DOI: 10.13140/RG.2.2.33004.54403
Soumendra Nath Thakur¹
¹Tagore's Electronic Lab. India¹
postmasterenator@gmail.com¹
postmasterenator@telitnetwork.in¹
18 September 2023
Chapter Abstract:
Redshift, a fundamental phenomenon in astrophysics and cosmology, is explored in detail through its governing equations. We delve into equations describing redshift as a function of wavelength and frequency changes, energy changes, and phase shifts. These equations provide insights into the behaviour of electromagnetic waves as sources move relative to observers. The mathematical rigor employed in deriving and interpreting these equations enhances our comprehension of redshift, its role in measuring celestial velocities and universe expansion, and its counterpart, blueshift. The interplay between frequency, wavelength, energy, and phase shift sheds light on this critical aspect of cosmological observation.
Keywords: Redshift, Blueshift, Phase Shift, Electromagnetic waves,
Introduction:
The fundamental understanding of electromagnetic wave behaviour and its relation to various phenomena has been instrumental in advancing astrophysics, cosmology, and telecommunications. This paper explores essential equations governing electromagnetic waves, including the redshift equation, which describes the change in wavelength and frequency as waves propagate through space. Additionally, the phase shift equation sheds light on how wave temporal behaviour is influenced by frequency, playing a critical role in fields like signal processing and telecommunications.
Methods:
In this study, we employ rigorous mathematical derivations to elucidate the key equations governing redshift and phase shift in electromagnetic waves. We analyze these equations, including their relationships with frequency, wavelength, energy changes, and phase shift, to provide a comprehensive understanding of their significance. Our methodology involves detailed mathematical derivations and interpretations to uncover the fundamental principles underlying these phenomena.
Equations and Descriptions:
1.1. Redshift Equation:
z = Δλ/λ;
z = f/Δf;
z represents the redshift factor. Δλ stands for the change in wavelength of light. λ represents the initial wavelength of light. f stands for the initial frequency of light. Δf represents the change in frequency of light. This equation relates the relative change in wavelength (Δλ/λ) to the relative change in frequency (f/Δf) for electromagnetic waves. It's essentially expressing the idea that as the wavelength of a wave changes, there is a corresponding change in its frequency, and vice versa, while maintaining a constant speed (c) as per the relationship c = λf, where c is the speed of light.
1.2. Phase shift Equation:
1° phase shift = T/360
Since, T = 1/f, we have:
1° phase shift = (1/f)/360
T (deg) = 1/ (360f);
T represents the period of the wave. f represents the frequency of the wave. T (deg) represents the period of the wave measured in degrees. The phase shift equation represented as "1° phase shift = T/360," plays a crucial role in understanding the temporal behaviour of waves in relation to their frequency (f). It elucidates that a 1-degree phase shift corresponds to a fraction of the wave's period (T), with the denominator 360 indicating that a full cycle of a wave consists of 360 degrees. To further explore this equation, we can express the wave's period (T) in terms of its frequency (f), leading to the equation "1° phase shift = (1/f)/360." This equation highlights that the phase shift, measured in degrees (°), is inversely proportional to the frequency (f) of the wave. As the frequency increases, the phase shift decreases, and vice versa.
2. Redshift as a Function of wavelength Change:
Δλ/λ
Δλ represents the change in wavelength. λ represents the initial or reference wavelength. Δλ/λ represents the phenomenon of redshift in the context of electromagnetic waves. Redshift occurs when an object emitting waves moves away from an observer, causing the waves to stretch or lengthen, resulting in an increase in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "Δλ" represents the change in wavelength, and "λ" represents the original wavelength of the waves. By calculating the ratio of Δλ to λ, you can determine the extent of redshift. If the value of Δλ/λ is greater than 1, it indicates that the wavelength has increased, which corresponds to a redshift. This is a fundamental concept in astrophysics and cosmology, as redshift is commonly used to measure the recessional velocities of distant celestial objects, such as galaxies, and to study the expansion of the universe.
