ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Each of the two Magellan telescopes has a diameter of $6.5 \mathrm{~m}$. In one configuration the effective focal length is $72 \mathrm{~m}$. Find the diameter of the image of a planet (in $\mathrm{cm}$ ) at this focus if the angular diameter of the planet at the time of the observation is $45^{\prime \prime}$.
Answer: 1.6
Problem: A white dwarf star has an effective temperature, $T_{e}=50,000$ degrees Kelvin, but its radius, $R_{\mathrm{WD}}$, is comparable to that of the Earth. Take $R_{\mathrm{WD}}=10^{4} \mathrm{~km}\left(10^{7} \mathrm{~m}\right.$ or $\left.10^{9} \mathrm{~cm}\right)$. Compute the luminosity (power output) of the white dwarf. Treat the white dwarf as a blackbody radiator. Give  your answer in units of ergs per second, to two significant figures.
Answer: 4.5e33
Problem: Preamble: A prism is constructed from glass and has sides that form a right triangle with the other two angles equal to $45^{\circ}$. The sides are $L, L$, and $H$, where $L$ is a leg and $H$ is the hypotenuse. A parallel light beam enters side $L$ normal to the surface, passes into the glass, and then strikes $H$ internally. The index of refraction of the glass is $n=1.5$.

Compute the critical angle for the light to be internally reflected at $H$.  Give your answer in degrees to 3 significant figures.
Answer: 41.8
Problem: A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \mathrm{Mpc}$, what will its apparent magnitude be?
Answer: 20.39
Problem: Find the gravitational acceleration due to the Sun at the location of the Earth's orbit (i.e., at a distance of $1 \mathrm{AU}$ ).  Give your answer in meters per second squared, and express it to one significant figure.
Answer: 0.006
Problem: Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\theta_w$ with respect to the surface normal.

Subproblem 0: If the index of refraction of water is $n=1.3$, find an expression for the angle of the light once it emerges from the water into the air, $\theta_a$, in terms of $\theta_w$.


Solution: Using Snell's law, $1.3 \sin{\theta_w} = \sin{\theta_a}$. So $\theta_a = \boxed{\arcsin{1.3 \sin{\theta_w}}}$.

Final answer: The final answer is \arcsin{1.3 \sin{\theta_w}}. I hope it is correct.

Subproblem 1: What is the critical angle, i.e., the critical value of $\theta_w$ such that the light will not emerge from the water?  Leave your answer in terms of inverse trigonometric functions; i.e., do not evaluate the function.
Answer: np.arcsin(10/13)
Problem: Find the theoretical limiting angular resolution (in arcsec) of a commercial 8-inch (diameter) optical telescope being used in the visible spectrum (at $\lambda=5000 \AA=500 \mathrm{~nm}=5 \times 10^{-5} \mathrm{~cm}=5 \times 10^{-7} \mathrm{~m}$).  Answer in arcseconds to two significant figures.
Answer: 0.49
Problem: A star has a measured parallax of $0.01^{\prime \prime}$, that is, $0.01$ arcseconds. How far away is it, in parsecs?
Answer: 100
Problem: An extrasolar planet has been observed which passes in front of (i.e., transits) its parent star. If the planet is dark (i.e., contributes essentially no light of its own) and has a surface area that is $2 \%$ of that of its parent star, find the decrease in magnitude of the system during transits.
Answer: 0.022
Problem: If the Bohr energy levels scale as $Z^{2}$, where $Z$ is the atomic number of the atom (i.e., the charge on the nucleus), estimate the wavelength of a photon that results from a transition from $n=3$ to $n=2$ in Fe, which has $Z=26$. Assume that the Fe atom is completely stripped of all its electrons except for one.  Give your answer in Angstroms, to two significant figures.
Answer: 9.6
Problem: If the Sun's absolute magnitude is $+5$, find the luminosity of a star of magnitude $0$ in ergs/s. A useful constant: the luminosity of the sun is $3.83 \times 10^{33}$ ergs/s.
Answer: 3.83e35
Problem: Preamble: A spectrum is taken of a single star (i.e., one not in a binary). Among the observed spectral lines is one from oxygen whose rest wavelength is $5007 \AA$. The Doppler shifted oxygen line from this star is observed to be at a wavelength of $5012 \AA$. The star is also observed to have a proper motion, $\mu$, of 1 arc second per year (which corresponds to $\sim 1.5 \times 10^{-13}$ radians per second of time). It is located at a distance of $60 \mathrm{pc}$ from the Earth. Take the speed of light to be $3 \times 10^8$ meters per second.

