celestial navigation astronomy lab

8/18/2019 Celestial navigation astronomy lab

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Celestial Navigation

Note the Due Date: OCTOBER 6 . Pro Tip: Don’t putit off.

Objectives

In this lab you will take measurements of the sun’s motion at around local noon . This datawill then be used to nd your position in latitude (N/S) and longitude (E/W) on the EarthThink of it as extremely low-tech GPS, yet good enough for navigation for centuries.

Materials

• Gnomon (a thin, round stick at least a foot long and capable of being put into theground or stood up vertically − a (clean!) toilet plunger works great)

• Big, blank piece of paper

• Watch

• Ruler

Procedure

Please submit a lab report of your work, the requested format can be found on the courseMoodle page just under these instructions. This section should help you know what toinclude in portions of your report.DATA• Report the height of your Gnomon and make a table giving for each observation: (1) thelocal time, (2) the length of shadow, and (3) the altitude angle of the Sun.• Make a graph showing the altitude angle of the sun (y-axis) plotted against time (x-axis),draw a smooth curve though the data, and identify the maximum.ANALYSIS • Use the altitude of the Sun at local noon to estimate the latitude (please include yourcalculation)• Use the time of local noon to estimate the longitude (please include your calculation)• Look up the actual latitude and longitude of your location.

• Where does the most uncertainty in your experiment comes from.

Setup

• Find a level, smooth, sunny location where your gnomon can be stuck or placed verticallyupright. If the gnomon cannot be easily stuck in the ground (as on asphalt), then a smallstand will help. A toilet plunger turns out to be ideal! Secure the piece of paper under thegnomon so you can mark your data directly on it and it won’t blow away. A possible setupis shown in Fig. 1.

Physics 182 1 – 1 The Universe


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Figure 1: Setup

• From about 11:30 am to about 12:30 pm EST accurately mark the position of the shadow’stip and the time of observation about every 5 minutes. The tip’s shadow will be fuzzy: youshould try to mark the point halfway between the end of the dark shadow and the end of the“fuzziness.” Try to be consistent in where in the shadow you are marking form measurementto measurement.

• The date, the time of your observations, the length of the shadow (from the center of thegnomon), and the height of the gnomon will be your data.

Finding the Latitude from Altitude Angle of the Sun

To nd the altitude of the Sun you need to nd the angle from the tip of the shadow up tothe tip of the gnomon as seen in Fig 2

Figure 2: Side view of the setup.



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Since you know two sides, the height of the gnomon and the length of the shadow, andone angle, the right 90 ◦ angle between the shadow on the ground and the gnomon, you cancompute any other side or angle. In this case the angle we are interested in is related to theknown sides by the tangent function

tan( α ) = hd

(1)

Where h is the height of the gnomon and d is the length of the shadow. The altitude of theSun (the angle α) can be extracted using the inverse tangent (or arctangent) function

α = arctanhd

(2)

Note: On some calculators this function will appear as tan − 1 . It is very important thatyou are working in degrees for this. To test this compute tan(45) and see which answeryou get

tan(45) = { 1 in degrees, you want this.1.6 in radians, you don’t want this.

Most calculators have a simple way to switch between degree and radian mode. If you areusing Google as your calculator you can specify that your angles are in degrees as seen inFig. 3 and Fig 4. This works the same with Bing . Don’t let the math get in your way if ithas been a while since you had it - we are using tan and arctan as two “tools” that will letyou determine unknown quantities on a triangle. Appendix 2 has a brief discussion of them.

Figure 3: Google syntax for calculating the tangent of an angle in degrees.

Figure 4: Google syntax for calculating the arctangent in degrees.

After converting the shadow lengths into altitude angles you should be able to estimate thehighest the sun ever gets in the sky. This maximum altitude, a max , of the Sun, correspondsto the point where the shadow is the shortest and can be used to calculate the latitude.



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Because of the earth’s tilt, the Sun’s declination (it’s height above or below the celes-tial equator) changes throughout the year from − 23.5◦ (northern winter solstice) to +23 .5◦

(northern summer solstice). For example the Sun’s declination on February 4th is -16.5(Appendix 1 has a table of declination for the time period of this lab). This means that theSun is 16.5◦ below the celestial equator on this date.To nd our Latitude (north/south position) on the Earth we need to account for the Sun’sdeclination in Eq 3:

Latitude = 90 ◦ − amax + declination (3)

Remember what happens when you add a negative number here.

Finding the Longitude from the time of “noon”

The time when the gnomon’s shadow is shortest is Local Apparent Noon or LAN . Thisis where the Sun is highest in the sky. This is the time you want to call 12:00, and you wouldhave called it 12:00 in pre-industrial society. The time on your watch will not agree with it.

Let’s call your watch time the Local Mean Time or LMT , since your watch attempts totell you the time locally in your time zone. If the Earth’s orbit were circular and uniformthen your could nd your longitude (East / West) just by the difference between these times.Because the earth does not orbit the sun in a perfect circle, a correction, called the Equationof Time correction, must be made. For February 4th the Equation of Time is -13,9 minutes.Since this is (-) negative, this indicates that the sun is “slow” compared to the mean timekept by your watch. This number is how many minutes after clock noon (on your watch)that solar noon would happen in the center of the time zone for which your watch is set.Correcting the LMT value for the Equation of Time will give you what we will call the localapparent time , LAT . In our case your watch is almost certainly set to the Eastern TimeZone, which is centered at 75 ◦ W longitude.Thus, to go from your value of LMT to LAT you calculate:

LAT = LMT + EOT correction (4)

where LAT is now the corrected value of noon that you will use to nd your longitude, LMTis what you’ve found from your gnomon data, and the EOT correction is the Equation of Time for the date in question. Don’t forget what happens when you add a negativenumber here. If you were to do this lab during daylight savings time you would needadjust an hour to move noon back to where it should be.After applying this corrections, any difference between your value for LAT and 12:00 is now

due to the fact that you are not at the center of the time zone (75◦

longitude West). Thistime difference can now be used to calculate the longitude of your location, rememberingthat for each four minutes that your local noon is observed to be later than localnoon for the time zone, your location must be 1 ◦ of longitude farther west .

