Orbits of the Galilean Moons - Revision history

Matthew: moved image of moon and jupiter to conclusion

2022-04-30T04:39:04Z

moved image of moon and jupiter to conclusion

← Older revision		Revision as of 04:39, 30 April 2022
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	==Discussion==		==Discussion==

	[[File:MWMoonJupiterConjunction.jpg\|right\|thumb\|300px\|Conjunction between the moon and the Jovian system, showing their relative sizes; the four Galilean moons are also visible in the full-sized image]]

	===Validity of Model===		===Validity of Model===
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	==Conclusion==		==Conclusion==

			[[File:MWMoonJupiterConjunction.jpg\|right\|thumb\|300px\|Conjunction between the moon and the Jovian system, showing their relative sizes; the four Galilean moons are also visible in the full-sized image]]

	Through the course of this project, the four largest moons of Jupiter were surveyed over the course of two weeks. Therefrom, the following results were obtained:		Through the course of this project, the four largest moons of Jupiter were surveyed over the course of two weeks. Therefrom, the following results were obtained:

Matthew: /* Inclination */

2022-04-30T04:37:09Z

Inclination

← Older revision		Revision as of 04:37, 30 April 2022
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	====Inclination====		====Inclination====

			[[File:MWInclinationAnimation.gif\|right\|thumb\|450px\|
			'''Top''': profile view of the orbital plane as viewed from the side with Earth on the left; the orange dot is Jupiter while the red dot is one of its moons<br>'''Middle''': apparent positions as viewed by a distant observer from Earth<br>'''Bottom''': the apparent angular separation in arbitrary units and its plot over time]]

	The model assumes that the [https://en.wikipedia.org/wiki/Orbital_inclination inclination] of the moons' orbits relative to the line of sight with the Earth is zero. In other words, it assumes the moon should pass directly over the center of Jupiter when in actuality it will not, leading to inaccurate estimations of the orbital phase, especially so when the moon appears near to the center of Jupiter.		The model assumes that the [https://en.wikipedia.org/wiki/Orbital_inclination inclination] of the moons' orbits relative to the line of sight with the Earth is zero. In other words, it assumes the moon should pass directly over the center of Jupiter when in actuality it will not, leading to inaccurate estimations of the orbital phase, especially so when the moon appears near to the center of Jupiter.

Matthew: /* Conclusion */

2022-04-29T18:53:57Z

Conclusion

← Older revision		Revision as of 18:53, 29 April 2022
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	* Kepler's third law was validated to within 2.022%.		* Kepler's third law was validated to within 2.022%.

	The fact that these observations were performed with consumer equipment not specifically designed for astronomical observations, yet ~~that~~ still managed to produce results reasonably close to the established literature values, shows that astronomy is among the most accessible in all sub-fields of physics. It requires no specialized equipment available only to scientists in laboratories, only access to the night sky, the [https://en.wikipedia.org/wiki/Common_heritage_of_humanity common heritage of humanity].		The fact that these observations were performed with consumer equipment not specifically designed for astronomical observations, that yet still managed to produce results reasonably close to the established literature values, shows that astronomy is among the most accessible in all sub-fields of physics. It requires no specialized equipment available only to scientists in laboratories, only access to the night sky, the [https://en.wikipedia.org/wiki/Common_heritage_of_humanity common heritage of humanity].

	==Citations==		==Citations==

Matthew: /* Conclusion */

2022-04-29T18:50:15Z

Conclusion

← Older revision		Revision as of 18:50, 29 April 2022
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	* Kepler's third law was validated to within 2.022%.		* Kepler's third law was validated to within 2.022%.

	The fact that these observations were performed with consumer equipment not specifically designed for astronomical observations, yet still ~~managing~~ to produce results reasonably close to the established literature values, shows that astronomy is among the most accessible in all sub-fields of physics. It requires no specialized equipment available only to scientists in laboratories, only access to the night sky, the [https://en.wikipedia.org/wiki/Common_heritage_of_humanity common heritage of humanity].		The fact that these observations were performed with consumer equipment not specifically designed for astronomical observations, yet that still managed to produce results reasonably close to the established literature values, shows that astronomy is among the most accessible in all sub-fields of physics. It requires no specialized equipment available only to scientists in laboratories, only access to the night sky, the [https://en.wikipedia.org/wiki/Common_heritage_of_humanity common heritage of humanity].

	==Citations==		==Citations==

Matthew: /* Conclusion */

2022-04-29T18:48:59Z

Conclusion

← Older revision		Revision as of 18:48, 29 April 2022
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	==Conclusion==		==Conclusion==

			Through the course of this project, the four largest moons of Jupiter were surveyed over the course of two weeks. Therefrom, the following results were obtained:

			* The orbital periods of the four Galilean moons, Io, Europa, Ganymede, and Callisto were found to be, respectively, <math>T_\mathrm{Io}=1.770\;\mathrm{d}</math>, <math>T_\mathrm{Europa}=3.555\;\mathrm{d}</math>, <math>T_\mathrm{Ganymede}=7.196\;\mathrm{d}</math>, and <math>T_\mathrm{Callisto}=16.92\;\mathrm{d}</math>, within 1.407% of literature values.