3. Blueshift as a Function of wavelength Change:
-Δλ/λ
-Δλ represents the negative change in wavelength. λ represents the initial or reference wavelength. -Δλ/λ represents the phenomenon of blueshift in the context of electromagnetic waves; Blueshift occurs when an object emitting waves moves toward an observer, causing the waves to compress or shorten, resulting in a decrease in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "Δλ" represents the change in wavelength, and "λ" represents the original wavelength of the waves. By calculating the ratio of -Δλ to λ, you can determine the extent of blueshift. If the value of -Δλ/λ is less than 0 (negative), it indicates that the wavelength has decreased, which corresponds to a blueshift.
4. Redshift as a Function of Frequency Change:
z = f/Δf
z represents the redshift factor. f is the observed frequency of light. Δf is the change in frequency from the source to the observer. f/Δf describes the phenomenon of redshift in the context of electromagnetic waves. Redshift occurs when an object emitting waves moves away from an observer, causing the waves to stretch or lengthen, resulting in an increase in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "f" represents the frequency of the waves, and "Δf" represents the change in frequency. By calculating the ratio of "f" to "Δf," you can determine the extent of redshift. If the value of "f/Δf" is greater than 1, it indicates that the frequency has decreased, which corresponds to a redshift.
5. Blueshift as a Function of Frequency Change:
z = f/-Δf
z represents the redshift (or blueshift) factor. f is the observed frequency of light. -Δf is the negative change in frequency from the source to the observer. f/-Δf describes the phenomenon of blueshift in the context of electromagnetic waves. Blueshift occurs when an object emitting waves moves toward an observer, causing the waves to compress or shorten, resulting in a decrease in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "f" represents the frequency of the waves, and "-Δf" represents the change in frequency. By calculating the ratio of "f" to "-Δf," you can determine the extent of blueshift. If the value of "f/-Δf" is greater than 1, it indicates that the frequency has increased, which corresponds to a blueshift.
6. Redshift as a Function of Positive Energy Change:
z = ΔE/E
z represents the redshift factor. ΔE is the change in energy of the radiation. E is the initial energy of the radiation. ΔE/E describes the phenomenon of redshift in the context of electromagnetic waves when there is a positive change in energy (ΔE). Redshift occurs when an object emitting waves is moving away from an observer, causing the waves to stretch or lengthen, resulting in an increase in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "ΔE" represents the change in energy, and "E" represents the initial energy of the electromagnetic waves. By calculating the ratio of "ΔE" to "E," you can determine the extent of redshift. If the value of "ΔE/E" is greater than zero (indicating a positive change in energy), it signifies that the wavelength has increased, corresponding to a redshift.
7. Blueshift as a Function of Negative Energy Change:
z = -ΔE/E
-ΔE The negative sign indicates a decrease or reduction in energy, ΔE represents the change in energy. -ΔE/E describes the phenomenon of blueshift in the context of electromagnetic waves when there is a negative change in energy (ΔE). Blueshift occurs when an object emitting waves is moving toward an observer, causing the waves to compress or shorten, resulting in a decrease in wavelength (Δλ) compared to the original wavelength (λ). In this equation, "ΔE" represents the change in energy, and "E" represents the initial energy of the electromagnetic waves. By calculating the ratio of "ΔE/E," you can determine the extent of blueshift. If the value of "ΔE/E" is less than zero (indicating a negative change in energy), it signifies that the wavelength has decreased, corresponding to a blueshift.
8. Redshift (z) as a Function of Phase Shift T(deg):
z = 360 * T(deg) * ΔE/h
z represents the redshift. T(deg) represents an angle measured in degrees. ΔE represents the change in energy. h represents Planck's constant. 360 * T(deg) * ΔE/h describes the relationship between redshift (z) and phase shift T(deg) in the context of electromagnetic waves and energy changes. The equation suggests that redshift (z) is directly related to phase shift T(deg), the change in energy (ΔE), and the Planck constant (h). When the phase shift or energy changes increases, it can lead to a corresponding increase in redshift. Conversely, when the phase shift or energy changes decreases, it may result in a decrease in redshift. This equation has several components.
