Seeking Expression For Radial Velocity Of An Object In An Elliptical Orbit (with Focus At Coordinate Origin) As Function Of Radius

Introduction

In classical mechanics, the study of orbital motion is crucial in understanding the behavior of celestial bodies and their interactions. One of the fundamental aspects of orbital motion is the radial velocity of an object, which is the velocity component directed towards or away from the focus of the orbit. In this article, we will focus on deriving an expression for the radial velocity of an object in an elliptical orbit, with the focus at the coordinate origin.

Problem Formulation

Consider a planar, elliptical orbit in a simplified two-body, $\frac{K}{r^2}$ central attractive force problem. In this scenario, we assume that the mass of the primary body ($m_1$) is significantly larger than the mass of the secondary body ($m_2$), so the focus $f_1$ is effectively at $m_1$, with $m_2$ at point $p\left(x, y\right)$. The central attractive force is given by the equation $\vec{F} = -\frac{K}{r^2} \hat{r}$, where $K$ is a constant, $r$ is the distance between the two bodies, and $\hat{r}$ is the unit vector pointing from $m_2$ to $m_1$.

Derivation of Radial Velocity

To derive an expression for the radial velocity of an object in an elliptical orbit, we need to consider the equation of motion for the object. The equation of motion for an object in a central force field is given by:

$$\frac{d^2\vec{r}}{dt^2} = -\frac{K}{r^2} \hat{r}$$

where $\vec{r}$ is the position vector of the object. We can rewrite this equation in terms of the radial and transverse components of the position vector:

$$\frac{d^2r}{dt^2} = -\frac{K}{r^2}$$

$$\frac{d^2\theta}{dt^2} = 0$$

where $r$ is the radial distance from the focus, and $\theta$ is the angular position of the object.

Solution of Radial Equation

To solve the radial equation, we can use the following substitution:

$$u = \frac{1}{r}$$

This substitution transforms the radial equation into a second-order differential equation in $u$:

$$\frac{d^2u}{dt^2} + \frac{K}{u^3} \frac{du}{dt} = 0$$

This equation can be solved using the method of separation of variables. We can separate the variables $u$ and $t$ as follows:

$$\frac{du}{dt} = -\frac{K}{u^3} \frac{du}{dt}$$

$$\frac{du}{u^3} = -\frac{K}{u^3} dt$$

Integrating both sides of this equation, we get:

$$\int \frac{du}{u^3} = -\frac{K}{u^3} \int dt$$

$$-\frac{1}{2u^2} = -\frac{K}{u^3} t + C$$

where $C$ is a constant of integration.

Solution of Angular

The angular equation is a simple harmonic oscillator equation, which can be solved using the following solution:

$$\theta(t) = \theta_0 + \omega t$$

where $\theta_0$ is the initial angular position, and $\omega$ is the angular frequency.

Expression for Radial Velocity

Using the solutions for the radial and angular equations, we can derive an expression for the radial velocity of an object in an elliptical orbit. The radial velocity is given by:

$$v_r = \frac{dr}{dt} = -\frac{K}{r^2} \frac{dr}{dt}$$

Substituting the solution for $r(t)$, we get:

$$v_r = -\frac{K}{r^2} \left(-\frac{1}{2r^2} \right)$$

$$v_r = \frac{K}{2r^3}$$

This is the expression for the radial velocity of an object in an elliptical orbit.

Conclusion

In this article, we have derived an expression for the radial velocity of an object in an elliptical orbit, with the focus at the coordinate origin. The expression is given by $v_r = \frac{K}{2r^3}$, where $K$ is a constant, and $r$ is the radial distance from the focus. This expression is a fundamental result in classical mechanics, and has important implications for the study of orbital motion.

References

• [1] Goldstein, H. (1980). Classical Mechanics. Addison-Wesley.
• [2] Landau, L. D., & Lifshitz, E. M. (1976). The Classical Theory of Fields. Pergamon Press.
• [3] Taylor, E. F. (1963). Classical Mechanics. McGraw-Hill.

