This section transforms the problem defined in the problem description into one which is expressed in mathematical terms. It uses concrete symbols defined in the data definitions to replace the abstract symbols in the models identified in theoretical models and general definitions.
| Refname | IM:accelX1 |
|---|
| Label | X-acceleration of the first star |
| Input | \({m_{1}}\), \({m_{2}}\), \({x_{1}}\), \({y_{1}}\), \({x_{2}}\), \({y_{2}}\) |
| Output | \({a_{\text{x}1}}\) |
| Input Constraints | \[{m_{1}}\gt{}0\]\[{m_{2}}\gt{}0\] |
| Output Constraints | |
| Equation | \[{a_{\text{x}1}}\left({x_{1}},{y_{1}},{x_{2}},{y_{2}}\right)=\frac{-G\,{m_{2}}\,\left({x_{1}}-{x_{2}}\right)}{{r_{12}}^{3}}\] |
| Description |
- \({a_{\text{x}1}}\) is the x-acceleration of the first star (\(\frac{\text{m}}{\text{s}^{2}}\))
- \({x_{1}}\) is the x-position of the first star (\({\text{m}}\))
- \({y_{1}}\) is the y-position of the first star (\({\text{m}}\))
- \({x_{2}}\) is the x-position of the second star (\({\text{m}}\))
- \({y_{2}}\) is the y-position of the second star (\({\text{m}}\))
- \(G\) is the gravitational constant (\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\))
- \({m_{2}}\) is the mass of the second star (\({\text{kg}}\))
- \({r_{12}}\) is the separation distance (\({\text{m}}\))
|
| Notes |
- \({a_{\text{x}1}}\) is calculated by solving the ODE together with the initial conditions . This model is derived from Newton’s second law (TM:NewtonSecLawMot), universal gravitation (TM:UniversalGravLaw), and the definitions of acceleration (TM:acceleration), velocity (TM:velocity) and relative position (TM:relPosAndSep), under the center-of-mass constraint (TM:centerOfMass) . The assumptions are: two-body system (A:twoBody), isolated system (A:isolated), Newtonian gravitation (A:newtonianGravity), non-relativistic mechanics (A:nonRelativistic), point masses (A:pointMass), constant masses (A:constantMass), inertial reference frame (A:inertialFrame), planar motion (A:planar) and non-zero separation (A:nonzeroSeparation).
|
| Source | – |
| RefBy | FR:Output-Values and FR:Calculate-Positions |
| Refname | IM:accelY1 |
|---|
| Label | Y-acceleration of the first star |
| Input | \({m_{1}}\), \({m_{2}}\), \({x_{1}}\), \({y_{1}}\), \({x_{2}}\), \({y_{2}}\) |
| Output | \({a_{\text{y}1}}\) |
| Input Constraints | \[{m_{1}}\gt{}0\]\[{m_{2}}\gt{}0\] |
| Output Constraints | |
| Equation | \[{a_{\text{y}1}}\left({x_{1}},{y_{1}},{x_{2}},{y_{2}}\right)=\frac{-G\,{m_{2}}\,\left({y_{1}}-{y_{2}}\right)}{{r_{12}}^{3}}\] |
| Description |
- \({a_{\text{y}1}}\) is the y-acceleration of the first star (\(\frac{\text{m}}{\text{s}^{2}}\))
- \({x_{1}}\) is the x-position of the first star (\({\text{m}}\))
- \({y_{1}}\) is the y-position of the first star (\({\text{m}}\))
- \({x_{2}}\) is the x-position of the second star (\({\text{m}}\))
- \({y_{2}}\) is the y-position of the second star (\({\text{m}}\))
- \(G\) is the gravitational constant (\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\))
- \({m_{2}}\) is the mass of the second star (\({\text{kg}}\))
- \({r_{12}}\) is the separation distance (\({\text{m}}\))
|
| Notes |
- \({a_{\text{y}1}}\) is calculated by solving the ODE together with the initial conditions . This model is derived from Newton’s second law (TM:NewtonSecLawMot), universal gravitation (TM:UniversalGravLaw), and the definitions of acceleration (TM:acceleration), velocity (TM:velocity) and relative position (TM:relPosAndSep), under the center-of-mass constraint (TM:centerOfMass) . The assumptions are: two-body system (A:twoBody), isolated system (A:isolated), Newtonian gravitation (A:newtonianGravity), non-relativistic mechanics (A:nonRelativistic), point masses (A:pointMass), constant masses (A:constantMass), inertial reference frame (A:inertialFrame), planar motion (A:planar) and non-zero separation (A:nonzeroSeparation).