			* Their relative orbital radii, in multiples of the orbital radius of Io were found to be, respectively, <math>r_\mathrm{Io}=1\;r_\mathrm{Io}</math>, <math>r_\mathrm{Europa}=1.573\;r_\mathrm{Io}</math>, <math>r_\mathrm{Ganymede}=2.534\;r_\mathrm{Io}</math>, <math>r_\mathrm{Callisto}=4.490\;r_\mathrm{Io}</math>, within 1.079% of literature values.

			* Their relative orbital periods, in multiples of the orbital period of Io were found to be, respectively, <math>T_\mathrm{Io}=1\;T_\mathrm{Io}</math>, <math>T_\mathrm{Europa}=2.008\;T_\mathrm{Io}</math>, <math>T_\mathrm{Ganymede}=4.065\;T_\mathrm{Io}</math>, <math>T_\mathrm{Callisto}=9.561\;T_\mathrm{Io}</math>, within 1.343% of literature values, in so doing demonstrating the orbital resonance phenomenon.

			* Kepler's third law was validated to within 2.022%.

			The fact that these observations were performed with consumer equipment not specifically designed for astronomical observations, yet still managing to produce results reasonably close to the established literature values, shows that astronomy is among the most accessible in all sub-fields of physics. It requires no specialized equipment available only to scientists in laboratories, only access to the night sky, the [https://en.wikipedia.org/wiki/Common_heritage_of_humanity common heritage of humanity].

	==Citations==		==Citations==

Matthew: added discrepancy from mean column

2022-04-29T18:48:39Z

added discrepancy from mean column

← Older revision		Revision as of 18:48, 29 April 2022
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	{\| class="wikitable" style="text-align: center		{\| class="wikitable" style="text-align: center
	\|-		\|-
	! scope="col" style="width: 25%" \| Satellite		! scope="col" style="width: 20%" \| Satellite
	! scope="col" style="width: 25%" \| Observed Amplitude<br><math>a</math> (a.u.)		! scope="col" style="width: 20%" \| Observed Amplitude<br><math>a</math> (a.u.)
	! scope="col" style="width: 25%" \| Observed Period<br><math>T</math> (d)		! scope="col" style="width: 20%" \| Observed Period<br><math>T</math> (d)
	! scope="col" style="width: 25%" \| Constant of proportionality<br><math>\frac{a^3}{T^2}</math>		! scope="col" style="width: 20%" \| Constant of proportionality<br><math>\frac{a^3}{T^2}</math>
			! scope="col" style="width: 20%" \| Discrepancy from Mean
	\|-		\|-
	! scope="row"\| Io		! scope="row"\| Io
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	\| 1.770		\| 1.770
	\| 303 900		\| 303 900
			\| 1.539%
	\|-		\|-
	! scope="row"\| Europa		! scope="row"\| Europa
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	\| 3.555		\| 3.555
	\| 293 300		\| 293 300
			\| 2.022%
	\|-		\|-
	! scope="row"\| Ganymede		! scope="row"\| Ganymede
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	\| 7.193		\| 7.193
	\| 299 100		\| 299 100
			\| 0.065%
	\|-		\|-
	! scope="row"\| Callisto		! scope="row"\| Callisto
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	\| 16.92		\| 16.92
	\| 301 000		\| 301 000
			\| 0.547%
	\|}		\|}

Matthew: /* Imaging */

2022-04-29T18:16:34Z

Imaging

← Older revision		Revision as of 18:16, 29 April 2022
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	&=0.5119^\circ\end{align}</math>		&=0.5119^\circ\end{align}</math>

	Since the moon occupies <math>n=1124\;\mathrm{px}</math> of this image horizontally, the field of view would be <math display="inline">\phi=\frac{5184}{1124}\times0.5119^\circ=2.36^\circ</math>		Since the moon occupies <math>n=1124\;\mathrm{px}</math> of this image horizontally, the field of view would be <math display="inline">\phi=\frac{5184}{1124}\times0.5119^\circ=2.36^\circ</math>.

	====Discussion====		====Discussion====

Matthew: /* Angular Field of View */

2022-04-29T18:14:57Z

Angular Field of View

← Older revision		Revision as of 18:14, 29 April 2022
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	This is the angle corresponding to a single pixel in a captured image. Multiplying that with the number of pixels horizontally <math>n=5184\;\mathrm{px}</math><ref name="canoncamera"/>, the field of view would be <math>\phi=0.000412^\circ\times5184=2.13^\circ</math>.		This is the angle corresponding to a single pixel in a captured image. Multiplying that with the number of pixels horizontally <math>n=5184\;\mathrm{px}</math><ref name="canoncamera"/>, the field of view would be <math>\phi=0.000412^\circ\times5184=2.13^\circ</math>.