What is the component of the star's velocity parallel to its vector to the Earth (in kilometers per second)?
Answer: 300
Problem: The differential luminosity from a star, $\Delta L$, with an approximate blackbody spectrum, is given by:
\[
\Delta L=\frac{8 \pi^{2} c^{2} R^{2}}{\lambda^{5}\left[e^{h c /(\lambda k T)}-1\right]} \Delta \lambda
\]
where $R$ is the radius of the star, $T$ is its effective surface temperature, and $\lambda$ is the wavelength. $\Delta L$ is the power emitted by the star between wavelengths $\lambda$ and $\lambda+\Delta \lambda$ (assume $\Delta \lambda \ll \lambda)$. The star is at distance $d$. Find the star's spectral intensity $I(\lambda)$ at the Earth, where $I(\lambda)$ is defined as the power per unit area per unit wavelength interval.
Answer: \frac{2\pic^{2}R^{2}}{\lambda^{5}[e^{hc/(\lambdakT)}-1]d^{2}}
Problem: Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.

Subproblem 0: Find the luminosity of the star (in units of $\mathrm{erg} \cdot \mathrm{s}^{-1}$).


Solution: \[
  L=4 \pi D^{2} \text { Flux }_{\text {Earth }}=10^{-12} 4 \pi\left(800 \times 3 \times 10^{21}\right)^{2}=\boxed{7e37} \mathrm{erg} \cdot \mathrm{s}^{-1}
\]

Final answer: The final answer is 7e37. I hope it is correct.

Subproblem 1: Compute the star's radius in centimeters.
Answer: 8.7e8
Problem: A star is at a distance from the Earth of $300 \mathrm{pc}$. Find its parallax angle, $\pi$, in arcseconds to one significant figure.
Answer: 0.003
Problem: The Sun's effective temperature, $T_{e}$, is 5800 Kelvin, and its radius is $7 \times 10^{10} \mathrm{~cm}\left(7 \times 10^{8}\right.$ m). Compute the luminosity (power output) of the Sun in erg/s. Treat the Sun as a blackbody radiator, and give your answer to one significant figure.
Answer: 4e33
Problem: Use the Bohr model of the atom to compute the wavelength of the transition from the $n=100$ to $n=99$ levels, in centimeters. [Uscful relation: the wavelength of $L \alpha$ ( $\mathrm{n}=2$ to $\mathrm{n}=1$ transition) is $1216 \AA$.]
Answer: 4.49
Problem: Preamble: A radio interferometer, operating at a wavelength of $1 \mathrm{~cm}$, consists of 100 small dishes, each $1 \mathrm{~m}$ in diameter, distributed randomly within a $1 \mathrm{~km}$ diameter circle. 

What is the angular resolution of a single dish, in radians?
Answer: 0.01
Problem: Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\text {orb }}=10$ days. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is $v_{1}=20 \mathrm{~km} \mathrm{~s}^{-1}$. Star 2 is an X-ray pulsar and its orbital radius about the center of mass is $r_{2}=3 \times 10^{12} \mathrm{~cm}=3 \times 10^{10} \mathrm{~m}$.

Subproblem 0: Find the orbital radius, $r_{1}$, of the optical star (Star 1) about the center of mass, in centimeters.


Solution: \[
\begin{gathered}
v_{1}=\frac{2 \pi r_{1}}{P_{\text {orb }}} \\
r_{1}=\frac{P_{\text {orb }} v_{1}}{2 \pi}=\boxed{2.75e11} \mathrm{~cm}
\end{gathered}
\]

Final answer: The final answer is 2.75e11. I hope it is correct.

