Radiation astromathematics problems

This composite image shows an exoplanet (2M1207b, the red spot on the lower left), orbiting the brown dwarf 2M1207 (centre). 2M1207b is a Jupiter-like planet. It orbits the brown dwarf at a distance nearly twice as far as Neptune is from the Sun. Credit: ESO.{{free media}}

This radiation astromathematics problem set is a series of mathematical problems that use dimensions.

Notations

Notation: let the symbol $R_{\oplus }$ indicate the Earth's radius.

Notation: let the symbol $R_{J}$ indicate the radius of Jupiter.

Notation: let the symbol $R_{\odot }$ indicate the solar radius.

Problem 1

$R_{\odot }(equatorial)=696,342km$

$R_{J}(equatorial)=71,492km$

$R_{S}(equatorial)=60,268km$

$R_{U}(equatorial)=25,559km$

$R_{N}(equatorial)=24,764km$

Then,

$R_{J}(equatorial)=F_{J}*R_{\odot }(equatorial),$

$R_{S}(equatorial)=F_{S}*R_{\odot }(equatorial),$

$R_{U}(equatorial)=F_{U}*R_{\odot }(equatorial),$

$R_{N}(equatorial)=F_{N}*R_{\odot }(equatorial),$

1. what is the value of F_J?
2. if R_J has 5 significant digits and R_⊙ has 6, how many significant digits should F_J have?
3. if the angular size of the Sun (equatorial) is 32.7', what would be the angular size of Jupiter if it were the same distance from the Earth as the Sun?
4. let R_S (equatorial) represent the equatorial radius of Saturn, what is the value of F_S?
5. if R_S has 5 significant digits and R_⊙ has 6, how many significant digits should F_S have?
6. what would be the angular size of Saturn if it were the same distance from the Earth as the Sun?
7. let R_U (equatorial) be the equatorial radius of Uranus, what is the value of F_U?
8. if R_U has 5 significant digits and R_⊙ has 6, how many significant digits should F_U have?
9. what would be the angular size of Uranus if it were the same distance from the Earth as the Sun?
10. let R_N (equatorial) be the equatorial radius of Neptune, what is the value of F_N?
11. if R_N has 5 significant digits and R_⊙ has 6, how many significant digits should F_N have?
12. what would be the angular size of Neptune if it were the same distance from the Earth as the Sun?
13. using the equatorial radii of the four gas giants (J, S, U, and N), if these four giants combined to form one approximately spherical object (T), without loss of matter and assuming each has the same particle density, what are the values of R_T and F_T?
14. if the object (T) is the same distance from the Earth as the Sun, what is its angular size?

Problem 2

$\rho _{\odot }=1.408kg/m^{3}$

$\rho _{J}=1.326g/cm^{3}$

$\rho _{S}=0.687g/cm^{3}$

$\rho _{U}=1.27g/cm^{3}$

$\rho _{N}=1.638g/cm^{3}$

$\rho _{H_{2}}(0^{\circ }C,101.325kPa)=0.08988g/L$

1. what is the density of the Sun in g/cm³?
2. if the standard atomic weight of hydrogen is 1.008 and there are 6.022 x 10²³ hydrogen atoms in one gram of monatomic hydrogen, how many molecules of H₂ are there in 1 gram of the diatomic gas?
3. if 1 L (liter) is 10^-3 m³, how many cm³ is 1 L?
4. if each of the classical planets is only made of diatomic hydrogen gas, how many particles of diatomic hydrogen are within each astronomical object, using the radii from problem 1?
5. how many diatomic hydrogen particles are in T?
6. using R_T, what is the H₂ particle density of T, ρ_T,H2, in number of particles cm^-3?

Problem 3

Using the density of H₂ gas at 0°C and 101.325 kPa,

1. calculate the H₂ particle density for this pressure,
2. if the particle density scales with the inverse of the pressure, at what pressure would T have the same particle density as H₂ gas at 0°C and 101.325 kPa?
3. using the density of the Sun in g/cm³ calculated in Problem 2, what is the H₂ particle density for the Sun?
4. if the particle density scales with the inverse of the pressure, at what pressure would the Sun have the same particle density as H₂ gas at 0°C and 101.325 kPa?
5. what would R_T have to be so that T has the same H₂ particle density as the Sun?

Problem 4

T_J = 165 K

T_S = 135 K

T_U = 76 K

T_N = 72 K

Let the particle density scale approximately with temperature in kelvin. For example, 0°C equals 273.15 K. As the temperature increases for the same volume, the particle density remains constant but the pressure increases. In order for the temperature to have an effect, let the volume increase as the temperature increases.

1. the effective surface temperature of the Sun is 5778 K, use the H₂ particle density and the above temperatures to calculate the radii when these temperatures are increased to the surface temperature of the Sun.
2. use the new radii for the gas giants to combine them into a T', what is R_T'?
3. using R_T', what is its particle density and F_T'?
4. if the object T' is at the same distance from the Earth as the Sun, what is its angular size?

Problem 5

As temperature increases in an astronomical object composed of H₂, the molecules begin to dissociate.

"At a temperature of 8000 K, hydrogen gas is 99.99 percent monatomic."

$\rho _{H}=\rho _{H_{0}}e^{E_{T}/{kT}},$

where $\rho _{H_{0}}$ is an initial concentration [H] at low temperatures as partial particle density, $E_{T}$ is the dissociation energy 4.52 eV, k is Boltzmann's contant (8.6173324(78)×10⁻⁵ eV K^-1), and T is temperature in K.

Using

$[H]=70400e^{-4.52/(0.00008617T)}$

1. what is the concentration of H ([H]) at T = 8000 K?
2. what is [H] at T = 800 K?
3. at what temperature is [H] = 1?
4. what is [H] at T = 5778 K?

Hypotheses

Main resource: Hypotheses

1. The change in mathematics to go from mathematical astronomy to radiation astronomy/Mathematics involves the mathematics of radiation.

See also

• Angular momentum and energy
• Column densities
• Cosmic circuits
• Energy phantoms
• Furlongs per fortnight
• Planck's equation
• Radiation astronomy/Problem set
• Radiation dosage
• Radiation astromathematics problems
• Radiation mathematics
• Spectrographs
• Star jumping
• Synchrotron radiation
• Telescopes and cameras
• Unknown coordinate systems
• Unusual units
• Vectors and coordinates

External links

• International Astronomical Union
• NASA/IPAC Extragalactic Database - NED
• NASA's National Space Science Data Center
• Office of Scientific & Technical Information
• The SAO/NASA Astrophysics Data System
• Scirus for scientific information only advanced search
• SDSS Quick Look tool: SkyServer
• SIMBAD Astronomical Database
• SIMBAD Web interface, Harvard alternate
• Spacecraft Query at NASA
• Universal coordinate converter

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