9. Blueshift (z) as a Function of Phase Shift T(deg):
z = -Δf * 360 * T(deg)
z represents the redshift. Δf represents the change in frequency. T(deg) represents an angle measured in degrees. -Δf * 360 * T(deg) describes the relationship between blueshift (z) and phase shift T(deg) in the context of electromagnetic waves and frequency changes. The equation suggests that blueshift (z) is directly related to phase shift T(deg) and the change in frequency (Δf).
When the phase shift or frequency changes increases, it can lead to a corresponding increase in blueshift. Conversely, when the phase shift or frequency changes decreases, it may result in a decrease in blueshift. This equation has several components.
10. Phase Shift T(deg) as a Function of Redshift (z):
T(deg) = h / (-360 * z * E)
T(deg) represents an angle measured in degrees. h is Planck's constant. z represents the redshift. E represents energy. h / (-360 * z * E) describes the relationship between phase shift T(deg) and redshift (z) in the context of electromagnetic waves and energy changes. The equation suggests that phase shift T(deg) is inversely related to redshift (z) and the energy (E) of electromagnetic waves.
When redshift increases (indicating that the source is moving away), phase shift decreases, and vice versa. Additionally, the energy of the waves is involved in this relationship, affecting the extent of the phase shift. This equation has several components:
11. Phase Shift T(deg) as a Function of Blueshift (z):
T(deg) = h / (-360 * z * E)
T(deg): This represents an angle measured in degrees. h: Planck's constant. z: Redshift. E: Energy.
h / (-360 * z * E) describes the relationship between phase shift T(deg) and blueshift (z) in the context of electromagnetic waves and energy changes. The equation suggests that phase shift T(deg) is inversely related to blueshift (z) and the energy (E) of electromagnetic waves. As blueshift increases (indicating that the source is approaching), phase shift decreases, and vice versa. Additionally, the energy of the waves is involved in this relationship, influencing the extent of the phase shift. This equation involves several key components.
Discussion:
The redshift equation (z = Δλ/λ; z = f/Δf) is a cornerstone in astrophysics and cosmology. It relates the relative change in wavelength (Δλ/λ) to the relative change in frequency (f/Δf) of electromagnetic waves. This equation reveals that as a source emitting waves moves away from an observer, the wavelength increases, resulting in a redshift. Conversely, blueshift occurs when the source approaches, causing a decrease in wavelength.
The phase shift equation (1° phase shift = T/360; T (deg) = 1/(360f)) provides insight into wave behaviour concerning frequency. It demonstrates that a 1-degree phase shift corresponds to a fraction of the wave's period (T), inversely proportional to the frequency (f). This equation is pivotal in telecommunications and signal processing, where precise control of phase is crucial for data transmission and modulation.
Furthermore, we discuss redshift and blueshift in the context of wavelength and frequency changes. Redshift (Δλ/λ) occurs when an object moves away, causing wavelength elongation. Blueshift (-Δλ/λ) arises when an object approaches, leading to wavelength compression. Similarly, redshift (z = f/Δf) and blueshift (z = f/-Δf) are explored concerning frequency changes. These phenomena are instrumental in determining the recessional velocities of celestial objects and are vital for understanding the universe's expansion.
Additionally, we delve into redshift and blueshift as functions of energy changes (ΔE/E). Positive energy changes lead to redshift, reflecting a stretching of waves, while negative energy changes result in blueshift, indicating wave compression. We also discuss the relationship between redshift (z) and phase shift T(deg), highlighting the role of energy changes (ΔE) and the Planck constant (h).
Finally, we examine the relationship between phase shift T(deg) and redshift (z) and blueshift (z), emphasizing their dependence on energy (E) and the Planck constant (h).
Conclusion:
In conclusion, the equations governing redshift, blueshift, and phase shift in electromagnetic waves are essential tools in astrophysics, cosmology, and telecommunications. The redshift equation, with its links to wavelength and frequency changes, provides crucial insights into the expansion of the universe and the velocity of celestial objects. The phase shift equation is fundamental in controlling phase in various applications, from data transmission to signal modulation. Understanding these equations enhances our comprehension of wave behaviour and its implications across diverse scientific disciplines.
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