Subproblem 1: What is the total orbital separation between the two stars, $r=r_{1}+r_{2}$ (in centimeters)?
Answer: 3.3e12
Problem: If a star cluster is made up of $10^{4}$ stars, each of whose absolute magnitude is $-5$, compute the combined apparent magnitude of the cluster if it is located at a distance of $1 \mathrm{Mpc}$.
Answer: 10
Problem: A galaxy moves directly away from us with a speed of $3000 \mathrm{~km} \mathrm{~s}^{-1}$. Find the wavelength of the $\mathrm{H} \alpha$ line observed at the Earth, in Angstroms. The rest wavelength of $\mathrm{H} \alpha$ is $6565 \AA$.  Take the speed of light to be $3\times 10^8$ meters per second.
Answer: 6630
Problem: The Spitzer Space Telescope has an effective diameter of $85 \mathrm{cm}$, and a typical wavelength used for observation of $5 \mu \mathrm{m}$, or 5 microns. Based on this information, compute an estimate for the angular resolution of the Spitzer Space telescope in arcseconds.
Answer: 1.2
Problem: It has long been suspected that there is a massive black hole near the center of our Galaxy. Recently, a group of astronmers determined the parameters of a star that is orbiting the suspected black hole. The orbital period is 15 years, and the orbital radius is $0.12$ seconds of arc (as seen from the Earth). Take the distance to the Galactic center to be $8 \mathrm{kpc}$. Compute the mass of the black hole, starting from $F=m a$. Express your answer in units of the Sun's mass; i.e., answer the question `what is the ratio of masses between this black hole and our Sun'? Give your answer to 1 significant figure. (Assume that Newton's law of gravity is applicable for orbits sufficiently far from a black hole, and that the orbiting star satisfies this condition.)
Answer: 3e6
Problem: Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.

Find the luminosity of the star (in units of $\mathrm{erg} \cdot \mathrm{s}^{-1}$).
Answer: 7e37
Problem: A large ground-based telescope has an effective focal length of 10 meters. Two astronomical objects are separated by 1 arc second in the sky. How far apart will the two corresponding images be in the focal plane, in microns?
Answer: 50
Problem: The equation of state for cold (non-relativistic) matter may be approximated as:
\[
P=a \rho^{5 / 3}-b \rho^{4 / 3}
\]
where $P$ is the pressure, $\rho$ the density, and $a$ and $b$ are fixed constants. Use a dimensional analysis of the equation of hydrostatic equilibrium to estimate the ``radius-mass'' relation for planets and low-mass white dwarfs whose material follows this equation of state. Specifically, find $R(M)$ in terms of $G$ and the constants $a$ and $b$. You should set all constants of order unity (e.g., $4, \pi, 3$, etc.) to $1.0$. [Hint: solve for $R(M)$ rather than $M(R)$ ]. You can check your answer by showing that for higher masses, $R \propto M^{-1 / 3}$, while for the lower-masses $R \propto M^{+1 / 3}$.
Answer: \frac{aM^{1/3}}{GM^{2/3}+b}
Problem: Take the total energy (potential plus thermal) of the Sun to be given by the simple expression:
\[
E \simeq-\frac{G M^{2}}{R}
\]
where $M$ and $R$ are the mass and radius, respectively. Suppose that the energy generation in the Sun were suddenly turned off and the Sun began to slowly contract. During this contraction its mass, $M$, would remain constant and, to a fair approximation, its surface temperature would also remain constant at $\sim 5800 \mathrm{~K}$. Assume that the total energy of the Sun is always given by the above expression, even as $R$ gets smaller. By writing down a simple (differential) equation relating the power radiated at Sun's surface with the change in its total energy (using the above expression), integrate this equation to find the time (in years) for the Sun to shrink to $1 / 2$ its present radius.  Answer in units of years.
Answer: 7.5e7
Problem: Preamble: Once a star like the Sun starts to ascend the giant branch its luminosity, to a good approximation, is given by:
\[
L=\frac{10^{5} L_{\odot}}{M_{\odot}^{6}} M_{\text {core }}^{6}
\]
where the symbol $\odot$ stands for the solar value, and $M_{\text {core }}$ is the mass of the He core of the star. Further, assume that as more hydrogen is burned to helium - and becomes added to the core - the conversion efficiency between rest mass and energy is:
\[
\Delta E=0.007 \Delta M_{\text {core }} c^{2} .
\]

