## Wednesday, December 17, 2014

### inertial forces and mechanical work

can you perform work against the centrifugal force? what about the Coriolis-force?

CQW wishes our readers a Merry Xmas and a happy New Year 2015, and we hope to welcome you back on the 14.Jan.2015. if you want to contribute to CQW, please drop us a mail at cosmologyquestionoftheweek["AT"]gmail["DOT"]com.

## Wednesday, December 10, 2014

### Eddington-limit for muons

the Eddington-limit is defined as the luminosity where gravity is balanced by radiation pressure. imagine now that an astrophysical object contains muons instead of electrons: would the Eddington-limit be higher or lower?

## Wednesday, December 3, 2014

### cosmic neutrinos in ICECUBE

CQW celebrates its third anniversary!

could you estimate how long one would need to wait for detecting a neutrino of the cosmic neutrino background in ICECUBE?

## Wednesday, November 26, 2014

### tidal stretching of Palomar 5

the globular cluster Palomar 5 orbits the Milky Way and is stretched by tidal forces. on what time-scale would the tidal arms reach all around the Milky Way?

## Wednesday, November 19, 2014

### a quantum of sunshine

the Planck-spectrum is dominated by Boltzmann-statistics at high energies and by Bose-statistics at low energies. could you estimate how much solar power we receive per unit area at Earth from the quantum-mechanical part of the Sun's spectrum?

## Wednesday, November 12, 2014

### tidal forces in the gravity train

earlier this year there was a question about the gravity train to New Zealand. a passenger would feel two kinds of tidal forces: firstly, there's a differential gravitational field between head and feet, and secondly, the shoulders would each follow trajectories pointing towards the Earth's centre. which of the two effects is larger?

## Wednesday, November 5, 2014

### anti-Earth

imagine there would be a planet identical in mass as our Earth circling the Sun in our orbit with a lag of exactly 6 months, i.e. it would always be behind the Sun and not directly visible for us. would there be nevertheless possibilities of detecting it?

## Wednesday, October 29, 2014

### correlation coefficient

can you provide a proof of the Cauchy-Schwarz inequality and show with it that the correlation coefficient

r = \frac{\langle xy\rangle}{\sqrt{\langle x^2\rangle\:\langle y^2\rangle}}

always ranges between $-1\leq r\leq +1$?

bonus question: can you compute the normalisation, variance and kurtosis of $p(x)\mathrm{d}x\propto\exp(-x^4)\mathrm{d}x$?

## Wednesday, October 22, 2014

### Eddington temperature

Astrophysical objects are limited in their luminosity by the Eddington luminosity, where radiation pressure exceeds gravity. Would it be sensible to define a limiting Eddington-temperature for Planckian spectra?

## Wednesday, October 15, 2014

### Boltzmann-tail of the Planck-spectrum

can you estimate the time how long would you have to wait until
1. the Hubble space telescope would see a CMB photon at visible energies?
2. XMM-Newton registers a CMB photon at $X$-ray energies?
3. Fermi-LAT would register a CMB photon at MeV-energies?

## Wednesday, October 8, 2014

### natural value of the Hubble constant

a motivation for investigating dark energy is that the vacuum energy density is 120 orders of magnitude smaller than expected (which btw applies to the density of everything, dark matter, photons and neutrinos alike): could one not simply say that we don't understand the value of the Hubble constant? what would be a natural value, and could you make an example of units which can be easily imagined?

## Wednesday, October 1, 2014

### Boltzmann-photons

can you estimate what fraction of photons in a Planck-spectrum are described by classical Boltzmann-statistics?

## Wednesday, July 30, 2014

### CQW holiday project: NFW-wave function

what's the wave-function of a quantum mechanical particle inside an NFW-potential?

with that question, we would like to wish all our readers a nice summer vacation and hope to see you again in October. if you'd like to contribute to CQW with a question, please contact us at cosmologyquestionoftheweek["AT"]gmail["DOT"]com.

## Wednesday, July 23, 2014

### Roche limit for a drop of water

can you compute the Roche limit, i.e. the closest distance at which an orbiting object can approach a central body without being disintegrated by tidal forces, for the case that the orbiting object is held together by surface tension instead of gravity?

## Wednesday, July 16, 2014

### periapsis recession

what would the shape of relativistic corrections to the gravitational potential need to be such that one would observe a periapsis recession instead of a periapsis advance in an orbiting object?

## Wednesday, July 9, 2014

### units of spectra

can you please show that the unit of the fluctuation spectrum $P_\phi$ of a field $\phi(\vec{x})$ is given by the unit of the variable $\vec{x}$ on which the field depends to the power of the dimension of the space on which the field is defined times the squared unit of the field amplitude?

## Wednesday, July 2, 2014

### a much smaller Planck-mass

could you define a set of Planck-units based on the speed of light $c$, the Planck-constant $\hbar$ and instead of the gravitational constant $G$ the cosmological constant $\Lambda$? what would the numerical values be?

## Wednesday, June 25, 2014

### Newton's Indian rope trick

how long would a vertical rope need to be so that it is in dynamical equilibrium in the Earth's gravitational field with the centrifugal force?

## Wednesday, June 18, 2014

### asymptotic Hubble-function

what value will the Hubble-parameter reach asymptotically in the future in a $\Lambda$CDM cosmology? when are we within $10^{-n}$, $n=1,2,3$ of that value?

## Wednesday, June 11, 2014

### cosmic Unruh-effect

please estimate the temperature of the quantum vacuum which we observe due to peculiar acceleration of the Milky Way. how large would an acceleration need to be such that its assigned temperature became comparable to the CMB-temperature?