Appendix

Derivation of Elliptical Orbit

To derive the expression for the radial velocity, we need to consider the equation of motion for the object in an elliptical orbit. The equation of motion for an object in a central force field is given by:

$$\frac{d^2\vec{r}}{dt^2} = -\frac{K}{r^2} \hat{r}$$

where $\vec{r}$ is the position vector of the object. We can rewrite this equation in terms of the radial and transverse components of the position vector:

$$\frac{d^2r}{dt^2} = -\frac{K}{r^2}$$

$$\frac{d^2\theta}{dt^2} = 0$$

where $r$ is the radial distance from the focus, and $\theta$ is the angular position of the object.

To solve the radial equation, we can use the following substitution:

$$u = \frac{1}{r}$$

This substitution transforms the radial equation into a second-order differential equation in $u$:

$$\frac{d^2u}{dt^2} + \frac{K}{u^3} \frac{du}{dt} = 0$$

This equation can be solved using the method of separation of variables. We can separate the variables $u$ and $t$ as follows:

$$\frac{du}{dt} = -\frac{K}{u^3} \frac{du}{dt}$$

$$\frac{du}{^3} = -\frac{K}{u^3} dt$$

Integrating both sides of this equation, we get:

$$\int \frac{du}{u^3} = -\frac{K}{u^3} \int dt$$

$$-\frac{1}{2u^2} = -\frac{K}{u^3} t + C$$

where $C$ is a constant of integration.

The solution to this equation is:

$$u(t) = \frac{1}{r(t)} = \frac{1}{\sqrt{a(1 - e^2)}} \left(1 + e \cos \left(\sqrt{\frac{K}{m}} t\right)\right)$$

where $a$ is the semi-major axis, $e$ is the eccentricity, and $m$ is the mass of the object.

Using this solution, we can derive an expression for the radial velocity of an object in an elliptical orbit. The radial velocity is given by:

$$v_r = \frac{dr}{dt} = -\frac{K}{r^2} \frac{dr}{dt}$$

Substituting the solution for $r(t)$, we get:

$$v_r = -\frac{K}{r^2} \left(-\frac{1}{2r^2} \right)$$

$$v_r = \frac{K}{2r^3}$$

This is the expression for the radial velocity of an object in an elliptical orbit.

Derivation of Angular Velocity

The angular velocity of an object in an elliptical orbit is given by:

$$\omega = \frac{d\theta}{dt}$$

Using the solution for $\theta(t)$, we get:

$$\omega = \frac{d}{dt} \left(\theta_0 + \omega t\right)$$

$$\omega = \omega$$

This is the expression for the angular velocity of an object in an elliptical orbit.

Derivation of Orbital Period

The orbital period of an object in an elliptical orbit is given by:

$$T = \frac{2\pi}{\omega}$$

Using the solution for $\omega$, we get:

$$T = \frac{2\pi}{\omega}$$

$$T = \frac{2\pi}{\sqrt{\frac{K}{m}}}$$

This is the expression for the orbital period of an object in an elliptical orbit.

Introduction

In our previous article, we derived an expression for the radial velocity of an object in an elliptical orbit, with the focus at the coordinate origin. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the radial velocity of an object in an elliptical orbit?

A: The radial velocity of an object in an elliptical orbit is the velocity component directed towards or away from the focus of the orbit. It is given by the expression $v_r = \frac{K}{2r^3}$, where $K$ is a constant, and $r$ is the radial distance from the focus.

Q: What is the significance of the radial velocity in an elliptical orbit?

A: The radial velocity is an important quantity in understanding the behavior of an object in an elliptical orbit. It determines the rate at which the object approaches or recedes from the focus of the orbit.

Q: How does the radial velocity change as the object moves along the orbit?

A: The radial velocity changes as the object moves along the orbit. It is maximum when the object is at the periapsis (closest point to the focus) and minimum when the object is at the apoapsis (farthest point from the focus).

Q: What is the relationship between the radial velocity and the orbital period?

A: The radial velocity is related to the orbital period by the expression $v_r = \frac{2\pi r}{T}$, where $T$ is the orbital period.

Q: How does the radial velocity affect the motion of the object in the orbit?

A: The radial velocity affects the motion of the object in the orbit by determining the rate at which the object approaches or recedes from the focus of the orbit. It also affects the shape of the orbit and the orbital period.

Q: Can the radial velocity be used to determine the mass of the object in the orbit?

A: Yes, the radial velocity can be used to determine the mass of the object in the orbit. By measuring the radial velocity and the orbital period, we can determine the mass of the object using the expression $m = \frac{4\pi^2 r^3}{GT^2}$, where $G$ is the gravitational constant.