Use these two expressions to write down a differential equation, in time, for $M_{\text {core }}$.  For ease of writing, simply use the variable $M$ to stand for $M_{\text {core }}$.  Leave your answer in terms of $c$, $M_{\odot}$, and $L_{\odot}$.
Answer: \frac{dM}{dt}=\frac{10^{5}L_{\odot}}{0.007c^{2}M_{\odot}^{6}}M^{6}
Problem: A star of radius, $R$, and mass, $M$, has an atmosphere that obeys a polytropic equation of state:
\[
P=K \rho^{5 / 3} \text {, }
\]
where $P$ is the gas pressure, $\rho$ is the gas density (mass per unit volume), and $K$ is a constant throughout the atmosphere. Assume that the atmosphere is sufficiently thin (compared to $R$ ) that the gravitational acceleration can be taken to be a constant.
Use the equation of hydrostatic equilibrium to derive the pressure as a function of height $z$ above the surface of the planet. Take the pressure at the surface to be $P_{0}$.
Answer: [P_{0}^{2/5}-\frac{2}{5}gK^{-3/5}z]^{5/2}
Problem: An eclipsing binary consists of two stars of different radii and effective temperatures. Star 1 has radius $R_{1}$ and $T_{1}$, and Star 2 has $R_{2}=0.5 R_{1}$ and $T_{2}=2 T_{1}$. Find the change in bolometric magnitude of the binary, $\Delta m_{\text {bol }}$, when the smaller star is behind the larger star. (Consider only bolometric magnitudes so you don't have to worry about color differences.)
Answer: 1.75
Problem: Preamble: It has been suggested that our Galaxy has a spherically symmetric dark-matter halo with a density distribution, $\rho_{\text {dark }}(r)$, given by:
\[
\rho_{\text {dark }}(r)=\rho_{0}\left(\frac{r_{0}}{r}\right)^{2},
\]
where $\rho_{0}$ and $r_{0}$ are constants, and $r$ is the radial distance from the center of the galaxy. For star orbits far out in the halo you can ignore the gravitational contribution of the ordinary matter in the Galaxy.

Compute the rotation curve of the Galaxy (at large distances), i.e., find $v(r)$ for circular orbits.
Answer: \sqrt{4\piG\rho_{0}r_{0}^{2}}
Problem: The Very Large Array (VLA) telescope has an effective diameter of $36 \mathrm{~km}$, and a typical wavelength used for observation at this facility might be $6 \mathrm{~cm}$.  Based on this information, compute an estimate for the angular resolution of the VLA in arcseconds
Answer: 0.33
Problem: Subproblem 0: A particular star has an absolute magnitude $M=-7$. If this star is observed in a galaxy that is at a distance of $3 \mathrm{Mpc}$, what will its apparent magnitude be? 


Solution: \[
\text { Given: } M=-7 \text { and } d=3 \mathrm{Mpc}
\]
\[
\begin{aligned}
  & \text { Apparent Magnitude: } m=M+5 \log \left[\frac{d}{10 \mathrm{pc}}\right]=-7+5 \log \left[\frac{3 \times 10^{6}}{10}\right]=\boxed{20.39} \\
\end{aligned}
\]

Final answer: The final answer is 20.39. I hope it is correct.

Subproblem 1: What is the distance modulus to this galaxy?
Answer: 27.39
Problem: Find the distance modulus to the Andromeda galaxy (M31). Take the distance to Andromeda to be $750 \mathrm{kpc}$, and answer to three significant figures.
Answer: 24.4
Problem: The Hubble Space telescope has an effective diameter of $2.5 \mathrm{~m}$, and a typical wavelength used for observation by the Hubble might be $0.6 \mu \mathrm{m}$, or 600 nanometers (typical optical wavelength). Based on this information, compute an estimate for the angular resolution of the Hubble Space telescope in arcseconds.
Answer: 0.05
Problem: Preamble: A collimated light beam propagating in water is incident on the surface (air/water interface) at an angle $\theta_w$ with respect to the surface normal.

If the index of refraction of water is $n=1.3$, find an expression for the angle of the light once it emerges from the water into the air, $\theta_a$, in terms of $\theta_w$.
Answer: \arcsin{1.3\sin{\theta_w}}
Problem: What fraction of the rest mass energy is released (in the form of radiation) when a mass $\Delta M$ is dropped from infinity onto the surface of a neutron star with $M=1 M_{\odot}$ and $R=10$ $\mathrm{km}$ ?
Answer: 0.15
Problem: Preamble: The density of stars in a particular globular star cluster is $10^{6} \mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average speed of $10 \mathrm{~km} \mathrm{sec}^{-1}$.