## Wednesday, June 4, 2014

### Dirac's large numbers

can you express the ratio between the size of the Universe and the size of the electron with fundamental constants and quote a numerical value?

## Wednesday, May 28, 2014

### temperature in a FLRW-universe

what temperature would you associate with gravity in a FLRW-universe and how would that temperature compare to the actual temperature or the Planck-temperature?

## Wednesday, May 21, 2014

### Sunyaev-Zel'dovich effects with neutrinos

would there be analogies of the Sunyaev-Zel'dovich effects due to electron-scattering with the cosmic neutrino background? can you point out similarities and differences and estimate the magnitude of the effects?

## Wednesday, May 14, 2014

### cluster cooling by immersion into the CMB

on what time scale would the intra-cluster medium of a cluster cool down by Compton-scattering of the cold cosmic microwave background (ignoring thermal replenishment by the negative specific heat)?

## Wednesday, May 7, 2014

### boundary to quantum cosmology

can you estimate the distance between two points in a FLRW-universe at which the quantum mechanical uncertainty in the velocity measurement is as large as the redshifting effect?

(this fun question is discussed in detail in the paper "the quantum indeterminancy of cosmic systems" by C.J. Hogan)

## Wednesday, April 30, 2014

### coasting dark energy universe

following up on the CQW from last week: is it possible to construct a dark energy component with an equation of state that would provide a stable balance between acceleration and deceleration and lead to a constant expansion speed?

bonus question: in what ways would this model be distinguishable from a universe with pure curvature?

## Wednesday, April 23, 2014

### Ricci-scalar in a FLRW-universe

CQW celebrates $10^{4.5}+$ views in 28 months of running the blog! we would like to thank our followers and remind you that if you'd like to propose a question yourself, you're invited to mail us your suggestions to cosmologyquestionoftheweek["AT"]gmail["DOT"]com.

can you express the Ricci-scalar in a FLRW-universe in terms of the Hubble function and the deceleration parameter?

## Wednesday, April 16, 2014

### Wien displacements in many dimensions

can you show that the Wien displacement law for frequencies for a Planck-spectrum in $n$ dimensions is identical to the displacement law for wave lengths in $n+2$ dimensions?

## Wednesday, April 9, 2014

### braking by scattering

can you estimate the time-scale on which a cluster of galaxies slows down due to the kinetic Sunyaev-Zel'dovich effect? (PS: there's something remarkable about the solution if one assumes small velocities and if CQW didn't make a mistake)

## Wednesday, April 2, 2014

### Planck, Bose and Boltzmann

the Planck-spectrum is determined by the Bose-distribution for indistinguishable integer-spin particles and not by the Boltzmann-distribution for classical distinguishable particles. can you show that the difference between the Stefan-Boltzmann-laws derived with both distributions vanishes in the limit of many dimensions? can you show this mathematically and give an intuitive argument?

## Wednesday, March 26, 2014

### fundamentals of cosmology

can you give an intuitive explanation why one needs general relativity for describing the dynamics of the Universe?

## Wednesday, March 19, 2014

### fat Hubble sphere

the Hubble sphere is a spherical surface around an observer in a FLRW-universe at which the recession velocity becomes larger than the speed of light. at what scale factor are the recession velocities stationary?

## Wednesday, March 12, 2014

### astronomy units in cosmology

can you express the flux density of the cosmic microwave background in Jansky per squared arcminute?

## Wednesday, March 5, 2014

### axioms of Lagrangian mechanics

imagine you'd want to formulate mechanics based on an extremal principle. what would you need to require such that all three Newton-axioms are fulfilled?

## Wednesday, February 26, 2014

### quantum statistics

can you write the Fermi-Dirac-distribution as a superposition of Bose-Einstein-distributions at different temperatures? is the inverse possible as well?

## Wednesday, February 19, 2014

### cosmic slow-motion

could you explain cosmological observations by postulating that space is static and time is passing faster today than in the past?

## Wednesday, February 12, 2014

### Eddington's number

can you estimate Eddington's number, the total number of protons in the observable Universe?

## Wednesday, February 5, 2014

### gravitational Rydberg constant

CQW celebrates 100 posts (and looks back on almost 3e4 views)!

what's the equivalent of the Rydberg-constant for a gravitationally bound hydrogen-analogue in Newtonian gravity (i.e. the ground state energy of "Keplerium")? would a similar construction work for Schwarzschildium?

## Wednesday, January 29, 2014

### magic of the Gaussian distribution

the width $\sigma$ of a Gaussian distribution

p(x)\mathrm{d}x =
\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{x^2}{2\sigma^2}\right)\mathrm{d}x

can be obtained by integration of the second moment, $\langle x^2\rangle = \int\mathrm{d}x\:p(x)x^2 = \sigma^2$ or by differentiation and solving $\mathrm{d}^2p/\mathrm{d}x^2=0$. can you show that this property is unique for the Gaussian distribution?

## Wednesday, January 22, 2014

### relativistic natural system of units

can you define fundamental scales for length, time and mass from the speed of light $c$, the gravitational coupling constant $G$ and the cosmological constant $\Lambda$? please compare size, age and density of the Universe to these fundamental scales.

## Wednesday, January 15, 2014

### subway to New Zealand

imagine there's a tunnel drilled across the centre of the Earth where a capsule moves under gravity serving as transport between Europe and the Antipodes. can you estimate the travel time? what would be the largest velocity? is there something funny about the result if you assume homogeneity?

## Wednesday, January 8, 2014

### nucleosynthesis with unstable nuclei

what qualitative changes would there be in the abundance of light elements if (1) neutrons had a shorter life-time or if (2) deuterium would be unstable?