Q: What are some of the applications of the radial velocity in an elliptical orbit?

A: The radial velocity has many applications in astronomy and space exploration. It is used to determine the mass of celestial bodies, study the motion of planets and stars, and understand the behavior of galaxies and galaxy clusters.

Q: How can the radial velocity be measured in an elliptical orbit?

A: The radial velocity can be measured using various techniques, including spectroscopy, interferometry, and astrometry. These techniques involve measuring the shift in the wavelength of light emitted or absorbed by the object, or the change in the position of the object in the sky.

Q: What are some of the challenges associated with measuring the radial velocity in an elliptical orbit?

A: Measuring the radial velocity in an elliptical orbit can be challenging due to various factors, including the small size of the radial velocity, presence of noise and systematic errors, and the need for high-precision measurements.

Conclusion

In this article, we have answered some of the most frequently asked questions related to the radial velocity of an object in an elliptical orbit. We hope that this article has provided a useful overview of this topic and has helped to clarify some of the concepts and ideas involved.

References

• [1] Goldstein, H. (1980). Classical Mechanics. Addison-Wesley.
• [2] Landau, L. D., & Lifshitz, E. M. (1976). The Classical Theory of Fields. Pergamon Press.
• [3] Taylor, E. F. (1963). Classical Mechanics. McGraw-Hill.

Appendix

Derivation of Radial Velocity

To derive the expression for the radial velocity, we need to consider the equation of motion for the object in an elliptical orbit. The equation of motion for an object in a central force field is given by:

$$\frac{d^2\vec{r}}{dt^2} = -\frac{K}{r^2} \hat{r}$$

where $\vec{r}$ is the position vector of the object. We can rewrite this equation in terms of the radial and transverse components of the position vector:

$$\frac{d^2r}{dt^2} = -\frac{K}{r^2}$$

$$\frac{d^2\theta}{dt^2} = 0$$

where $r$ is the radial distance from the focus, and $\theta$ is the angular position of the object.

To solve the radial equation, we can use the following substitution:

$$u = \frac{1}{r}$$

This substitution transforms the radial equation into a second-order differential equation in $u$:

$$\frac{d^2u}{dt^2} + \frac{K}{u^3} \frac{du}{dt} = 0$$

This equation can be solved using the method of separation of variables. We can separate the variables $u$ and $t$ as follows:

$$\frac{du}{dt} = -\frac{K}{u^3} \frac{du}{dt}$$

$$\frac{du}{u^3} = -\frac{K}{u^3} dt$$

Integrating both sides of this equation, we get:

$$\int \frac{du}{u^3} = -\frac{K}{u^3} \int dt$$

$$-\frac{1}{2u^2} = -\frac{K}{u^3} t + C$$

where $C$ is a constant of integration.

The solution to this equation is:

$$u(t) = \frac{1}{r(t)} = \frac{1}{\sqrt{a(1 - e^2)}} \left(1 + e \cos \left(\sqrt{\frac{K}{m}} t\right)\right)$$

where $a$ is the semi-major axis, $e$ is the eccentricity, and $m$ is the mass of the object.

Using this solution, we can derive an expression for the radial velocity of an object in an elliptical orbit. The radial velocity is given by:

$$v_r = \frac{dr}{dt} = -\frac{K}{r^2}frac{dr}{dt}$$

Substituting the solution for $r(t)$, we get:

$$v_r = -\frac{K}{r^2} \left(-\frac{1}{2r^2} \right)$$

$$v_r = \frac{K}{2r^3}$$

This is the expression for the radial velocity of an object in an elliptical orbit.

Derivation of Angular Velocity

The angular velocity of an object in an elliptical orbit is given by:

$$\omega = \frac{d\theta}{dt}$$

Using the solution for $\theta(t)$, we get:

$$\omega = \frac{d}{dt} \left(\theta_0 + \omega t\right)$$

$$\omega = \omega$$

This is the expression for the angular velocity of an object in an elliptical orbit.

Derivation of Orbital Period

The orbital period of an object in an elliptical orbit is given by:

$$T = \frac{2\pi}{\omega}$$

Using the solution for $\omega$, we get:

$$T = \frac{2\pi}{\omega}$$

$$T = \frac{2\pi}{\sqrt{\frac{K}{m}}}$$

This is the expression for the orbital period of an object in an elliptical orbit.