Find the mean free path for collisions among stars.  Express your answer in centimeters, to a single significant figure.
Answer: 2e27
Problem: For a gas supported by degenerate electron pressure, the pressure is given by:
\[
P=K \rho^{5 / 3}
\]
where $K$ is a constant and $\rho$ is the mass density. If a star is totally supported by degenerate electron pressure, use a dimensional analysis of the equation of hydrostatic equilibrium:
\[
\frac{d P}{d r}=-g \rho
\]
to determine how the radius of such a star depends on its mass, $M$.  Specifically, you will find that $R$ is proportional to some power of $M$; what is that power?
Answer: \frac{-1}{3}
Problem: A galaxy moves directly away from us with speed $v$, and the wavelength of its $\mathrm{H} \alpha$ line is observed to be $6784 \AA$. The rest wavelength of $\mathrm{H} \alpha$ is $6565 \AA$. Find $v/c$.
Answer: 0.033
Problem: A candle has a power in the visual band of roughly $3$ Watts. When this candle is placed at a distance of $3 \mathrm{~km}$ it has the same apparent brightness as a certain star. Assume that this star has the same luminosity as the Sun in the visual band $\left(\sim 10^{26}\right.$ Watts $)$. How far away is the star (in pc)?
Answer: 0.5613
Problem: Preamble: A galaxy is found to have a rotation curve, $v(r)$, given by
\[
v(r)=\frac{\left(\frac{r}{r_{0}}\right)}{\left(1+\frac{r}{r_{0}}\right)^{3 / 2}} v_{0}
\]
where $r$ is the radial distance from the center of the galaxy, $r_{0}$ is a constant with the dimension of length, and $v_{0}$ is another constant with the dimension of speed. The rotation curve is defined as the orbital speed of test stars in circular orbit at radius $r$.

Find an expression for $\omega(r)$, where $\omega$ is the angular velocity.  The constants $v_{0}$ and $r_{0}$ will appear in your answer.
Answer: \frac{v_{0}}{r_{0}}\frac{1}{(1+r/r_{0})^{3/2}}
Problem: Preamble: Orbital Dynamics: A binary system consists of two stars in circular orbit about a common center of mass, with an orbital period, $P_{\text {orb }}=10$ days. Star 1 is observed in the visible band, and Doppler measurements show that its orbital speed is $v_{1}=20 \mathrm{~km} \mathrm{~s}^{-1}$. Star 2 is an X-ray pulsar and its orbital radius about the center of mass is $r_{2}=3 \times 10^{12} \mathrm{~cm}=3 \times 10^{10} \mathrm{~m}$.

Find the orbital radius, $r_{1}$, of the optical star (Star 1) about the center of mass, in centimeters.
Answer: 2.75e11
Problem: Preamble: The density of stars in a particular globular star cluster is $10^{6} \mathrm{pc}^{-3}$. Take the stars to have the same radius as the Sun, and to have an average speed of $10 \mathrm{~km} \mathrm{sec}^{-1}$.

Subproblem 0: Find the mean free path for collisions among stars.  Express your answer in centimeters, to a single significant figure.


Solution: \[
\begin{gathered}
\ell \simeq \frac{1}{n \sigma}=\frac{1}{10^{6} \mathrm{pc}^{-3} \pi R^{2}} \\
\ell \simeq \frac{1}{3 \times 10^{-50} \mathrm{~cm}^{-3} \times 1.5 \times 10^{22} \mathrm{~cm}^{2}} \simeq \boxed{2e27} \mathrm{~cm}
\end{gathered}
\]

Final answer: The final answer is 2e27. I hope it is correct.

Subproblem 1: Find the corresponding mean time between collisions. (Assume that the stars move in straight-line paths, i.e., are not deflected by gravitational interactions.)  Answer in units of years, to a single significant figure.
Answer: 6e13
Problem: Preamble: A radio interferometer, operating at a wavelength of $1 \mathrm{~cm}$, consists of 100 small dishes, each $1 \mathrm{~m}$ in diameter, distributed randomly within a $1 \mathrm{~km}$ diameter circle. 