	====~~Empirically~~====		====Imaging====

	[[File:MWMoonCapture.jpg\|right\|thumb\|300px\|Image of the full moon used to calculate the angular field of view of the setup]]		[[File:MWMoonCapture.jpg\|right\|thumb\|300px\|Image of the full moon used to calculate the angular field of view of the setup]]
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	&=0.5119^\circ\end{align}</math>		&=0.5119^\circ\end{align}</math>

	Since the moon occupies <math>n=1124\;\mathrm{px}</math> of this image horizontally, the field of view would be <math display="inline">\phi=\frac{5184}{1124}0.5119^\circ=2.36^\circ</math>		Since the moon occupies <math>n=1124\;\mathrm{px}</math> of this image horizontally, the field of view would be <math display="inline">\phi=\frac{5184}{1124}\times0.5119^\circ=2.36^\circ</math>

			====Discussion====

			These methods assume that the image contains no [https://en.wikipedia.org/wiki/Distortion_(optics) lens distortion], especially at the fringes of the image, far from the [https://en.wikipedia.org/wiki/Optical_axis optical axis], which may not be true and lead to the discrepancies between the derived values above.

	==Conclusion==		==Conclusion==
	==Citations==		==Citations==

Matthew: /* Discussion */

2022-04-29T18:07:40Z

Discussion

← Older revision		Revision as of 18:07, 29 April 2022
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	==Discussion==		==Discussion==

			[[File:MWMoonJupiterConjunction.jpg\|right\|thumb\|300px\|Conjunction between the moon and the Jovian system, showing their relative sizes; the four Galilean moons are also visible in the full-sized image]]

	===Validity of Model===		===Validity of Model===

Matthew: /* Angular Field of View */

2022-04-29T18:00:12Z

Angular Field of View

← Older revision		Revision as of 18:00, 29 April 2022
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	===Angular Field of View===		===Angular Field of View===
	A few methods can be used to ~~infer~~ the ~~correlation between pixels in~~ the ~~image~~ and ~~angular separation on~~ the ~~celestial sphere~~.		A few methods can be used to determine the horizontal field of view of the setup. Being able to measure angles precisely is an important part of astronomy and can be used to infer physical distances using trigonometry assuming the absolute distances to the objects of interest are known.

	====Manufacturer's Claim====		====Manufacturer's Claim====
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	&=0^\circ\,0'\,1.48''\end{align}</math>		&=0^\circ\,0'\,1.48''\end{align}</math>

	Multiplying that with the number of pixels horizontally <math>n=5184\;\mathrm{px}</math><ref name="canoncamera"/>, the field of view would be <math>\phi=0.000412^\circ\times5184=2.13^\circ</math>.		This is the angle corresponding to a single pixel in a captured image. Multiplying that with the number of pixels horizontally <math>n=5184\;\mathrm{px}</math><ref name="canoncamera"/>, the field of view would be <math>\phi=0.000412^\circ\times5184=2.13^\circ</math>.

	====Empirically====		====Empirically====
	~~====Comparing with Theoretical Values====~~

	~~During~~ the ~~observation~~ on 1 Feb, ~~Callisto~~ and ~~Jupiter were~~		[[File:MWMoonCapture.jpg\|right\|thumb\|300px\|Image of the full moon used to calculate the angular field of view of the setup]]

			A photo of the full moon was taken on 17 Feb. Knowing the distance to the moon <math>r_\mathrm{Moon}=389\,110\;\mathrm{km}</math><ref>Wolfram Research (2014) [https://reference.wolfram.com/language/ref/PlanetaryMoonData.html <code>PlanetaryMoonData</code>], Wolfram Language function</ref> and the diameter of the moon <math>d_\mathrm{Moon}=3476.2\;\mathrm{km}</math><ref>NASA, [https://en.wikipedia.org/wiki/Goddard_Space_Flight_Center GSFC], [https://en.wikipedia.org/wiki/NASA_Space_Science_Data_Coordinated_Archive NSSDC] (December 2021) [https://nssdc.gsfc.nasa.gov/planetary/factsheet/galileanfact_table.html ''Moon Fact Sheet'']</ref>, the angular diameter of the moon <math>\theta_\mathrm{Moon}</math> can be calculated by

			<math display="block">\begin{align}\theta_\mathrm{Moon}&=\arctan\left(\frac{3476.2}{389\,110}\right)\\
			&=0.5119^\circ\end{align}</math>

			Since the moon occupies <math>n=1124\;\mathrm{px}</math> of this image horizontally, the field of view would be <math display="inline">\phi=\frac{5184}{1124}0.5119^\circ=2.36^\circ</math>

	==Conclusion==		==Conclusion==
	==Citations==		==Citations==