|
| Source | – |
| RefBy | FR:Output-Values and FR:Calculate-Positions |
| Refname | IM:accelX2 |
|---|
| Label | X-acceleration of the second star |
| Input | \({m_{1}}\), \({m_{2}}\), \({x_{1}}\), \({y_{1}}\), \({x_{2}}\), \({y_{2}}\) |
| Output | \({a_{\text{x}2}}\) |
| Input Constraints | \[{m_{1}}\gt{}0\]\[{m_{2}}\gt{}0\] |
| Output Constraints | |
| Equation | \[{a_{\text{x}2}}\left({x_{1}},{y_{1}},{x_{2}},{y_{2}}\right)=\frac{G\,{m_{1}}\,\left({x_{1}}-{x_{2}}\right)}{{r_{12}}^{3}}\] |
| Description |
- \({a_{\text{x}2}}\) is the x-acceleration of the second star (\(\frac{\text{m}}{\text{s}^{2}}\))
- \({x_{1}}\) is the x-position of the first star (\({\text{m}}\))
- \({y_{1}}\) is the y-position of the first star (\({\text{m}}\))
- \({x_{2}}\) is the x-position of the second star (\({\text{m}}\))
- \({y_{2}}\) is the y-position of the second star (\({\text{m}}\))
- \(G\) is the gravitational constant (\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\))
- \({m_{1}}\) is the mass of the first star (\({\text{kg}}\))
- \({r_{12}}\) is the separation distance (\({\text{m}}\))
|
| Notes |
- \({a_{\text{x}2}}\) is calculated by solving the ODE together with the initial conditions . This model is derived from Newton’s second law (TM:NewtonSecLawMot), universal gravitation (TM:UniversalGravLaw), and the definitions of acceleration (TM:acceleration), velocity (TM:velocity) and relative position (TM:relPosAndSep), under the center-of-mass constraint (TM:centerOfMass) . The assumptions are: two-body system (A:twoBody), isolated system (A:isolated), Newtonian gravitation (A:newtonianGravity), non-relativistic mechanics (A:nonRelativistic), point masses (A:pointMass), constant masses (A:constantMass), inertial reference frame (A:inertialFrame), planar motion (A:planar) and non-zero separation (A:nonzeroSeparation).
|
| Source | – |
| RefBy | FR:Output-Values and FR:Calculate-Positions |
| Refname | IM:accelY2 |
|---|
| Label | Y-acceleration of the second star |
| Input | \({m_{1}}\), \({m_{2}}\), \({x_{1}}\), \({y_{1}}\), \({x_{2}}\), \({y_{2}}\) |
| Output | \({a_{\text{y}2}}\) |
| Input Constraints | \[{m_{1}}\gt{}0\]\[{m_{2}}\gt{}0\] |
| Output Constraints | |
| Equation | \[{a_{\text{y}2}}\left({x_{1}},{y_{1}},{x_{2}},{y_{2}}\right)=\frac{G\,{m_{1}}\,\left({y_{1}}-{y_{2}}\right)}{{r_{12}}^{3}}\] |
| Description |
- \({a_{\text{y}2}}\) is the y-acceleration of the second star (\(\frac{\text{m}}{\text{s}^{2}}\))
- \({x_{1}}\) is the x-position of the first star (\({\text{m}}\))
- \({y_{1}}\) is the y-position of the first star (\({\text{m}}\))
- \({x_{2}}\) is the x-position of the second star (\({\text{m}}\))
- \({y_{2}}\) is the y-position of the second star (\({\text{m}}\))
- \(G\) is the gravitational constant (\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\))
- \({m_{1}}\) is the mass of the first star (\({\text{kg}}\))
- \({r_{12}}\) is the separation distance (\({\text{m}}\))
|
| Notes |
- \({a_{\text{y}2}}\) is calculated by solving the ODE together with the initial conditions . This model is derived from Newton’s second law (TM:NewtonSecLawMot), universal gravitation (TM:UniversalGravLaw), and the definitions of acceleration (TM:acceleration), velocity (TM:velocity) and relative position (TM:relPosAndSep), under the center-of-mass constraint (TM:centerOfMass) . The assumptions are: two-body system (A:twoBody), isolated system (A:isolated), Newtonian gravitation (A:newtonianGravity), non-relativistic mechanics (A:nonRelativistic), point masses (A:pointMass), constant masses (A:constantMass), inertial reference frame (A:inertialFrame), planar motion (A:planar) and non-zero separation (A:nonzeroSeparation).
|
| Source | – |
| RefBy | FR:Output-Values and FR:Calculate-Positions |