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.

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

## Notations

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

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

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

## Problem 1

${\displaystyle R_{\odot }(equatorial)=696,342km}$
${\displaystyle R_{J}(equatorial)=71,492km}$
${\displaystyle R_{S}(equatorial)=60,268km}$
${\displaystyle R_{U}(equatorial)=25,559km}$
${\displaystyle R_{N}(equatorial)=24,764km}$

Then,

${\displaystyle R_{J}(equatorial)=F_{J}*R_{\odot }(equatorial),}$
${\displaystyle R_{S}(equatorial)=F_{S}*R_{\odot }(equatorial),}$
${\displaystyle R_{U}(equatorial)=F_{U}*R_{\odot }(equatorial),}$
${\displaystyle R_{N}(equatorial)=F_{N}*R_{\odot }(equatorial),}$
1. what is the value of FJ?
2. if RJ has 5 significant digits and R has 6, how many significant digits should FJ 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 RS (equatorial) represent the equatorial radius of Saturn, what is the value of FS?
5. if RS has 5 significant digits and R has 6, how many significant digits should FS have?
6. what would be the angular size of Saturn if it were the same distance from the Earth as the Sun?
7. let RU (equatorial) be the equatorial radius of Uranus, what is the value of FU?
8. if RU has 5 significant digits and R has 6, how many significant digits should FU have?
9. what would be the angular size of Uranus if it were the same distance from the Earth as the Sun?
10. let RN (equatorial) be the equatorial radius of Neptune, what is the value of FN?
11. if RN has 5 significant digits and R has 6, how many significant digits should FN 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 RT and FT?
14. if the object (T) is the same distance from the Earth as the Sun, what is its angular size?

## Problem 2

${\displaystyle \rho _{\odot }=1.408kg/m^{3}}$
${\displaystyle \rho _{J}=1.326g/cm^{3}}$
${\displaystyle \rho _{S}=0.687g/cm^{3}}$
${\displaystyle \rho _{U}=1.27g/cm^{3}}$
${\displaystyle \rho _{N}=1.638g/cm^{3}}$
${\displaystyle \rho _{H_{2}}(0^{\circ }C,101.325kPa)=0.08988g/L}$
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 RT, what is the H2 particle density of T, ρT,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 RT have to be so that T has the same H2 particle density as the Sun?

## Problem 4

TJ = 165 K

TS = 135 K

TU = 76 K

TN = 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 RT'?
3. using RT', what is its particle density and FT'?
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."[1]

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

where ${\displaystyle \rho _{H_{0}}}$ is an initial concentration [H] at low temperatures as partial particle density, ${\displaystyle E_{T}}$ 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

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

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