Subproblem 0: What is the angular resolution of a single dish, in radians?


Solution: The angular resolution of a single dish is roughly given by the wavelength over its radius, in this case $\boxed{0.01}$ radians.

Final answer: The final answer is 0.01. I hope it is correct.

Subproblem 1: What is the angular resolution of the interferometer array for a source directly overhead, in radians?
Answer: 1e-5
Problem: If a star cluster is made up of $10^{6}$ stars whose absolute magnitude is the same as that of the Sun (+5), compute the combined magnitude of the cluster if it is located at a distance of $10 \mathrm{pc}$.
Answer: -10
Problem: A certain red giant has a radius that is 500 times that of the Sun, and a temperature that is $1 / 2$ that of the Sun's temperature. Find its bolometric (total) luminosity in units of the bolometric luminosity of the Sun.
Answer: 15625
Problem: Suppose air molecules have a collision cross section of $10^{-16} \mathrm{~cm}^{2}$. If the (number) density of air molecules is $10^{19} \mathrm{~cm}^{-3}$, what is the collision mean free path in cm? Answer to one significant figure.
Answer: 1e-3
Problem: Two stars have the same surface temperature. Star 1 has a radius that is $2.5$ times larger than the radius of star 2. Star 1 is ten times farther away than star 2. What is the absolute value of the difference in apparent magnitude between the two stars, rounded to the nearest integer?
Answer: 3
Problem: What is the slope of a $\log N(>F)$ vs. $\log F$ curve for a homogeneous distribution of objects, each of luminosity, $L$, where $F$ is the flux at the observer, and $N$ is the number of objects observed per square degree on the sky?
Answer: \frac{-3}{2}
Problem: Preamble: Comparison of Radio and Optical Telescopes.

The Very Large Array (VLA) is used to make an interferometric map of the Orion Nebula at a wavelength of $10 \mathrm{~cm}$. What is the best angular resolution of the radio image that can be produced, in radians? Note that the maximum separation of two antennae in the VLA is $36 \mathrm{~km}$.
Answer: 2.7778e-6
Problem: A globular cluster has $10^{6}$ stars each of apparent magnitude $+8$. What is the combined apparent magnitude of the entire cluster?
Answer: -7
Problem: Preamble: A very hot star is detected in the galaxy M31 located at a distance of $800 \mathrm{kpc}$. The star has a temperature $T = 6 \times 10^{5} K$ and produces a flux of $10^{-12} \mathrm{erg} \cdot \mathrm{s}^{-1} \mathrm{cm}^{-2}$ at the Earth. Treat the star's surface as a blackbody radiator.

Subproblem 0: Find the luminosity of the star (in units of $\mathrm{erg} \cdot \mathrm{s}^{-1}$).


Solution: \[
  L=4 \pi D^{2} \text { Flux }_{\text {Earth }}=10^{-12} 4 \pi\left(800 \times 3 \times 10^{21}\right)^{2}=\boxed{7e37} \mathrm{erg} \cdot \mathrm{s}^{-1}
\]

Final answer: The final answer is 7e37. I hope it is correct.

Subproblem 1: Compute the star's radius in centimeters.


Solution: \[
  R=\left(L / 4 \pi \sigma T^{4}\right)^{1 / 2}=\boxed{8.7e8} \mathrm{~cm}=0.012 R_{\odot}
\]

Final answer: The final answer is 8.7e8. I hope it is correct.

Subproblem 2: At what wavelength is the peak of the emitted radiation? Answer in $\AA$.
Answer: 48
Problem: A Boolean function $F(A, B)$ is said to be universal if any arbitrary boolean function can be constructed by using nested $F(A, B)$ functions. A universal function is useful, since using it we can build any function we wish out of a single part. For example, when implementing boolean logic on a computer chip a universal function (called a 'gate' in logic-speak) can simplify design enormously. We would like to find a universal boolean function. In this problem we will denote the two boolean inputs $A$ and $B$ and the one boolean output as $C$. 
First, to help us organize our thoughts, let's enumerate all of the functions we'd like to be able to construct. How many different possible one-output boolean functions of two variables are there? I.e., how many functions are there of the form $F(A, B)=C ?$
Answer: 16