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Change of angle of aberration of light / change of velocity

by kmarinas86
Tags: aberration, angle, light, velocity
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kmarinas86
#1
Mar1-12, 02:08 PM
P: 1,011
Quote Quote by Wikipedia
Suppose, in the reference frame of the observer, the source is moving with speed [itex]v\,[/itex] at an angle [itex]\theta_s\,[/itex] relative to the vector from the observer to the source at the time when the light is emitted. Then the following formula, which was derived by Einstein in 1905, describes the aberration of the light source, [itex]\theta_o\,[/itex], measured by the observer:

[tex]\cos \theta_o=\frac{\cos \theta_s-\frac{v}{c}}{1-\frac{v}{c} \cos \theta_s} \,[/tex]
[itex]\theta_o-\theta_s[/itex] does not vary linearly with changes of [itex]v/c[/itex].

cos (y+b) = ((cos (y) - x)/(1 - x cos(y)))											
x=v/c											
y=angle of the source											
b=angular change											
											
		-x=1.00	-x=0.80	-x=0.60	-x=0.40	-x=0.20	x=0.00	x=0.20	x=0.40	x=0.60	x=0.80	x=1.00
y=0.00	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=0.63	-0.6283	-0.4125	-0.3062	-0.2091	-0.1097	0.0000	0.1291	0.2931	0.5242	0.9169	2.5133
y=1.26	-1.2566	-0.7814	-0.5597	-0.3687	-0.1858	0.0000	0.1977	0.4182	0.6794	1.0247	1.8850
y=1.88	-1.8850	-1.0247	-0.6794	-0.4182	-0.1977	0.0000	0.1858	0.3687	0.5597	0.7814	1.2566
y=2.51	-2.5133	-0.9169	-0.5242	-0.2931	-0.1291	0.0000	0.1097	0.2091	0.3062	0.4125	0.6283
y=3.14	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=3.77	-3.7699	-2.1736	-1.7808	-1.5497	-1.3858	-1.2566	-1.1470	-1.0475	-0.9504	-0.8441	-0.6283
y=4.40	-4.3982	-3.5379	-3.1927	-2.9314	-2.7110	-2.5133	-2.3275	-2.1446	-1.9535	-1.7318	-1.2566
y=5.03	-5.0265	-4.5513	-4.3297	-4.1386	-3.9557	-3.7699	-3.5722	-3.3517	-3.0905	-2.7452	-1.8850
y=5.65	-5.6549	-5.4391	-5.3328	-5.2357	-5.1362	-5.0265	-4.8974	-4.7335	-4.5023	-4.1096	-2.5133
y=6.28	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832
											
change in angle per additional 0.1 increase of v/c											
		-x=1.00	-x=0.80	-x=0.60	-x=0.40	-x=0.20	x=0.00	x=0.20	x=0.40	x=0.60	x=0.80	x=1.00
y=0.00	N/A		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=0.63	N/A		0.2158	0.1063	0.0971	0.0995	0.1097	0.1291	0.1639	0.2311	0.3927	1.5964
y=1.26	N/A		0.4752	0.2217	0.1910	0.1829	0.1858	0.1977	0.2205	0.2613	0.3452	0.8603
y=1.88	N/A		0.8603	0.3452	0.2613	0.2205	0.1977	0.1858	0.1829	0.1910	0.2217	0.4752
y=2.51	N/A		1.5964	0.3927	0.2311	0.1639	0.1291	0.1097	0.0995	0.0971	0.1063	0.2158
y=3.14	N/A		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=3.77	N/A		1.5964	0.3927	0.2311	0.1639	0.1291	0.1097	0.0995	0.0971	0.1063	0.2158
y=4.40	N/A		0.8603	0.3452	0.2613	0.2205	0.1977	0.1858	0.1829	0.1910	0.2217	0.4752
y=5.03	N/A		0.4752	0.2217	0.1910	0.1829	0.1858	0.1977	0.2205	0.2613	0.3452	0.8603
y=5.65	N/A		0.2158	0.1063	0.0971	0.0995	0.1097	0.1291	0.1639	0.2311	0.3927	1.5964
y=6.28	N/A		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
											
When using relativistically corrected velocities, we get:											
		-x=0.76	-x=0.66	-x=0.54	-x=0.38	-x=0.20	x=0.00	x=0.20	x=0.38	x=0.54	x=0.66	x=0.76
y=0.00	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=0.63	-0.3904	-0.3384	-0.2754	-0.1994	-0.1083	0.0000	0.1273	0.2744	0.4408	0.6238	0.8186
y=1.26	-0.7343	-0.6255	-0.4978	-0.3502	-0.1834	0.0000	0.1950	0.3946	0.5911	0.7772	0.9475
y=1.88	-0.9475	-0.7772	-0.5911	-0.3946	-0.1950	0.0000	0.1834	0.3502	0.4978	0.6255	0.7343
y=2.51	-0.8186	-0.6238	-0.4408	-0.2744	-0.1273	0.0000	0.1083	0.1994	0.2754	0.3384	0.3904
y=3.14	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=3.77	-2.0753	-1.8805	-1.6974	-1.5310	-1.3839	-1.2566	-1.1483	-1.0572	-0.9812	-0.9183	-0.8663
y=4.40	-3.4608	-3.2905	-3.1044	-2.9079	-2.7083	-2.5133	-2.3299	-2.1631	-2.0155	-1.8877	-1.7790
y=5.03	-4.5042	-4.3955	-4.2677	-4.1201	-3.9533	-3.7699	-3.5749	-3.3753	-3.1788	-2.9927	-2.8224
y=5.65	-5.4169	-5.3649	-5.3019	-5.2260	-5.1349	-5.0265	-4.8993	-4.7522	-4.5858	-4.4027	-4.2079
y=6.28	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832	-6.2832
											