Longitude = 75 ◦ + LAT − 12 : 00

4 (5)



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Appendix 1: Equation of Time and Declination of the Sun

The following tables shows the EOT corrections for the days you might take data on. Whenthe time is - that means the Sun is fast compared to a watch, and + indicates the Sun isslow compared to a watch. If you do the lab on days where the sun is fast you will need toadd the time shift because the Sun is actually ahead of “clock time” on those days. All theother days are “Slow” where the sun is lagging behind a properly set clock, so one would

subtract off the time shift when the Sun is slow.

Note that both the EoT and the Declination are given in decimal form. -13.9 in the EoTis 13.9 minutes, or 13 minutes and 54 seconds. -17.3 of Declination is 17.3 degrees south(below the equator)

Date EoT (hms) Declination Date EoT (hms) Declination(degrees) (degrees)

1-Sep -0m 5s 8.28 1-Oct 10m 12s -2.582-Sep 0m 13s 8.06 2-Oct 10m 32s -3.223-Sep 0m 32s 7.44 3-Oct 10m 51s -3.454-Sep 0m 52s 7.22 4-Oct 11m 9s -4.085-Sep 1m 12s 7.00 5-Oct 11m 27s -4.316-Sep 1m 32s 6.38 6-Oct 11m 45s -4.547-Sep 1m 52s 6.16 7-Oct 12m 3s -5.178-Sep 2m 13s 5.53 8-Oct 12m 20s -5.409-Sep 2m 33s 5.31 9-Oct 12m 37s -6.0310-Sep 2m 54s 5.08 10-Oct 12m 53s -6.2611-Sep 3m 15s 4.45 11-Oct 13m 8s -6.4912-Sep 3m 36s 4.22 12-Oct 13m 24s -7.1113-Sep 3m 57s 3.60 13-Oct 13m 38s -7.34

14-Sep 4m 19s 3.37 14-Oct 13m 53s -7.5615-Sep 4m 40s 3.14 15-Oct 14m 6s -8.1816-Sep 5m 1s 2.51 16-Oct 14m 20s -8.4117-Sep 5m 23s 2.27 17-Oct 14m 32s -9.0318-Sep 5m 44s 2.04 18-Oct 14m 44s -9.2519-Sep 6m 5s 1.41 19-Oct 14m 56s -9.4620-Sep 6m 27s 1.18 20-Oct 15m 6s -10.0821-Sep 6m 48s 0.54 21-Oct 15m 16s -10.3022-Sep 7m 9s 0.31 22-Oct 15m 26s -10.5123-Sep 7m 30s 0.08 23-Oct 15m 35s -11.12

24-Sep 7m 51s -0.15 24-Oct 15m 43s -11.3325-Sep 8m 12s -0.38 25-Oct 15m 50s -11.5426-Sep 8m 32s -1.02 26-Oct 15m 57s -12.1527-Sep 8m 53s -1.25 27-Oct 16m 3s -12.3528-Sep 9m 13s -1.48 28-Oct 16m 8s -12.5629-Sep 9m 33s -2.12 29-Oct 16m 12s -13.1630-Sep 9m 53s -2.35 30-Oct 16m 16s -13.36

31-Oct 16m 19s -13.55



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Appendix 2: Tan? Arctan? What? Math was a while ago...

What the heck are these functions? Well in the case of a right triangle there exists a verynice relationship between the sides of the triangle and the other angles. These relationshipshave been studied and used since antiquity, reaching their modern form under Euler in hisIntroductio in analysin innitorum in 1748.Take a right triangle (a triangle where one of the angles is 90 ◦ ) such as in Fig. 5.

Adjacent

Opposite Hypotenuse

x

Figure 5: A right triangle.

The trigonometric functions are functions that relate the angle x to any two of the threesides of the triangle. As “viewed” from the angle x we have labeled the sides Adjacent (theshorter of the two sides that form the angle) Opposite (the side ”facing” the angle) andHypotenuse (the longest side on the triangle).The three main trigonometric functions with respect to Fig 5 are:

sin(x) = OppositeHypotenuse

cos(x) = AdjacentHypotenuse

tan( x) = OppositeAdjacent

So computing Tan(x) is asking “what is the ratio of the opposite side to the adjacent side

of a right triangle with an angle of x◦

?” Useful if you know the angle and want to nd thelength of the sides.

Each of these functions has an inverse that “undoes” the function. These are the arc-functions. They are useful if you know the length of two sides of the triangle, and want toknow an angle.

arcsin( OppositeHypotenuse

) = x



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arccos( AdjacentHypotenuse

) = x

arctan(OppositeAdjacent

) = x

The arctan function here takes the ratio of the Opposite side to Adjacent side and reportsback the angle that makes it possible. Another way of thinking about the arctan function is

to read something like arctan(0.5) = x as “x is the angle whose tangent is 0.5” .In the case of this assignment we want the altitude angle of the Sun. We have the oppositeside to the angle (the height of the gnomon) and the adjacent side to the angle (the lengthof the shadow). In order to nd the angle α we would need to use:

arctanGnomonShadow

= Altitude angle of the Sun.


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