEarth’s surface. From this, the right angle of thetriangle in figure 4 becomes slightly less than 90◦,meaning another term appears in the equation.However, this is small enough over this small timeperiod to be negligible with the current error dueto the time measurement’s limitations [11].

Another limitation is that when the experi-ment is performed the ecliptic, the path of theSun in the sky, should be approximately normal tothe western horizon. When the data in this paperwas collected the ecliptic was about 10◦ off thevertical. Further work on analysis of the timing,such as the development of a program takingvideo input for digital imaging, would lead toan even more accurate result. If necessary, thevalue for the Earth’s radius can be correctedby taking into account the latitude of the placeof observation, the declination of the Sun andrefraction error [8].

Received 18 June 2013, in final form 18 July 2013doi:10.1088/0031-9120/48/6/731

References[1] Eratosthenes 2010 Eratosthenes’ ‘Geography’

(Princeton, NJ: Princeton University Press)(Engl. transl. D W Roller)

[2] Lide D R 2000 CRC Handbook of Chemistry andPhysics; A Ready-reference Book of Chemicaland Physical Data 81st edn (Boca Raton, FL:CRC)

[3] Gangadharan D 2009 Measuring the radius of theEarth from a mountain top overlooking theocean Phys. Teach. 47 533–5

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Using a video camera to measure the radius of the Earth

[4] Goodman F O 1993 Measure the Earth’s radiuswhile boating on one of its lakes Am. J. Phys.61 378–9

[5] French A P 1982 How far away is the horizon?Am. J. Phys. 50 795–9

[6] O’Keefe R and Ghavimi-Aiagha B 1992 TheWorld Trade Centre and the distance to theworld’s center Am. J. Phys. 60 183–5

[7] Rawlins D 1979 Doubling your sunsets or howanyone can measure the Earth’ size withwristwatch and meterstick Am. J. Phys.47 126–8

[8] Aravind P K 2008 Sunsets, tall buildings and theEarth’s radius arXiv:0812.3911v2

[9] Abramoff M D 2007 ImageJ as an imageprocessing tool and library Microsc.Microanal. 13 1672–3

[10] Hughes S W 2006 Measuring the orbital periodof the Moon using a digital camera Phys.Educ. 41 144–50

[11] Uhl R and Reddmann T 2004 Divergence ofsun-rays by atmospheric refraction at largesolar zenith angles Atmos. Chem. Phys.4 1399–405

Joshua Carroll has a science degree(majoring in physics and astrophysics)and is a current master’s student at theQueensland University of Technology(QUT) in Brisbane, Australia. He isresearching certain applications forstand-off Raman spectroscopy withlasers.

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