change in angle per additional 0.1 increase of rapidity											
		-x=0.76	-x=0.66	-x=0.54	-x=0.38	-x=0.20	x=0.00	x=0.20	x=0.38	x=0.54	x=0.66	x=0.76
y=0.00	N/A		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=0.63	N/A		0.0520	0.0630	0.0760	0.0911	0.1083	0.1273	0.1471	0.1664	0.1831	0.1948
y=1.26	N/A		0.1087	0.1277	0.1476	0.1668	0.1834	0.1950	0.1996	0.1965	0.1861	0.1703
y=1.88	N/A		0.1703	0.1861	0.1965	0.1996	0.1950	0.1834	0.1668	0.1476	0.1277	0.1087
y=2.51	N/A		0.1948	0.1831	0.1664	0.1471	0.1273	0.1083	0.0911	0.0760	0.0630	0.0520
y=3.14	N/A		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
y=3.77	N/A		0.1948	0.1831	0.1664	0.1471	0.1273	0.1083	0.0911	0.0760	0.0630	0.0520
y=4.40	N/A		0.1703	0.1861	0.1965	0.1996	0.1950	0.1834	0.1668	0.1476	0.1277	0.1087
y=5.03	N/A		0.1087	0.1277	0.1476	0.1668	0.1834	0.1950	0.1996	0.1965	0.1861	0.1703
y=5.65	N/A		0.0520	0.0630	0.0760	0.0911	0.1083	0.1273	0.1471	0.1664	0.1831	0.1948
y=6.28	N/A		0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
Let:
r=rapidity

Solution for b':
http://www.quickmath.com/webMathemat...29-y&v2=r&v6=1

Solution for b'':
http://www.quickmath.com/webMathemat...2%29&v2=r&v6=1

Solution for b''=0:
http://www.quickmath.com/webMathemat...v11=0&v12=2*pi

Rapidity (Horizontal) vs. Theta_s (Vertical) (for the condition that b'' with respect to r eq 0).png
See the larger context of the plot at http://www.quickmath.com/webMathemat...v11=-20&v12=20

Observing the second derivative of the aberration angle with respect to changes in rapidity r, represented above as b'' (=[itex]\left(\theta_o-\theta_s\right)''[/itex]), does not require knowledge of the "true" b, and nor does it require that one choose a specific frame of reference to determine the initial value of r, as changes in r are invariant with respect to inertial frames. For constant force, the derivative of r with respect to time is a constant. Thus, a constant value of b', where b''=0, would be observed by taking a sample of b at equally-spaced time intervals and noting the minimal change in slope that results from plotting b over that slope.

Observing a condition where b''=0 also does not require knowing the value of y ([itex]\theta_s[/itex]). However, if one can find [itex]\theta_s[/itex] and r, one can also find b.

There are upper and lower bounds for r for which there are no real solutions. A test device can therefore be sent to determine an interval of r in which solutions exist. The length of the interval of r containing the real solutions does not depend on the frame of reference of an external observer. It is well defined by the above solution (http://www.physicsforums.com/attachm...1&d=1330650117). To go from one extreme solution to another requires specific changes of [itex]\theta_s[/itex]. In this way, [itex]\theta_s[/itex] can be known simply by alternating between: 1) observing the rate change of observed b' under constant linear acceleration, and 2) changing direction. From there, r can be known. Thus, v/c can also be known. The value of b can thus be known. Therefore, b is totally independent of the frame of reference. In this sense, this value of b is a "real" aberration angle. v/c in turn is a definitive velocity with a direction of an angle [itex]\theta_s[/itex] away from the light path. Thus, this definitive velocity between the light and the interceptor of that light is a kind that is independent of a distant observer, and thus, it is also independent of the frame of reference. A specific set of frames of reference would observe an orthogonal velocity between the light and the interceptor matching that of this definitive orthogonal velocity. This would indeed be a special set of inertial reference frames. If we were to repeat this example with variations in x and y, each having their own corresponding special sets of inertial reference frames, it would stand to reason that a single inertial reference frame would be a member of all of these sets.

The above suggests that rapidity provides a mathematical loophole in relativity in which a preferred frame could actually be found in principle. However, there are unknowns that will always be unknown (or relative to the observer) if, on the contrary, neither the angles nor the orthogonal velocity were definitive:

Caption:
"Comoving Star and Planet"
"Observer 1 sees…."
"Observer 2 sees…."
"How do we know which is the actual path?"
"Can we “really” know the angle between the light path and the object’s velocity?"

Answer to question #1 would be, "We don't."
Answer to question #2 would be, "We can't."
Attached Thumbnails
Comoving Star and Planet.png   Rapidity (Horizontal) vs. Theta_s (Vertical) (for the condition that b'' with respect to r eq 0).png  
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